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Thermodynamics is a science of bulk matter (solid, liquid, gaseous) presuming no detailed information on the microscopic structure of matter: it provides mathematical relations between macroscopic observable properties that are invariable to all changes in microscopic concepts. Thermodynamic theory is applicable to all types of macroscopic matter, irrespective of its chemical composition and independent of molecule-based information – this is the strength of thermodynamics and the basis of its enormous generality. Using an axiomatic approach, this chapter presents systematically and concisely essential parts of classical thermodynamics applicable to non-electrolyte fluids, pure and mixed. All fundamental property relations in the internal energy representation as well as in the entropy representation are derived via Legendre transformation. Residual properties in (T,P,x)-space, in (T,V,x)-space and in (T,ρ,x)-space are presented and their relations to fugacities and fugacity coefficients are established. In addition, property changes on mixing, excess properties and Lewis–Randall activity coefficients are discussed. Finally, several topics of current interest in molecular thermodynamics are considered, such as internal pressure, solubility parameter and equations of state.

Tyger, Tyger, burning bright,

In the forests of the night

What immortal hand or eye,

Could frame thy fearful symmetry?

William Blake

(London, 28 November 1757–12 August 1827)

The poem The Tyger (six stanzas in length, each stanza four lines long) was published in 1794 as part of Blake's illuminated book Songs of Experience (a sequel to Songs of Innocence, published in 1789). Reproduced from Songs of Experience, quotation in the public domain.

The two fundamental disciplines representing the scientific basis of chemistry are quantum mechanics and thermodynamics. Whereas the former focuses on the properties and behaviour of the microscopic constituents of matter, that is, on atoms, molecules and electrons in the realm of chemistry, the latter is concerned with macroscopic or bulk properties and behaviour and does not consider the microscopic state of matter at all. By bridging the gap between them, statistical mechanics is used to reconcile these two extremes. Based entirely on experiments on macroscopic systems, that is, on fundamental laws extracted therefrom, thermodynamics is a formalised phenomenological theory of enormous generality in the following sense:

  1. The remarkable feature of thermodynamics is its independence from any microscopic assumptions: it provides us with mathematical relations between macroscopic properties that are invariable to all changes in microscopic molecular models. Thermodynamic theory is applicable to all types of macroscopic matter, irrespective of its chemical composition and independent of molecule-based information, i.e. systems are treated as “black boxes” and the concepts used ignore the existence of molecules and indeed do not need it: thermodynamic relations would correctly describe macroscopic reality if matter were continuous. Although thermodynamics alone does not provide any molecular information, this is not a disadvantage. Consider, for instance, (biological) systems that are too complicated to be adequately described by molecule-based theory; yet regardless of molecular complexity, thermodynamics may still be applied and the results obtained remain exact.

  2. Thermodynamics does not provide ab initio predictions of numerical values for thermodynamic properties – these have to be found by measurement; nor can it provide by itself any microscopic, molecule-based details. It provides, however, a mathematical network of equations (and a few inequalities) that yields exact relations between measurable macroscopic quantities and restricts the behaviour of any natural process. While thermodynamics does not supply numerical values of bulk properties, when statistical-mechanical modelling is applied to macroscopic systems, the results have to be consistent with thermodynamics: its discriminative power is an extremely valuable tool for evaluating molecule-based model theories.

Thermodynamics is complementary to statistical mechanics: its independence of the molecular details of physical systems is its strength and is responsible for its generality. This aspect has been epitomised by Einstein's view on thermodynamics:1,2 

A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates and the more extended is its area of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content concerning which I am convinced that, within the framework of applicability of its basic concepts, it will never be overthrown.

Thermodynamics is a beautiful subject of great intellectual attractiveness: from a few selected postulates, all thermodynamic relations are derivable by deduction, informally known as top-down reasoning. It orders and explicates already existing knowledge, thereby leading to predictions that may be corroborated experimentally (or, in principle, falsified3 ): a theory has no value in science unless it is possible to test it experimentally. The scope of chemical thermodynamics was succinctly summarised by McGlashan:4 

What then is the use of thermodynamic equations to the chemist? They are indeed useful, but only by virtue of their use for the calculation of some desired quantity which has not been measured or which is difficult to measure, from others which have been measured or which are easier to measure.

This aspect alone is already of the greatest value for applications. However, experimental results not only corroborate existing knowledge or facilitate acquisition of materials properties: since experiments, molecule-based theory and computer simulations represent the three pillars of science,5 augmenting the formal framework of thermodynamic equations with molecule-based models of material behaviour, i.e. by using concepts from statistical mechanics for model building, experimental thermodynamic data discussed in terms of such models may then contribute decisively towards a better understanding of molecular interactions and thus lead to an improved description of Nature. After amplifying, logically ordering and generalising our experimental observations, inductive reasoning, known informally as bottom-up reasoning, leads to hypotheses and then theories and thus to truly new knowledge. This field of molecular thermodynamics (the term was coined by Prausnitz6  more than four decades ago) is of great academic fascination and has become indispensable in (bio-)physical chemistry and chemical engineering. It corroborates Freeman Dyson's point of view:7 

Science is not a collection of truths. It is a continuing exploration of mysteries …an unending argument between a great multitude of voices.

The growth of molecular thermodynamics has been stimulated by the continuously increasing need for thermodynamic property data and phase equilibrium data8–31  in the applied sciences and it has profited from advances in experimental techniques,32–48  from modern formulations of chemical thermodynamics,49–68  from advances in statistical thermodynamics and the theory of fluids6,69–97  and from advances in computer simulations of model systems.98–102 

The most popular heuristic principle to guide hypothesis/theory testing is known as Occam's razor, named after the Franciscan friar William of Ockham (England, ca. 1285–1349). Also called the principle of parsimony or the principle of the economy of thought, it states that the number of assumptions to be incorporated into an adequate model should be kept minimal. Although this is the preferred approach, the model with the fewest assumptions may turn out to be wrong. More elaborate versions of Occam's razor have been introduced by modern scientists and for in-depth philosophical discussions see Mach,103  Popper,3  Katz104  and Sober.105 

Most approaches to thermodynamics are either historical or postulatory (axiomatic),56  that is, the experimentally established facts are formalised by introducing a set of axioms inspired by observation that cannot be proved from more basic principles. The only proof of their validity lies in the absence of any conflict between derived relations and experiment. In fact, for about a century and a half such tests have been carried out with complete success. The ultimate justification of this approach rests solely on its usefulness. In this introductory chapter, after a few historical and introductory remarks, a postulatory approach lucidly formulated by Van Ness and Abbott55  will be adopted.

Thermodynamics is a physical science concerned with energy and its transformations attending physical and/or chemical processes. Historically, it was developed to improve the understanding of steam engines at the beginning of the industrial age, the focus then being on the convertibility of heat into useful work. The concepts of work, heat and energy were developed over centuries,106  and the formulation of the principle of conservation of energy has been one of the most important achievements in physics. The initial key players in the field of thermodynamics were Sadi Carnot (1796–1832), Julius Robert Mayer (1814–1878), James Prescott Joule (1818–1889), Hermann von Helmholtz (1821–1894), Rudolf Clausius (1822–1888), William Thomson (1824–1907), who became Baron Kelvin of Largs in 1892, and Josiah Willard Gibbs (1839–1903). In the late 1800s, Wilhelm Ostwald,107,108  the “father of physical chemistry” and founding Editor-in-Chief of the first worldwide journal devoted exclusively to physical chemistry (the first issue of Zeitschrift für physikalische Chemie appeared on February 15, 1887) became deeply interested in thermodynamics and thus in the work of Gibbs, which at that time was largely neglected. He suggested to Gibbs a translation into German (and publication as a book) of his thermodynamic treatises, in particular his fundamental study On the Equilibrium of Heterogeneous Substances,109  together with two earlier papers.110,111  As evidenced by the Ostwald–Gibbs correspondence (a number of letters written between 1887 and 1895),112  with persistence and psychological finesse Ostwald succeeded, thereby making readily accessible to the scientific community the hitherto virtually inaccessible work of Gibbs.113  In his autobiography,114  Ostwald says that the English and Americans had to read Gibbs in German until Yale University published a collected edition115  of his scattered contributions to science that had appeared in society transactions and various scientific journals. However, Gibbs's last work, Elementary Principles in Statistical Mechanics, had been printed as a volume of the Yale Bicentennial Series in 1902.116 

Essentially all applications of thermodynamics in this introductory chapter will focus on the thermodynamic properties of macroscopic homogeneous samples of fluids, i.e. of phases, on relations among them and their dependence on measurable conditions specified, for instance, by thermodynamic temperature T, pressure P and composition, in the absence of extraneous influences. Such thermodynamic systems are generally referred to as simple systems: they are macroscopically homogeneous, isotropic, uncharged, non-reactive and large enough to neglect surface effects. In addition, simple systems are not acted upon, for instance, by electrostatic or magnetic fields and the fluid samples are small enough that the influence of the Earth's gravitational field is not detected in a variation of properties with the height of the vessel containing the fluid. Pressure is the only mechanical force considered, causing contraction or dilation of the fluid. As pointed out, classical thermodynamics makes no assumptions about the microscopic, molecular structure, nor does it reveal any molecular mechanism. In fact, essential parts of thermodynamics were developed before the internal structure of matter was established and a logically consistent theory can be developed without assuming the existence of molecules or atoms. Thus, the composition of a phase may always be characterised by the set of mass fractions {wi}, defined by

Equation 1.1

where mi denotes the mass of component i, is the total mass of the phase, and for a pure fluid wi = 1. However, we do have reliable theories concerning the molecular structure of matter, that is, for properties of molecules and their interactions, and molecule-based models allow, in principle, the calculation of macroscopic properties via statistical mechanics. Because of this connection, for systems of known molecular composition, i.e. where the molar mass mm,i of each component i is known, the composition of a phase is preferably characterised by a set of mole fractions {xi}, defined by

Equation 1.2

where ni denotes the amount (of substance) of component i, is the total amount in the phase, and for a pure fluid xi = 1. Since mi = nimm,i is the total mass of component i, we have

Equation 1.3

Temperature, pressure and composition are perceived as the principal intensive thermodynamic coordinates for homogeneous fluids, that is, they do not depend on the quantity of fluid and have the valuable bonus of being (in principle) easily measured, monitored and controlled. In contradistinction, the total (superscript t) volume Vt of a phase is an extensive property that does depend on the quantity of material and may thus be alternatively expressed either by

Equation 1.4

with the proportionality factor being either the specific volume v or the molar volume V, respectively; they are intensive properties, independent of the quantity of fluid present. Entirely analogous definitions apply to other extensive properties. Note that the quotient of any two extensive properties is an intensive property. Thus, an extensive property is transformed into an intensive specific property by dividing by the total mass and into an intensive molar property by dividing by the total amount (of substance); a density is obtained on dividing by the total volume. Commonly used densities are the mass density ρmm/Vt and in (statistical) thermodynamics, the amount (of substance) density ρn/Vt = 1/V and the number density ρNN/Vt.

Near the liquid–vapour critical point, many thermodynamic properties (and also transport properties) show anomalies linked to the divergence of the fluid's isothermal compressibility:56,79,117–119 

Equation 1.5

that is, for a pure fluid with critical pressure Pc, critical molar volume Vc and critical temperature Tc the divergence of βT is expressed by the power law (simple scaling)118,119 

Equation 1.6

where γ = 1.239 ± 0.002 denotes the critical exponent (the classical value is 1), and ρc is the critical amount density. Thus, associated with the large compressibility of a near-critical fluid, the presence of the terrestrial gravitational field will cause the local value of the fluid density ρ to vary with height, i.e. macroscopic density gradients will develop due to compression under the fluid's own weight (gravitational sedimentation). The formation of such gradients implies that the fluid sample will be at its critical density at only one point along its height and over a few centimetres this may result in density variations of the order of 10%.120–122  Measurements made under such conditions will therefore average over a range of heights. Note that the divergence of the isothermal compressibility gives rise to the divergence of the isobaric expansivity (expansion coefficient),

Equation 1.7

and to the divergence of the molar isobaric heat capacity,

Equation 1.8

where CV denotes the molar isochoric heat capacity and (∂P/∂T)V is the isochoric thermal pressure coefficient (for details see Section 1.3.1). Measurements close to the critical point thus become significantly distorted. Recent results on the development of a complete scaling theory123–125  of critical phenomena are discussed in ref. 119 and 126.

For experiments on fluid systems under terrestrial laboratory conditions and sufficiently removed from the critical region, the influence of the Earth's gravitational field is generally ignored. However, for interpreting precision measurements in the critical region and to test theoretical predictions for critical phenomena, for the reasons outlined above, experiments taking advantage of the microgravity environment of space laboratories are indispensable and have indeed been performed.127 

Keeping these caveats in mind, simple systems are known as PVT systems (or PVTx systems) and the appropriate first fundamental postulate reads as follows:

  • Postulate 1: The macroscopic properties of homogeneous fluids in equilibrium states (closed PVT or PVTx systems) are functions of temperature, pressure and composition only.

Work, W, and heat, Q, represent energy transfers; they are both energy in transit between the system of interest and its surroundings. The transfer of energy represented by a quantity of work W is a result of the existence of unbalanced forces between system and surroundings and W is not a system property. The transfer of energy represented by a quantity of heat Q is a result of the existence of a temperature difference between system and surroundings and Q is not a system property: W and Q are defined only for processes transferring energy across a system boundary (see Table 1.1); they represent energy flows to or from a system.

Table 1.1

The most important systems, boundaries and interactions

System, boundariesInteractions with surroundings
Isolated No exchange of energy or matter between the system and the surroundings 
Mechanically enclosed, diathermic walls Cannot exchange matter with surroundings; no work is done on it, but thermal interaction is possible, i.e. exchange of energy through the boundary as a result of a temperature difference 
Adiabatically enclosed, adiabatic walls (thermally insulated) Cannot exchange matter with surroundings; does not permit the transfer of heat through its boundary. The state of the adiabatically enclosed system remains unchanged unless work is done on it 
Closed Cannot exchange matter with its surroundings (constant mass), but exchange of energy between system and surroundings is possible 
Open Permits exchange of matter and energy with its surroundings 
System, boundariesInteractions with surroundings
Isolated No exchange of energy or matter between the system and the surroundings 
Mechanically enclosed, diathermic walls Cannot exchange matter with surroundings; no work is done on it, but thermal interaction is possible, i.e. exchange of energy through the boundary as a result of a temperature difference 
Adiabatically enclosed, adiabatic walls (thermally insulated) Cannot exchange matter with surroundings; does not permit the transfer of heat through its boundary. The state of the adiabatically enclosed system remains unchanged unless work is done on it 
Closed Cannot exchange matter with its surroundings (constant mass), but exchange of energy between system and surroundings is possible 
Open Permits exchange of matter and energy with its surroundings 

Consider a homogeneous equilibrium fluid in a closed PVT system, that is, a system with a boundary that restricts only the transfer of matter (constant mass system), while energy exchanges with its surroundings involve only heat Q and/or work W. For such a system, the existence of a form of energy called total internal energy Ut is postulated that is an extensive material property and a function of T, P and mass or amount of substance . Internal energy refers to the molecules of the bulk fluid, that is, it reflects the N-body intermolecular potential energy and includes the kinetic energy of molecular translation and, except for monatomic fluids, overall molecular rotation and internal molecular modes of motion, such as molecular vibration, intramolecular rotation, intermolecular association, etc. These details are, however, of no concern to classical thermodynamics. This designation of internal energy distinguishes it from kinetic energy and potential energy that the system may possess macroscopically, that is, from external energy. The existence postulate reads as follows:

  • Postulate 2: There exists a form of energy, known as total internal energy Ut, that for homogeneous fluids at equilibrium in closed PVT systems is a material property and a function of temperature, pressure and composition.

It is prerequisite for the formulation of a conservation law of energy that includes, in addition to mechanical energy, heat and internal energy:

  • Postulate 3 (First Law of Thermodynamics): The total energy of a system and its surroundings is conserved; energy may be transferred from a system to its surroundings and, vice versa, it may be transformed from one form to another, but the total quantity remains constant.

Hence, for a closed PVT system (system of constant mass) at rest at constant elevation (no changes in kinetic and/or potential energy), according to Postulate 3, i.e. according to the First Law of Thermodynamics, changes in the total internal energy (with Δ signifying a finite change in the indicated quantity) are given by

Equation 1.9

where, in analogy with eqn (1.4), UUt/n denotes the molar internal energy of the fluid and uUt/m is the specific internal energy of the fluid. As suggested by IUPAC,128 Q > 0 and W > 0 indicate an increase in the energy of the system. For a differential change in the internal energy of the closed system, we have

Equation 1.10

Mathematically, dUt is an exact differential of the state function Ut: the change in the value of this extensive property for any process taking place between an initial equilibrium state 1 and a final equilibrium state 2 of a homogeneous fluid at rest at constant elevation is given by

Equation 1.11

and depends only on the two states (path independence). On the other hand, δQ and δW are inexact differentials, i.e. they represent infinitesimal amounts of heat and work. The integration of δQ and δW has to be carried out along some path to give finite amounts Q and W, respectively, the values of which are path dependent. The notation used in the first law, eqn (1.9), asserts that the sum of the two path functions Q and W always yields an extensive state function change ΔUt between two equilibrium state points, independent of the choice of path. However, two special cases have to be pointed out:

  • 1. In the case of a process where the system is adiabatically enclosed, Q = 0 and eqn (1.9) becomes

Equation 1.12

Measurement of the adiabatic work Wad is therefore a measurement of ΔU t and thus depends only on the initial and final equilibrium states (path independence). This is confirmed by adiabatic calorimetric experiments, thereby providing the primary evidence that Ut is indeed a state function.

  • 2. In the case of a process where the system is mechanically enclosed, yet equipped with diathermic walls (only thermal interactions connect the system with its surroundings), W = 0 and eqn (1.9) becomes

Equation 1.13

and Qmech depends only on the initial and final equilibrium states (path independence).

The energy conservation law can be generalised to include other types of work, such as work associated with a change in surface area A of a plane surface phase, that is, dW σ = σdA, where σ denotes the surface tension.129 

Eqn (1.9) does not provide an explicit definition of the internal energy. In fact, there is no known way to measure absolute values of Ut: the internal energy of a system is an extensive conceptual property. This is not a disadvantage since in thermodynamics only changes in internal energy are of interest and differences in internal energy can be measured. Consider now a series of experiments performed essentially reversibly on a homogeneous constant-composition fluid (closed system) along different paths from an initial equilibrium state 1 at (T1, P1) to a final equilibrium state 2 at (T2, P2). All measurements show that the experimentally determined sum Qrev + Wrev is constant, independent of the path selected, as it must be provided that Postulate 2 (the postulate asserting that internal energy is a material property) is valid. Thus, we have

Equation 1.14

as a special case of eqn (1.9), which for closed systems is generally applicable for reversible as well as irreversible processes between equilibrium states. In differential form, this equation reads

Equation 1.15

hence the measurability of any change of the internal energy follows from

Equation 1.16

regardless of the path 1 → 2. Since all experimental evidence to date has shown this relation to be true, we may safely assume that it is generally true, although the possibility of falsification, of course, remains.3  No further “proof” exists beyond the experimental evidence.

Two additional property changes are revealed by systematic experiments on homogeneous equilibrium fluids in closed PVT systems. One is already known, i.e. ΔVt, but this fact could also be demonstrated as follows. The work caused by a reversible volume change is given by

Equation 1.17

Multiplication by 1/P serving as an integrating factor yields an exact differential:

Equation 1.18

and upon integration of experimental data, a single value of the property change ΔVt of the extensive, measurable state function Vt is obtained, independent of the path 1 → 2:

Equation 1.19

Similarly, careful evaluation of systematic experiments on homogeneous equilibrium fluids in closed systems reveals that whereas δQrev is an inexact differential, multiplication by 1/T serving as an integrating factor yields an exact differential and identifies the so defined total entropy St as a state function:

Equation 1.20
Equation 1.21

This result is summarised by an additional postulate asserting the existence of an extensive state function St called the total entropy:

  • Postulate 4: There exists a material property called total entropy St, which for homogeneous fluids at equilibrium in closed PVT systems is a function of temperature, pressure and composition; differential changes of the total entropy are given by eqn (1.20).

The change of entropy between two equilibrium states 1 and 2 therefore depends solely on the difference between the values of St in these states and is independent of the path and irrespective of the process being reversible or irreversible. However, in order to calculate the difference ΔSt, a reversible path connecting the two equilibrium states must be selected.

Analogous to the statement associated with the total internal energy, i.e. the first law, eqn (1.9), eqn (1.21) does not give an explicit definition of the total entropy. In fact, classical thermodynamics does not provide one. As is the case with internal energy, this is not a disadvantage since in thermodynamics only changes in entropy are of interest and entropy differences can be measured: the entropy St of a system is also an extensive conceptual property. With the postulated existence of entropy, experimental results have led to the formulation of another general restriction, besides energy conservation, applying to all processes. This fifth postulate is known as the Second Law of Thermodynamics:

  • Postulate 5 (Second Law of Thermodynamics): All processes proceed in such a direction that the entire entropy change of any system and its surroundings, caused by the process, is positive; the limiting value zero is approached when the process approaches reversibility:

Equation 1.22

Following Van Ness and Abbott,55  this postulate completes the axiomatic basis upon which classical equilibrium thermodynamics rests. All that is now needed to develop the network of mathematical equations (and a few inequalities) that interrelate the thermodynamic properties of macroscopic equilibrium systems are formal definitions and mathematical deduction (top–down reasoning3 ).

So far, only homogeneous fluids in closed equilibrium PVT systems (simple systems) have been considered. For a closed heterogeneous equilibrium system consisting of p fluid phases α, β, …, each in itself a PVT system, any overall total property Mt is the sum of the total property values of the p constituent phases:

Equation 1.23

Hence the overall molar property is obtained from

Equation 1.24

where

Equation 1.25

denotes the total amount of substance in the entire closed system. All experiments performed on fluids in heterogeneous PVT systems yielded the same general results as obtained with homogeneous systems, hence the equations obtained so far may safely be extended to heterogeneous systems.

The first law of thermodynamics in differential form for any closed PVT system consisting either of a single phase (homogeneous system) or of several phases (heterogeneous system) is given by eqn (1.15). With eqn 18 and (1.20), the basic differential equation for closed systems reads

Equation 1.26

where U, S and V denote the molar internal energy, the molar entropy and the molar volume, respectively. Since Ut = nU, St = nS and Vt = nV are extensive state functions, eqn (1.26) is not restricted to reversible processes, although it was derived for the special case of such a process. It applies to any differential change in a closed multiphase PVT system from one equilibrium state to another, including mass transfer between phases or/and composition changes due to chemical reactions. However, for irreversible processes TdSt is not the heat transferred, nor is −PdVt the work done: δQirrev < δQrev and δWirrev > δWrev, but, of course, dUt = TdStPdVt = δQirrev + δWirrev, since Ut is a state function.

Application of the exact differential eqn (1.26) to a closed single-phase multicomponent system without chemical reactions (i.e. a constant mass, constant composition system) yields

Equation 1.27
Equation 1.28

where the subscript {ni} denotes constant amounts of all components.

When selecting nS as the dependent property, the alternative basic differential equation for a closed PVT system consisting either of a single phase (homogeneous system) or of several phases (heterogeneous system) reads

Equation 1.29

To reiterate: since Ut = nU, St = nS and Vt = nV are extensive state functions, eqn (1.29) is not restricted to reversible processes, although it was derived for the special case of such a process. It applies to any differential change in a closed multiphase PVT system from one equilibrium state to another, including mass transfer between phases or/and composition changes due to chemical reactions.

Application of the exact differential eqn (1.29) to a closed single-phase multicomponent system without chemical reactions (i.e. a constant mass, constant composition system) yields

Equation 1.30
Equation 1.31

A functional relation between all extensive system parameters is called a fundamental equation. Consider now an open single-phase multicomponent PVT system in which the amounts of substance ni may vary because of interchange of matter with its surroundings or because of chemical reactions within the system or both. Since Ut and St are the only conceptual properties, one may have either a fundamental equation in the internal energy representation:

Equation 1.32

or, equivalently, a fundamental equation in the entropy representation:

Equation 1.33

Taking into account eqn (1.27), (1.28), (1.30) and (1.31), the corresponding exact differential forms (total differentials) of the fundamental equations, also known as the fundamental equations for a change of the state of a phase or the fundamental property relations or the Gibbs equations, are

Equation 1.34

in the internal energy representation, and, equivalently,

Equation 1.35

in the entropy representation. eqn 34 and (1.35) apply to single-phase multicomponent PVT systems, either open or closed, where ni vary because of interchange of matter with the surroundings or because of chemical reactions within the systems or both. The intensive state function defined by the partial derivative of nU with respect to ni, at constant entropy and volume, is called the chemical potential of component i in the mixture:

Equation 1.36

From eqn (1.35), we obtain

Equation 1.37

Hence the fundamental property relation eqn (1.34) can be written in a more compact form:

Equation 1.38

while the fundamental property relation eqn (1.35) becomes

Equation 1.39

eqn 38 and (1.39) are fundamental because they completely specify all changes that can take place in single-phase, multicomponent PVT systems, either open or closed, and they form the bases of extremum principles predicting equilibrium states.

In the internal energy representation, the corresponding fundamental equation for an open, single-phase, multicomponent PVT systems reads

Equation 1.40

and in the entropy representation we have

Equation 1.41

To reiterate: in this introductory chapter only non-reacting fluid equilibrium systems will be considered.

eqn 40 and (1.41) are also known as the integrated forms of the fundamental equations for a change of the state of a phase and the state functions nU(nS, nV, {ni}) and nS(nU, nV, {ni}) are commonly known as primary functions, cardinal functions or thermodynamic potentials. They were obtained by integrating eqn 38 and (1.39), respectively, over the change in the amount of substance at constant values of the intensive quantities {T, −P, µi} or , respectively. These two primary functions (cardinal functions, thermodynamic potentials) are, of course, related:

Equation 1.42

Alternatively,eqn 40 and (1.41) can be regarded as a consequence of Euler's theorem, which asserts the following: if f (z1, z2,…) is a homogeneous function of degree k in the variables z1, z2,…, i.e. if it satisfies for any value of the scaling parameter λ the relation

Equation 1.43

it must also satisfy

Equation 1.44

In thermodynamics, only homogeneous functions of degree k = 0 and k = 1 are important. The former functions are known as intensive functions and the latter as extensive functions. Based on the homogeneous first-order properties of both Ut and St, that is,

Equation 1.45
Equation 1.46

the use of eqn (1.44) with k = 1, in conjunction with eqn 38 and (1.39), yields eqn 40 and (1.41), respectively. The corresponding variable sets, i.e. {nS, nV, n1, n2,⋯} for the internal energy representation and {nU, nV, n1, n2,⋯} for the entropy representation, are called the canonical or natural variables. With a satisfactory fundamental equation established, all thermodynamic equilibrium properties of a PVT phase can be calculated by fairly simple mathematical manipulations, that is, by combining appropriate derivatives of the corresponding primary function (thermodynamic potential); it is for this reason they are called fundamental equations.

As indicated by eqn (1.27), (1.28) and (1.36), T, −P and µi are partial derivatives of Ut(nS, nV, n1, n2,⋯). A functional relation expressing an intensive parameter in terms of the independent canonical extensive parameters of the system is called an equation of state (EOS). With the definitions indicated above, the fundamental equation in the internal energy representation implies three EOSs, that is, three zeroth-order homogeneous equations of the extensive parameters of the system, that is

Equation 1.47

A single EOS does not contain complete information on the thermodynamic properties of the system. However, the entire set of these three EOSs is equivalent to the fundamental equation and contains all thermodynamic information.56  Focusing now on the fundamental property relation in the entropy representation, analogous comments apply to the partial derivatives of St(nU, nV, n1, n2,⋯), see eqn (1.30), (1.31) and (1.37), yielding the corresponding EOSs:

Equation 1.48
Equation 1.49
Equation 1.50

For constant-composition fluids and thus also for pure fluids, T = T(nU, nV, n1, n2,⋯) is usually explicitly resolved for the internal energy, i.e.

Equation 1.51

This type of equation is known as the caloric equation of state. Clearly, by inserting eqn (1.51) into eqn (1.49) we obtain either a pressure-explicit thermal equation of state:

Equation 1.52

or, after rearrangement, a volume-explicit thermal equation of state:

Equation 1.53

Frequently, for the sake of brevity, the adjective thermal (having historical roots) is omitted. A well-known example of a volume-explicit (thermal) EOS is the virial equation in pressure:

Equation 1.54

where Z is known as the compression factor, B′(T, { yi}) denotes the second virial coefficient of the pressure series, C′(T, { yi}) is the third virial coefficient of the pressure series and so forth and yi is the mole fraction of component i in the gaseous phase, see eqn (1.2). Well-known examples of pressure-explicit (thermal) EOSs are the virial equation in amount density:

Equation 1.55

and the van der Waals (vdW) equation:

Equation 1.56

In eqn (1.55), B(T, { yi}) denotes the second virial coefficient of the amount density-series, C(T, { yi}) is the third virial coefficient of the amount density-series and so forth. Since P and ρ are related via P = ρRTZ, the virial coefficients of the two series are also related, that is,

Equation 1.57
Equation 1.58

In the vdW equation, the parameter b is known as the covolume and allows for the finite hard size of the molecules, and the averaged attractive intermolecular interaction in the real fluid leads to a correction of the pressure amounting to a/V2. Most EOSs in practical use are explicit in pressure.

In the fundamental property relations for an open single-phase PVT system in both the internal energy representation and the entropy representation, the extensive properties are the mathematically independent variables, whereas the intensive parameters are derived, which situation does not reflect experimental reality. The choice of nS and nV as independent extensive variables in eqn (1.38) and of nU and nV as independent extensive variables in eqn (1.39) is not convenient. Experiment-based experience shows that the conjugate intensive parameters {T, P} and {1/T, P/T}, respectively, are much more easily measured and controlled. Hence for describing the system behaviour in, say, isothermal or isobaric processes, alternative versions of the fundamental equations are necessary in which one or more of the extensive parameters are replaced with their conjugate intensive parameter(s) without loss of information. The appropriate generating method is the Legendre transformation.130–134  It is worth mentioning that the Legendre transformation is also useful in classical mechanics by providing the transition from the Lagrangian to the Hamiltonian formulation of the equations of motion.135 

Eqn (1.40) suggests the definition of useful alternative internal energy-based primary functions related to nU and with total differentials (fundamental property relations) consistent with eqn (1.38), but with a set of canonical variables different from {nS, nV, {ni}} and potentially more practical; and eqn (1.41) suggests the definition of useful alternative entropy-based primary functions related to nS and with total differentials (fundamental property relations) consistent with eqn (1.39), but with a set of canonical variables different from {nU, nV, {ni}} and potentially more practical.

Consider the exact (total) differential

Equation 1.59

pertaining to the base function f(0) of n independent variables Xi:

Equation 1.60

where

Equation 1.61

Consider now the function obtained by subtracting the product of X1 with its conjugate partial derivative c1 from the base function f(0), eqn (1.60):

Equation 1.62

The total differential reads

Equation 1.63

and with eqn (1.59) we obtain

Equation 1.64

Comparison of eqn (1.59) with eqn (1.64) shows that the original variable X1 and its conjugate c1 have interchanged their roles (and changed sign). For such an interchange, it suffices to subtract from the base function f(0) to obtain the first-order partial Legendre transform:

Equation 1.65

which is frequently identified by a bracket notation as indicated. This Legendre transform represents a new function with independent variables {c1, X2, X3,⋯, Xn}, being the canonical or natural, variables.

Analogously, the Legendre transformation of higher order p of the base function f(0) that introduces the partial derivatives {c1, c2,…, cp} into f(0) reads

Equation 1.66

and the associated total differential is

Equation 1.67

The complete Legendre transform, i.e. the transform of order p = n, replaces all variables by their respective conjugate partial derivatives and vanishes identically for any system, thus yielding the null function; this follows directly from the definition

Equation 1.68

The associated differential expression reads

Equation 1.69

In thermodynamic theory, the complete Legendre transform of the fundamental equation in the internal energy representation for an open, single-phase, multicomponent (c components) PVT system, eqn (1.40), has all extensive canonical variables replaced with their conjugate intensive variables, thus yielding the null function

Equation 1.70

and correspondingly

Equation 1.71

with canonical variables {T, P, { µi}}. This property of the complete Legendre transform gives rise to the Gibbs–Duhem equation, which represents an important relation between the intensive parameters T, P and { µi} characterising the system and shows that they are not independent of each other.

When focusing on the fundamental equation in the entropy representation for an open, single-phase, multicomponent (c components) PVT system, eqn (1.41), the complete Legendre transform has all extensive canonical variables replaced by their conjugate intensive variables, thus yielding the null function

Equation 1.72

and correspondingly

Equation 1.73

with canonical variables {1/T, P/T, { µi/T}}. This property of the complete Legendre transform gives rise to the entropy-based Gibbs–Duhem equation, which shows that the intensive parameters characterising the system, i.e. 1/T, P/T and {µi/T}, are not independent of each other.

As shown in eqn (1.66), a partial Legendre transform f( p) of order p of the base function f(0)(X1, X2,…, Xn), with 1 ≤ p ≤ (n − 1), is obtained via subtraction of p products of Xi with its conjugate partial derivative , i.e. via subtraction of . The number of partial Legendre transforms of order p is therefore given by the number of combinations without repetition, that is, by

Equation 1.74

The total number NLe,p of partial Legendre transforms, that is, the total number of equivalent alternatives to f(0), is thus obtained from

Equation 1.75

Since the total number NLe t of Legendre transforms includes the complete transform, it is given by

Equation 1.76

Application of the above results to the fundamental equations for an open multicomponent PVT phase either in the energy representation, eqn (1.40), or in the entropy representation, eqn (1.41), is now straightforward. Provided that the summation term in the former is treated as a single term (in this case n = 3), the entire number Nt of equivalent primary functions (equivalent thermodynamic potentials) related to the internal energy, including nU, and therefore the number of the corresponding equivalent fundamental property relations, that is, of the corresponding total differentials of these primary functions, is seven:

Equation 1.77

They are presented in Tables 1.2 and 1.3, respectively, together with the null function and the associated internal energy-based Gibbs–Duhem equation.134 

Table 1.2

Equivalent alternative extensive primary functions (thermodynamic potentials) related to the extensive internal energy nU, see eqn (1.40), applying to open, single-phase, multicomponent PVT systems. They are obtained via Legendre transformations of the fundamental equation in the internal energy representation. Also listed is the complete Legendre transform that vanishes identically, i.e. the null function

  • Primary function (thermodynamic potential)

  • Symbol

  • Name

  • Alternative expression

nU nU Internal energy  (1.40) 
nU + P(nVnH Enthalpy  (1.78) 
nUT(nSnF Helmholtz energy     (1.79) 
 nG Gibbs energy            (1.80) 
 nX Not named T(nS) − P(nV)        (1.81) 
 nY Not named T(nS)         (1.82) 
 nJ Grand canonical potential P(nV)       (1.83) 
 — Null function 0      (1.70) 
  • Primary function (thermodynamic potential)

  • Symbol

  • Name

  • Alternative expression

nU nU Internal energy  (1.40) 
nU + P(nVnH Enthalpy  (1.78) 
nUT(nSnF Helmholtz energy     (1.79) 
 nG Gibbs energy            (1.80) 
 nX Not named T(nS) − P(nV)        (1.81) 
 nY Not named T(nS)         (1.82) 
 nJ Grand canonical potential P(nV)       (1.83) 
 — Null function 0      (1.70) 
Table 1.3

Equivalent alternative forms of the fundamental property relation in the internal energy representation, see eqn (1.38). They represent total (exact) differentials of the primary functions (thermodynamic potentials) presented in Table 1.2 and thus apply to open, single-phase, multicomponent PVT systems. Also listed is the Gibbs–Duhem equation corresponding to the null function

Alternative fundamental property relationsCanonical variables
 nS, nV, {ni}       (1.38) 
 nS, P, {ni}         (1.84) 
 T, nV, {ni}           (1.85) 
 T, P, {ni}         (1.86) 
 nS, nV, { µi}          (1.87) 
 nS, P, { µi}           (1.88) 
 T, nV, { µi}      (1.89) 
 T, P, { µi}      (1.71) 
Alternative fundamental property relationsCanonical variables
 nS, nV, {ni}       (1.38) 
 nS, P, {ni}         (1.84) 
 T, nV, {ni}           (1.85) 
 T, P, {ni}         (1.86) 
 nS, nV, { µi}          (1.87) 
 nS, P, { µi}           (1.88) 
 T, nV, { µi}      (1.89) 
 T, P, { µi}      (1.71) 

Since the total differentials of the primary functions presented in Table 1.3 are all equivalent, alternatives to the definition of the chemical potential µi of component i by eqn (1.36) are possible:

Equation 1.90

where H denotes the molar enthalpy, F the molar Helmholtz energy and G the molar Gibbs energy. The last equality, that is,

Equation 1.91

represents the preferred working definition of the chemical potential because T and P are the most useful experimental thermodynamic coordinates.

In complete analogy, when treating the summation term in eqn (1.41), i.e. the fundamental equation in the entropy representation, as a single term (in this case again n = 3), the entire number Nt of equivalent primary functions (equivalent thermodynamic potentials) related to the entropy, including nS, and therefore the number of the corresponding equivalent fundamental property relations, that is, of the total differentials of these primary functions, is also seven. They are summarised in Tables 1.4 and 1.5, respectively, together with the appropriate null function and its associated entropy-based Gibbs–Duhem equation.134  The replacement of one or more of the extensive variables nU, nV, {ni} by the corresponding conjugate intensive variable(s) 1/T, P/T and µi/T, respectively, yields primary functions known as Massieu–Planck functions. Interestingly, such a Legendre transform was already reported by Massieu in 1869 and thus predates the Legendre transforms of the internal energy reported by Gibbs in 1875 (see Callen56 ).

Table 1.4

Equivalent alternative extensive primary functions (thermodynamic potentials) related to the extensive entropy nS, see eqn (1.41), applying to open, single-phase, multicomponent PVT systems. They are obtained via Legendre transformation of the fundamental equation in the entropy representation and are known as Massieu–Planck functions. Also listed is the complete Legendre transform that vanishes identically, i.e. the null function

Primary functionSymbolNameAlternative expression
nS nS Entropy  (1.41) 
  Not named              (1.92) 
  Massieu function             (1.93) 
  Planck function          (1.94) 
  Not named     (1.95) 
  Not named          (1.96) 
  Kramers function          (1.97) 
 — Null function 0           (1.72) 
Primary functionSymbolNameAlternative expression
nS nS Entropy  (1.41) 
  Not named              (1.92) 
  Massieu function             (1.93) 
  Planck function          (1.94) 
  Not named     (1.95) 
  Not named          (1.96) 
  Kramers function          (1.97) 
 — Null function 0           (1.72) 
Table 1.5

Equivalent alternative forms of the fundamental property relation in the entropy representation, see eqn (1.39). They represent total (exact) differentials of the primary functions (thermodynamic potentials) presented in Table 1.4 and thus apply to open, single-phase, multicomponent PVT systems. They are known as fundamental Massieu–Planck property relations. Also listed is the GibbsDuhem equation corresponding to the null function

Alternative fundamental property relationsCanonical variables
 nU, nV, {ni}           (1.39) 
             (1.98) 
             (1.99) 
         (1.100) 
       (1.101) 
       (1.102) 
        (1.103) 
              (1.73) 
Alternative fundamental property relationsCanonical variables
 nU, nV, {ni}           (1.39) 
             (1.98) 
             (1.99) 
         (1.100) 
       (1.101) 
       (1.102) 
        (1.103) 
              (1.73) 

Division of the extensive primary functions listed in Tables 1.2 and 1.4 by the total amount of substance n yields the corresponding molar functions:

Equation 1.104

Although not always recognized, the (molar) Massieu–Planck functions are simply related to the (molar) primary functions (thermodynamic potentials):134 

Equation 1.105
Equation 1.106
Equation 1.107
Equation 1.108
Equation 1.109
Equation 1.110
Equation 1.111

The primary functions/fundamental property relations presented so far are all equivalent, although each is associated with a different set of canonical variables. The selection of any primary thermodynamic function/fundamental property relation depends on deciding which set of independent variables simplifies the problem to be solved. In physical chemistry and chemical engineering, the most useful sets of variables are {T, P, {ni}} and {T, nV, {ni}}, since they are easily measured and controlled. Hence the total Gibbs energy Gt(T, P, {ni}) and the total Helmholtz energy Ft(T, nV, {ni}) are particularly important.

Partial derivatives of a total property with respect to ni at constant T, P and nji are ubiquitous in chemical thermodynamics, hence a brief survey of relevant definitions and relations is presented below.

Focusing now on mixtures/solutions in single-phase equilibrium PVT systems, any extensive total property is a function of the amounts (of substance) {ni} of all the components present:

Equation 1.112

where is the total amount contained in the phase and M(T, P, {xi}) is the corresponding intensive molar property (it may also represent dimensionless properties such as the compression factor Z or the dimensionless ratio G/RT). The total differential of any extensive property of a homogeneous fluid at equilibrium in a PVT system may thus be expressed by

Equation 1.113

where the subscript {ni} indicates that the amounts of all components i and thus the composition {xi} are/is held constant. The summation term of eqn (1.113) is important for the thermodynamic description of mixtures of variable composition and extent. The derivatives are response functions known as partial molar properties Mi and are defined by

Equation 1.114

The partial molar property Mi(T, P, {xi}) is an intensive state functions: it quantifies the change (response) of the total (extensive) property Mt = nM when an infinitesimal amount (of substance) dni of component i is added to the solution at constant temperature and pressure, while keeping the amounts of all the other components, i.e. nji, constant. Note that when the amounts ni are replaced with the masses mi of the components, for instance, because their molar masses mm,i are unknown (ni = mi/mm,i), and n with , eqn (1.114) yields partial specific properties and the composition of the mixture is then characterised by weight fractions wi as defined by eqn (1.1). In the literature, partial molar properties are frequently characterised by an overbar and identified by a subscript, e.g. M̄i.

With eqn (1.114), the exact differential eqn (1.113) can be written in a more compact form:

Equation 1.115

Eqn (1.115) applies to single-phase equilibrium PVT system, either open or closed, with the last term giving the differential variation of Mt = nM caused by amount-of-substance transfer across the phase boundary or by chemical reactions or both.

From experiments, we know that Mt is homogeneous of the first degree in the amounts of substance, hence Euler's theorem, eqn (1.44), yields

Equation 1.116

Evidently, the partial derivatives are the partial molar properties Mi just defined by eqn (1.114), and at constant T and Peqn (1.116) may therefore be written as follows:

Equation 1.117

Division by the total amount of substance n gives the molar property

Equation 1.118

eqn 117 and (1.118) are known as summability relations. To reiterate: since M(T, P, {xi}) is an intensive property, the partial molar property Mi(T, P, {xi}) is also intensive. Characterising a single-phase pure-substance property with an asterisk (*) and, if needed, identifying it by a subscript, we find in general

Equation 1.119

However, from eqn (1.118),

Equation 1.120

We now recognise that the chemical potential of component i, cf.eqn (1.91), is the partial molar Gibbs energy of component i:

Equation 1.121

For homogeneous PVT fluids, eqn (1.117) is generally valid, hence the total differential is given by

Equation 1.122

Eqn (1.115) provides an alternative expression for d(nM). Thus, it follows that

Equation 1.123

and division by n yields the most general form of the Gibbs–Duhem equation:

Equation 1.124

applicable to any molar property M in a homogeneous system. This equation is of central importance in chemical thermodynamics. For changes at constant T and P it simplifies to

Equation 1.125

which shows the constraints on changes of mixture composition. It is important to note that a partial molar property Mi is an intensive property referring to the entire mixture, it is not a property of component i: partial molar properties must be evaluated for each mixture at any T and P, at each composition of interest. However, a partial molar property defined by eqn (1.114) can always be used to provide a systematic formal subdivision of the extensive property Mt = nM into a sum of contributions ascribed to the individual species i and constrained by eqn (1.117) or a systematic formal subdivision of the intensive property M into a sum of contributions ascribed to the individual species i and constrained by eqn (1.118). Hence one may use partial molar properties as though they possess property values referring to the individual constituent species in solution. Such a formal subdivision may also be based on mass instead of amounts of components, in which case partial specific properties are obtained with similar physical significance.

In summary, with few exceptions, the following general system of notation will be used throughout this chapter:

  • Solutions/mixtures are of prime interest and a molar property of a single-phase multicomponent solution, such as the molar volume V, is represented by the plain symbol M; additional superscripts (such as L or V, identifying a liquid phase or a vapour phase property, respectively, or pg, signifying the perfect gas state) will be attached as needed.

  • A total property of a single-phase multicomponent solution is represented by the product nM, where denotes the total amount (of substance) and ni is the amount of component i, or alternatively by the symbol Mt.

  • Pure-substance properties are characterised by an asterisk (*) and identified by a subscript, e.g. Mi* is a molar property of pure component i = 1, 2, …; additional superscripts/subscripts will be attached as needed.

  • Partial molar properties referring to component i in solution are identified by a subscript, e.g. Mi, i = 1, 2, …; additional superscripts/subscripts will be attached as needed.

For homogeneous fluids of constant composition in a closed PVT system, the fundamental property relations corresponding to Legendre transforms excluding the chemical potentials are readily obtained and for one mole of such a homogeneous constant composition fluid (this includes pure fluids) the following four energy-based property relations apply:

Equation 1.126
Equation 1.127
Equation 1.128
Equation 1.129

They are exact differentials, hence

Equation 1.130
Equation 1.131
Equation 1.132
Equation 1.133

These relations establish links between the independent canonical variables S, V, P and T and the internal energy-based molar properties U, H, F and G. For simplicity's sake, subscripts {ni} or {xi}, indicating constant amounts or constant composition, respectively, will be omitted. Because eqn (1.126)–(1.129) are exact differentials, application of the reciprocity relation (Schwarz's theorem) yields Maxwell relations for a homogeneous constant-composition fluid (PVT system):

Equation 1.134
Equation 1.135
Equation 1.136
Equation 1.137

The last two relations, derived from eqn 128 and (1.129), respectively, are the most useful Maxwell relations, since they replace derivatives of the entropy, which is not a directly measurable property, with derivatives of the measurables pressure and volume. Maxwell relations also form part of the thermodynamic basis of the relatively new experimental technique known as scanning transitiometry.136,137 

When investigating thermodynamic system properties experimentally, we are usually interested in measuring how the properties of homogeneous constant-composition fluids respond to changes of temperature, pressure or volume. That is, we are interested in exploiting linear relations between cause and effect, say, for Mt(T, P) = nM:

Equation 1.138

Since the partial derivatives measure the change of the total property caused by a small change of T or P (sufficiently small for linear approximation), the partial derivatives are collectively known as response functions.

As indicated by eqn (1.130)–(1.133), the first-order derivatives of the primary functions are thermodynamically important, although it is the second-order derivatives that are indispensable for an in-depth discussion of material properties. For simple, constant-composition fluids (PVT systems), the most frequently selected set of three basic derivatives (to which many others can be related56,134 ) consists of the molar heat capacity at constant pressure (molar isobaric heat capacity):134,138–140 

Equation 1.139

the isothermal compressibility:134,139,140 

Equation 1.140

and the isobaric expansivity:134,139,140 

Equation 1.141

The conventional choice of the set {CP, βT, αP} is based on the experimentally useful alternative internal energy-based fundamental property relation eqn (1.86), see Table 1.3, which for one mole of a homogeneous fluid of constant composition is given by eqn (1.129). In (T, P) space, the basic set {CP, βT, αP} contains the only independent second-order derivatives of the Gibbs potential.56,134  As demonstrated by Callen,56  this set may be used advantageously for the “reduction of derivatives”, that is, for expressing any desired thermodynamic first-order derivative in terms of these (easily) measurable properties. As pointed out by McGlashan4  (see Section 1.1, Introduction), this is one of the major accomplishments of thermodynamics. Although deriving thermodynamic relations involves only fairly straightforward mathematics, the difficulty lies in the large number of possible alternative representations and in the selection of the optimal sequence of steps. One of the first to tackle this problem was Bridgman, who presented extensive tables,141  although later the use of Jacobians for systematising the reduction of partial thermodynamic derivatives attracted increased attention.142–148 

Alternatively, we may base our choice on the equivalent internal energy-based fundamental property relation eqn (1.85), see Table 1.3, which for one mole of a homogeneous constant-composition fluid is given by eqn (1.128). The corresponding three second-order derivatives of the Helmholtz potential are the molar heat capacity at constant volume (molar isochoric heat capacity):134,138–140 

Equation 1.142

the isothermal compressibility:134,139,140 

Equation 1.143

and the isochoric thermal pressure coefficient:134,139,140 

Equation 1.144

In (T, V) space, the set {CV, 1/βT, γV} contains the only independent second-order derivatives of the Helmholtz potential,134  and it may as well be used as the basic set for the “reduction of derivatives”, that is, for expressing any desired thermodynamic first-order derivative in terms of these three (easily) measurable properties. This approach is particularly convenient when dealing with properties derived from pressure-explicit EOSs.

For the sake of completeness, the corresponding relations for the remaining two thermodynamic potentials U(S, V), see eqn (1.28), and H(S, P), see eqn (1.29), are summarised as follows:

Equation 1.145

for the inverse isentropic compressibility134,138–140  and

Equation 1.146

for the isentropic expansivity134,139,140 

Equation 1.147

and {1/CV, 1/βS, 1/αS} is the set of independent second-order derivatives in (S, V) space.134  Based on H(S, P), for a constant-composition fluid we have

Equation 1.148
Equation 1.149

and the isentropic thermal pressure coefficient134,139 

Equation 1.150

thus providing the set {1/CP, βS, 1/γS}.134 

Since for a constant-composition phase the three mutual derivatives of P, V and T satisfy the triple product rule:

Equation 1.151

the three mechanical coefficients are related as follows:

Equation 1.152

and

Equation 1.153

Additional useful relations for a closed constant-composition phase may now be established systematically between members of the fundamental set and of the alternative sets of second-order partial derivatives of the primary functions. However, as in preceding volumes,134,137,138  here I adopt another approach by placing the emphasis on discussing some selected responses of U, H, F, G, etc., to changes in T and P or T and V, respectively, and introducing appropriate relations between second-order partial derivatives en route, whenever convenient.

Focusing now on the molar Helmholtz energy F and the molar Gibbs energy G, their definitions in Table 1.2 [cf. eqn (1.79) and (1.80), respectively] in conjunction with eqn (1.133) directly yield the Gibbs–Helmholtz equations:

Equation 1.154
Equation 1.155

Simple mathematical manipulations lead to the following alternative forms:

Equation 1.156
Equation 1.157

and

Equation 1.158
Equation 1.159

Eqn (1.158) suggests an alternative to the fundamental property relation eqn (1.86) by introducing the dimensionless property G/RT:

Equation 1.160a

and thus

Equation 1.160b

Eqn (1.160a) is of considerable utility: knowledge of nG/RT as a function of the canonical variables T, P and ni allows the calculation of all thermodynamic properties of the mixture. All terms have the dimension of amount-of-substance and, in contradistinction to eqn (1.86), the enthalpy rather than the entropy appears in the first term on the right-hand side of this exact differential eqn (1.160a), with obvious benefits for discussing experimental results.

An analogous equation may be derived involving the Helmholtz energy. Introducing the dimensionless property F/RT, the fundamental property relation alternative to eqn (1.85) reads

Equation 1.161a

and thus

Equation 1.161b

In contradistinction to eqn (1.85), the internal energy rather than the entropy appears in the first term on the right-hand side of eqn (1.161a).

The volume dependence of U and the pressure dependence of H are conveniently derived as follows: differentiation of the appropriate Gibbs–Helmholtz equation, eqn 154 and (1.155), respectively, yields

Equation 1.162
Equation 1.163

Note that both equations can be contracted:

Equation 1.164
Equation 1.165

(∂U/∂V)T is a useful property in liquid-state physical chemistry,149–154  and has been given a special symbol:

Equation 1.166

and a special name, internal pressure. It may be directly determined at any pressure viaeqn (1.162) by measuring γV ≡ (∂P/∂T)V with a piezometer or by using experimental results for the isobaric expansivity αP and the isothermal compressibility βT in conjunction with eqn (1.152). Somewhat surprisingly, the former direct approach is nowadays rarely pursued, although the various methods reported in the literature can be carried out with high precision.155–162  For a pure liquid (L), the internal pressure reads

Equation 1.167

At ambient temperatures and pressures, usually PVL,*, notable exceptions being liquid water and liquid heavy water in the temperature range from the respective triple point, Ttr(H2O) = 273.16 K and Ttr(D2O) = 276.97 K, to temperatures slightly above that of the respective density maximum, i.e. Tmax(H2O) = 277.13 K and Tmax(D2O) = 284.35 K. Recent measurements and calculations of the internal pressure of saturated and compressed fluid phases of several hydrocarbons, carbon dioxide, methanol and water over large temperature ranges, including near- and supercritical conditions, have been discussed by Abdulagatov et al.153  For instance, the curves of Π versus P along the critical isotherm all show a similar behaviour: first, Π increases with increase in pressure, then passes through a maximum and decreases (less pronounced) at higher pressures. For n-butane, Πmax ≈ 190 MPa at an external pressure of about 150 MPa. For CO2, Πmax ≈ 345 MPa at an external pressure of about 235 MPa. Note, however, that the statistical-mechanical definitions of the internal energy [their eqn (16.3)] and of the internal pressure [their eqn (16.4)], as presented by Abdulagatov et al., are incorrect.

The isodimensional cohesive energy density cL,*(T, P) was introduced to liquid-state physical chemistry by Scatchard163  and is defined by (see Wilhelm164,165 )

Equation 1.168

where EcohL,*(T, P) denotes the cohesive energy per mole of pure liquid, the molar isobaric residual internal energy is defined by UR,L,*(T, P) ≡ UL,*(T, P) − Upg,*(T), with the superscript R characterising a residual property in (T, P, {xi}) space (for details see Section 1.3.2), and the superscript pg indicates the perfect-gas state. Expanding van Laar's ideas on non-electrolyte solubility,166  Hildebrand167,168  and Scatchard163  formulated the basic concepts of regular solution theory1, and in 1950 Hildebrand and Scott, in their influential monograph The Solubility of Nonelectrolytes, 3rd edition,69  introduced the term solubility parameter with the symbol δ:

Equation 1.169

which has been in general use since then; conventionally, the superscripts L and * are omitted with δ(T). We note, however, that the values of most solubility parameters of liquids reported in the literature for ambient pressure and for temperatures well below the critical temperature of the liquid are based on the approximate relation

Equation 1.170

where the subscript σ indicates saturation conditions (vapour–liquid equilibrium, VLE), that is, Pσ = Pσ(T) denotes the vapour pressure and ΔvapH*(T) is the molar enthalpy of vaporisation. In this form, essentially only the effect of temperature is explicitly considered. Evidently, for use in the design and evaluation of supercritical fluid extraction (SCFE) processes, where T > Tc and P > Pc, eqn (1.170) is not applicable: under supercritical conditions, saturation properties, such as the enthalpy of vaporisation, are devoid of meaning. The exact definition provided by eqn (1.168), however, is valid for liquid and gaseous phases, hence after dropping the superscript L it may be used together with

Equation 1.171

for calculating the supercritical solubility parameter. In fact, any appropriate pressure-explicit EOS may be used to obtain the molar residual internal energy (see ref. 165). However, using the Lee–Kesler equation,174  which is based on Pitzer's three-parameter corresponding states theorem (CST),175–178  Pang and McLaughlin179  derived a CST formulation for the square of the reduced solubility parameter at TrT/Tc and PrP/Pc:

Equation 1.172

and presented tables of (δ2/Pc)(0), (δ2/Pc)(1) and (δ2/Pc)(2) as functions of Tr and Pr for ranges of practical interest in SCFE work, i.e. for 1 ≤ Tr ≤ 4 and 0.2 ≤ Pr ≤ 10. Here, Pitzer's acentric factor ω is defined by

Equation 1.173

where Pσ,rPσ/Pc is the reduced vapour pressure. Applications were given concerning the solubility of fluorene (solid) in supercritical ethene and carbon dioxide; comparisons with experimental data obtained at up to 70 °C and 48.35 MPa were satisfactory throughout. They also performed explorative SCFE experiments at 300 °C on bitumen from tar sands with a variety of solvents (carbon dioxide, methanol, acetone, etc.).

Although the internal pressure Π* ≡ (∂U*/∂V)T = V*(T, P) − P and the cohesive energy density c* ≡ −UR,*/V* possess the same dimensions, that is, in SI units we have Pa = J m−3, the defining equations, i.e.eqn 167 and (1.168), respectively, immediately reveal the macroscopic differences between these two properties (and, of course, the close connection): as an integral property, c* ≡ −UR,*/V* measures the total molecular cohesion of the fluid, whereas Π* ≡ (∂U*/∂V)T = (∂UR,*/∂V)T characterises its isothermal differential change with volume.

Cohesive energy density and internal pressure do not reflect the same fluid property although, unfortunately, many researchers have failed to discriminate between them. From a microscopic point of view, the difference is, perhaps, best seen when comparing the appropriate statistical-mechanical expressions.165  Noting that the perfect-gas value of the internal energy is independent of P and V, with the usual assumptions the molar residual internal energy for a fluid is obtained as

Equation 1.174

Hence the cohesive energy density is given by

Equation 1.175

The total pressure is given by

Equation 1.176

hence the internal pressure is obtained as

Equation 1.177

where ρN*L/V* = * is the pure-fluid number density, L is the Avogadro constant2 and u(r) denotes the pair-potential energy (pairwise additivity of the potential energy). The pair-distribution function for spherically symmetric molecules, g(r, T, ρN*), has been written explicitly as a function of the interparticle distance r and also of T and the pure-fluid number density ρN* to emphasise its dependence on these state variables. Comparison of eqn (1.175) with eqn (1.177) clearly shows that the cohesive energy density (and thus the square of the solubility parameter) c*(T, P) = [δ(T, P)]2 and the internal pressure Π*V*P are two different, although isodimensional, properties.

As already indicated in Section 1.2, the volumetric properties of fluids occupy an important position in physics, physical chemistry and chemical engineering and many distinguished scientists have contributed to this subject, that is, they contributed to the development of pressure–volume–temperature-composition relations that would eventually lead to reliable PVTx EOSs, applicable to both gaseous and liquid phases. These EOSs relate the variables in either a volume-explicit form, nV = Vt(T, P, n1, n2,…), as does the virial equation in pressure, eqn (1.54), or in a pressure-explicit form, P = P(T, nV, n1, n2,…), as does the virial equation in amount density, eqn (1.55), although most realistic EOSs are pressure explicit, i.e. T, V or ρ = 1/V, and the compositions are the independent variables. In particular, experimental vapour- or gas-phase PVTx data at low densities/low pressures have provided a large body of second virial coefficients B and third virial coefficients C,10  and have thus contributed enormously80,182–186  to our knowledge of intermolecular interactions. For a pure fluid with a spherically symmetric potential-energy function u(r) for a pair of molecules, the second virial coefficient is given by

Equation 1.178

where kB is the Boltzmann constant and the quantity is commonly known as the Mayer f-function. Eqn (1.178) provides access to the fundamentally important pair potential-energy function u(r), which is frequently approximated by a Mie (n, m)-type function, introduced in 1903:187,188 

Equation 1.179

The positive constants n and m (n > m) are associated with molecular repulsion and attraction, respectively, ε is an intermolecular energy parameter characterizing the well depth of the interaction energy function, i.e. u(rmin) = −ε, and σ is an intermolecular distance parameter characterised by u(σ) = 0. Special cases of the Mie (n, m) function were introduced by Lennard-Jones in 1924 and connected with gas viscosities,189  the EOSs of real gases,190  X-ray measurements on crystals191  and quantum mechanics.192  The most common form of the Lennard-Jones (12,6) function is192 

Equation 1.180

where σ = 2−1/6rmin.

(∂H/∂P)T is a useful property for determining second virial coefficients of gases and vapours (subcritical conditions) and is known as the isothermal Joule–Thomson coefficient, that is,

Equation 1.181

which in turn is related to the isenthalpic Joule–Thomson coefficient:

Equation 1.182

by

Equation 1.183

The three quantities φ, µJT and CP of gases/vapours can be measured by flow calorimetry.193–195  Since for perfect gases P = 1, φ = 0 and µJT = 0, the real-gas values of these coefficients are directly related to molecular interactions. Flow calorimetry has the advantage over compression experiments that adsorption errors are avoided and measurements can therefore be made at low temperatures where conventional techniques are difficult to apply. Specifically, in an isothermal throttling experiment the quantity measured can be expressed in terms of virial coefficients and their temperature derivatives:

Equation 1.184

where 〈P〉 is the mean experimental pressure. The zero-pressure value of φ is thus given by

Equation 1.185

and integration between a suitable reference temperature Tref and T yields196 

Equation 1.186

This relation has been used for the determination of B of vapours. The isothermal Joule–Thomson coefficient of steam, the most important vapour on Earth, was measured by McGlashan and Wormald194  in the temperature range 313–413 K and derived values of φ0 were compared with results from the 1984 NBS/NRC steam tables,197  with data of Hill and MacMillan198  and with values derived from the IAPWS-95 formulation for the thermodynamic properties of water.12 

The isothermal pressure dependence of U of a constant-composition fluid:

Equation 1.187

is obtained viaeqn (1.162) and the chain rule, and eqn (1.163) plus the chain rule yields

Equation 1.188

We now turn to the temperature derivatives of U and H, i.e. to the heat capacities of constant-composition fluids. Recalling that the molar isochoric heat capacity is defined by eqn (1.142), from eqn (1.162) we obtain directly

Equation 1.189

The molar heat capacity at constant pressure is defined by eqn (1.139) and with eqn (1.163) we obtain

Equation 1.190

In high-pressure research,136–140,199–206 eqn 189 and (1.190) are particularly interesting. For instance, the pressure dependence of CP of a constant-composition fluid may be determined either from PVT data alone or by high-pressure calorimetry or by transitiometry,136,137  or by measuring the speed of ultrasound at sufficiently low frequency as a function of P and T,138,199,201,202,204–211  and the consistency of the experimental results can be ascertained in various ways.

Starting from

Equation 1.191

we obtain

Equation 1.192

and thus, with eqn (1.141)–(1.144), (1.146) and (1.154), the alternative relations

Equation 1.193
Equation 1.194

and

Equation 1.195

Note that the difference between CP and CV depends on volumetric properties only. The heat capacity difference may therefore also be expressed by138–140 

Equation 1.196

where the compression factor Z is defined by eqn (1.54).

For a constant-composition fluid, the functional dependence of the molar internal energy and the molar entropy on T and V and of the molar enthalpy and the molar entropy on T and P, respectively, can be expressed as follows:

Equation 1.197
Equation 1.198
Equation 1.199
Equation 1.200

where use was made of eqn (1.136), (1.137), (1.139), (1.142), (1.162) and (1.163). From eqn (1.198), in conjunction with eqn (1.193), the following relation is obtained:

Equation 1.201

while eqn (1.200) in conjunction with eqn (1.193) yields

Equation 1.202

Combination of eqn (1.201) and (1.202) results in

Equation 1.203

thus complementing eqn 198 and (1.200).

The ratio of the molar heat capacities, κCP/CV , is accessible viaeqn 139 and (1.142) in conjunction with the chain rule, i.e.

Equation 1.204

and the triple product rule:

Equation 1.205

Thus, for homogeneous constant-composition fluids we obtain

Equation 1.206

thereby establishing the experimentally and theoretically important ultrasonics connection.32,212–217  Using eqn (1.206) together with

Equation 1.207

leads to

Equation 1.208

which is one of the most important equations in fluid phase thermophysics. Here, ρmmm/V = mm ρ = mm ρN/L denotes the mass density and v0 = v0(T, P, {xi}) is the speed of ultrasound at sufficiently low frequency and small amplitude. To an excellent approximation, i.e. neglecting dissipative processes due to shear viscosity ηs, thermal conductivity λ, bulk viscosity ηv, etc., v0 may be treated as an intensive thermodynamic equilibrium property32,138,207–215  related to βSviaeqn (1.207). Alternatively, by using the relations provided by eqn 131 and (1.132), we have

Equation 1.209
Equation 1.210
Equation 1.211

and

Equation 1.212

Other equivalent equations may be found by straightforward applications of relations between βS and βT introduced below, e.g.

Equation 1.213

Whereas sufficiently small amplitudes of sound waves are readily realised, sufficiently low frequencies f constitute a more delicate problem.138,218  Here, only a few aspects are mentioned to alert potential users to the fact that not all sound speed data reported in the literature are true thermodynamic data that can be used, say, with eqn 207 and (1.208). When sound waves propagate through molecular liquids, several mechanisms help dissipate the acoustic energy. The classical mechanisms that cause absorption, i.e. those due to shear viscosity ηs and thermal conductivity λ, are described by the Kirchhoff–Stokes equation:212 

Equation 1.214

where αcl denotes the classical amplitude absorption coefficient. Sound dispersion due to classical absorption is almost always negligible and the product v(  f  )v02 in eqn (1.214) is usually replaced with v03. However, bulk viscosity, thermal molecular relaxation and structural relaxation may contribute to make the experimental absorption coefficient significantly larger than classically predicted. Relaxation processes cause absorption and dispersion, i.e. the experimental speed of sound v(  f  ) is larger than v0 (for details, consult the classical monograph of Herzfeld and Litovitz212 ). At higher frequencies, many liquids show dispersion of the speed of sound,32,138,208,209,212–218  but particular care must be exercised when investigating liquids with molecules that exhibit rotational isomerism, where ultrasonic absorption experiments indicate rather low relaxation frequencies.

At temperatures well below the critical temperature,118,119,219,220 γV of liquids is large and the direct calorimetric determination of CV is not easy. It requires sophisticated instrumentation,221  as evidenced by the careful work of Magee at NIST,222,223  although it becomes more practicable near the critical point where γV is much smaller.

From the equations for the difference CPCV of a constant-composition fluid, it follows that

Equation 1.215

and

Equation 1.216

We note that heat capacities may be determined by measuring only isobaric expansivities and isothermal and isentropic compressibilities. Combining eqn (1.194)(1.215) and (1.216) yields

Equation 1.217

and

Equation 1.218

Eqn (1.217) establishes a link with Rayleigh–Brillouin light scattering.208,209  For liquid rare gases, the ratio of the integrated intensity of the central unshifted Rayleigh peak, IR, and of the two Brillouin peaks, 2IB, is given by the Landau–Placzek ratio:

Equation 1.219

From eqn (1.208), the difference between βT and βS may be expressed as

Equation 1.220

and for the difference of the reciprocals we have

Equation 1.221

Isentropic changes on the PVT surface are described in terms of the isentropic compressibility βS, eqn (1.146), the isentropic expansivity αS, eqn (1.147), and the isentropic thermal pressure coefficient γS, eqn (1.150). The three isentropic coefficients are related by

Equation 1.222

Useful relations with more conventional second-order derivatives are given below:

Equation 1.223
Equation 1.224

For the isentropic thermal pressure coefficient, we have, with the triple product rule and eqn 137 and (1.220)

Equation 1.225
Equation 1.226

Most values of βS are calculated from measured speeds of sound using eqn (1.198), although they can also be measured directly. Direct measurement of isentropic compressibility was pioneered by Tyrer in 1913/1914.224  The principle of the method consists simply of subjecting the liquid contained in a suitable vessel (usually a piezometer made of glass) to a change in pressure of about 1 bar and then measuring the volume change on release of the pressure (instead of on compression). The rapidity of the operation ensures an almost true adiabatic condition. However, surprisingly few researchers followed up his work, such as Philip,225  Staveley and co-workers,226,227  Harrison and Moelwyn-Hughes228  and Nývlt and Erdös.229 

Burlew's piezo-thermometric method230  for determining CP is based on eqn (1.225), i.e. on measuring (∂T/∂P)S and (∂V/∂T)P = P, similarly to the approach of Richards and Wallace.231 

As pointed out by Rowlinson and Swinton,79  the mechanical coefficients αP, βT and γV are determined, to a high degree of accuracy, solely by intermolecular forces, while the isentropic coefficients αS, βS and γS, with which they are related through the thermal coefficients, i.e. the heat capacities, and the heat capacities themselves depend also on internal molecular properties.

The perfect gas (ideal gas) is a hypothetical substance with the following three defining characteristics:

  1. Its constituent molecules exert no forces on each other (non-interacting molecules), hence the total intermolecular potential energy is independent of the positions of the molecules and may be taken to be zero.

  2. Because of their unimpeded movement the molecules possess kinetic energy and there is temperature.

  3. Randomly moving molecules hit the container walls and are elastically reflected; the average force exerted per unit wall area is equal to the gas pressure, so there is pressure (and volume).

The summary definition given above implies the perfect-gas EOS:

Equation 1.227

For any substance that obeys this EOS, eqn 162 and (1.163) show that internal energy and the enthalpy depend only on temperature:

Equation 1.228

While no real fluid conforms to this model, nevertheless the concept is eminently useful, because (a) the associated simple equations may frequently be used as reasonable approximations of real-gas behaviour at low pressures and (b) the model serves as the basis for the definition of an extremely useful class of auxiliary functions known as residual functions, that play a central role in the description of real-fluid behaviour. Note, that in the limit of vanishingly small pressure, real-gas behaviour in many respects approaches perfect-gas behaviour, but not in all, a fact which is frequently overlooked.138  Specifically, for a constant composition gas (this includes a pure gas), we have

Equation 1.229

The thermodynamic equations formally introduced in Section 1.2 establish exact relations between system properties and judiciously selected independent variables, the most convenient being the sets {T, P, {xi}} and {T, V, {xi}}. However, they do not provide numerical values for any thermodynamic property: reliable experimental data and/or reliable models are necessary to reach this goal. For comparing real-fluid properties with perfect-gas properties at the same temperature, same pressure and same composition, say, the actual molar volume V to Vpg, two obvious choices exist: one may quantify deviations in terms of a ratio measure, here the compression factor:

Equation 1.230

or in terms of a difference measure, here the molar residual volume:

Equation 1.231

The two functions are, of course, related:

Equation 1.232

and since both the numerator (Z − 1) and the denominator P vanish as the pressure reaches its limiting value zero, the zero-pressure limit of VR becomes indeterminate, necessitating application of de l’Hôpital's rule:

Equation 1.233

Since experiments show that (∂Z/∂P)T generally remains finite (and not zero) in the limit of vanishingly small pressure, VR also remains generally finite: as shown by eqn (1.54), it is given by the second virial coefficient, i.e.

Equation 1.234

Hence VR is non-zero except at the Boyle temperature at which B = 0. Eqn (1.234) identifies an experimentally accessible macroscopic property as a key thermophysical quantity establishing an important link to the intermolecular pair-potential energy function u(r) as indicated by eqn (1.178) and the cited ref. 80 and 182–186.

In analogy with VR, molar isobaric residual properties MR of a single-phase pure fluid or constant-composition fluid mixture are defined similarly by55,68,79,165 

Equation 1.235

where the superscript R identifies a residual function in (T, P, {xi}) space. The Ms denote molar values of any extensive thermodynamic property nM (T, P, {xi}), such as U, H, S, V, G or F. M (T, P, {xi}) is the actual molar property value of the fluid at the temperature, pressure and composition of interest and Mpg (T, P, {xi}) is the molar property value for the fluid in its perfect-gas state at the same T, P and {xi}. It is important to note that if the temperature and the pressure are the same for the real fluid and the perfect gas, the molar volume is not the same, and if the temperature and the molar volume are the same, the pressure is different. It is reiterated that the perfect-gas state is hypothetical except in the zero-pressure limit, where the perfect-gas EOS eqn (1.227) is valid (that is, for this real perfect-gas state PV = RT). Residual properties are the most direct measures of the effects of the intermolecular forces. The computation of values M of any thermodynamic fluid property is based on

Equation 1.236

From the defining equation, eqn (1.235), we have for a differential change in state of MR at constant T and constant {xi}

Equation 1.237

and integration from P = 0 to the pressure of interest P yields

Equation 1.238

In contradistinction to the observed limiting behaviour of the residual volume, see eqn (1.234), experimental evidence indicates that for the pivotal properties internal energy and enthalpy the zero-pressure terms can be set equal to zero (at constant T and {xi}):

Equation 1.239
Equation 1.240

Thus, for the molar isobaric residual enthalpy HR(T, P, {xi}) we obtain, in conjunction with and ,

Equation 1.241

For the molar isobaric residual entropy we obtain, in conjunction with and ,

Equation 1.242

Although each term in the first integrand diverges for P → 0, these divergences cancel and the integral in eqn (1.242) is bounded.

Since

Equation 1.243

and the pressure dependence of CP is given by and , the molar isobaric residual constant-pressure heat capacity is obtained from

Equation 1.244

For the remaining molar isobaric residual properties, we have

Equation 1.245

The two terms of the integrand in the GR equation cancel each other for P → 0 and no divergence is observed. Equivalent, albeit slightly different, expressions are presented in the Appendix.

The isobaric residual properties are the conventional forms since they are advantageously based on {T, P, {xi}} as independent variables. They have been most useful in applications to real gases and gas mixtures, although their suitability in dealing with liquid systems has been greatly furthered by the application of Pitzer's3

Equation 1.246

three-parameter corresponding-states theorem (CST);176,177,233–235  this theorem is firmly based on statistical mechanics and occupies a leading position in the field of property estimation.6,8,55,68,178  For pure fluids, that is, for gases, vapours and liquids, the compression factor Z* = PV*/RT is expressed as a function of reduced temperature TrT/Tc, reduced pressure PrP/Pc and Pitzer's acentric factor ω, which is defined by eqn (1.173). Specifically, in the key three-parameter CST correlation

Equation 1.247

Z(0) represents the simple-fluid contribution to Z* that is based on experimental PVT data determined for Ar, Kr and Xe for which ω is essentially zero (and thus a two-parameter CST correlation suffices). Z(1) represents the non-simple-fluid contribution to Z*; it is determined via experimental PVT data of selected fluids with ω ≠ 0 (quantum fluids, strongly polar fluids and fluids with strong hydrogen bonds are excluded). Critically evaluated values of the ωs for many fluids, together with values for Tc and Pc, are tabulated in ref. 8. The most popular Pitzer-type correlation is that developed by Lee and Kesler,174  who presented tables for the contributions Z(0)(Tr, Pr) and Z(1)(Tr, Pr), and also for derived functions for both liquid and vapour phases, covering large temperature and pressure ranges, i.e. 0.30 ≤ Tr ≤ 4.00 and 0.01 ≤ Pr ≤ 10.00.

The general relations introduced above apply to both pure substances and mixtures. In order to use generalised CST correlations for mixtures, the conventional practice is based on the assumption that mixture properties can be represented by the same types of correlation developed for pure fluids, although with appropriately defined values for the corresponding-states scaling parameters of the mixture, that is, by essentially empirically averaging pure-component parameters Tc,i, Pc,i and ωi to obtain pseudocritical temperatures Tpc, pseudocritical pressures Ppc and pseudo-acentric factors ωp referring to the mixture. This is accomplished by using recipes known as mixing rules. Thus, a three-parameter CST correlation for the compression factor Z of the mixture, in the one-fluid approximation, may be written as

Equation 1.248

where the pseudoreduced temperature Tpr and the pseudoreduced pressure Ppr are defined by

Equation 1.249

respectively.

The simplest set of mixing rules for pseudocritical parameters are those of Kay.236  They are defined as mole fraction-weighted sums of the pure-component values and so is ωp, i.e.

Equation 1.250

Although simple to apply, for mixtures of molecularly noticeably dissimilar fluids, Kay's rules are often inadequate and more flexible and therefore more elaborate recipes must be introduced,8  such as quadratic mixing rules (reminiscent of those used in the multicomponent vdW model):

Equation 1.251

where Tc,ii and Tc, jj denote the critical temperatures of the pure components i and j, respectively, Pc,ii and Pc,jj are their critical pressures, respectively, and for the evaluation of the cross parameters Tc,ij and Pc,ij empirical recipes known as combining rules are required.6,8 

However, it is emphasised again that temperature and volume or, alternatively, amount density ρ ≡ 1/V or number density L/V are the commonly used variables in statistical mechanics; in addition, on the practical side, most PVTx EOSs, such as cubic vdW-type equations, are pressure explicit.6,8,55,63,68,237–241  With T and V (and for mixtures, of course, also the composition {xi}) being the natural independent (canonical) variables, the focus is on the Helmholtz energy as generating function. Thus, for a single-phase pure fluid or constant-composition mixture one may also define molar residual properties in (T, V, {xi}) space, i.e. molar isochoric (isometric) residual properties:

Equation 1.252

To distinguish them from isobaric residual properties, they are indicated throughout by a superscript lower-case r. Again, the Ms denote molar properties of any extensive thermodynamic property nM(T, nV, {xi}) of the fluid, for instance U, H, S, G or F, and Mpg(T, V, {xi}) is the corresponding molar property of the fluid in its hypothetical perfect-gas state at the same temperature, the same molar volume and the same composition. Note the important fact that if the temperature and the volume are the same for the real-fluid state and the perfect-gas state, the pressure is not the same. We note, however, that eqn (1.252) may also be used for the definition of a residual pressure, an intensive property:

Equation 1.253

With the availability of a pressure-explicit EOS, molar isochoric residual properties are the properties of direct interest. In complete analogy with eqn (1.238), we have

Equation 1.254

Again, experimental evidence indicates that for the pivotal properties internal energy and entropy, the infinite-volume terms (zero-pressure terms) can be set equal to zero (at constant T and {xi}), that is,

Equation 1.255

Thus, for the molar isochoric residual internal energy Ur(T, V, {xi}) we obtain, in conjunction with , and eqn (1.255),

Equation 1.256

and for the molar isochoric residual enthalpy Hr(T, V, {xi}) we obtain, with , , and eqn (1.240),

Equation 1.257

For the molar isochoric residual entropy Sr(T, V, {xi}), we obtain, in conjunction with , and eqn (1.255),

Equation 1.258

Since CV of a real gas approaches CVpg for V → ∞ (i.e. for P → 0), together with and , see eqn (1.189), the molar isochoric residual constant-volume heat capacity CVr(T, V, {xi}) may be calculated via

Equation 1.259

In addition, we have

Equation 1.260

Equivalent, albeit slightly different, expressions are presented in the Appendix. For the computation of values M of any thermodynamic fluid property,

Equation 1.261

is used.

The two types of residual functions, i.e. MR (T, P, {xi}) and Mr (T, V, {xi}), are rigorously related:164 

Equation 1.262

It is important to realise that P and V are parameters associated with the state of the real fluid system at temperature T and constant composition {xi} and they are therefore not related by the perfect-gas law: the lower integral limit denotes the gas pressure P = RT/V for which the molar volume of the perfect-gas mixture has the same value V as that of the real mixture at T and {xi}. Alternatively, we have

Equation 1.263

Note that Vr(T, V, {xi}) and PR(T, P, {xi}) are identically zero.

Since at constant composition the perfect-gas properties Upg, Hpg, CPpg  and CVpg   are all functions of temperature only, i.e. the first-law properties are independent of pressure and of volume, the equality

Equation 1.264

holds for M = U, H, CP and CV. In contradistinction, the second-law perfect-gas properties Spg, Gpg and Fpg are functions of temperature and they do depend on pressure:

Equation 1.265
Equation 1.266

and they do depend on volume:

Equation 1.267
Equation 1.268

Hence, by virtue of eqn (1.262) or eqn (1.263), respectively, the following relations between the residual second-law properties are obtained:

Equation 1.269
Equation 1.270
Equation 1.271

An important property in solution chemistry is the fugacity coefficient ϕiπ(T, P, {xiπ}) of component i in solution in phase π.242  It is related to the molar isobaric residual chemical potential, which is obtained by applying the partial molar derivative prescription eqn (1.114) to the expression for the molar isobaric residual Gibbs energy of the mixture in any phase π, taking into account that for the model perfect-gas mixture we have

Equation 1.272

Hence

Equation 1.273

and the partial molar isobaric residual Gibbs energy/the isobaric residual chemical potential of component i in solution in phase π reads

Equation 1.274

As shown later, eqn (1.274) provides a rigorous basis for the definition of the fugacity coefficient ϕiπ (T, P, {xiπ }) of component i in solution in phase π.

The fugacity concept was introduced by G. N. Lewis in 1901.243,244  It serves to maintain the simple formal structure of thermodynamic equations applicable to perfect-gas (ideal-gas) systems, while avoiding the troublesome behaviour of the chemical potential when either P or xiπ approaches zero. Thus, in analogy with the expression for an isothermal change of the molar Gibbs energy of a homogeneous pure real fluid i in phase π:

Equation 1.275

which becomes for a pure perfect gas with Vipg,*        = RT/P

Equation 1.276

we keep the simple structure of eqn (1.276) and, following Lewis, replace P with a new function fi* that makes, by definition, this differential relation generally valid. Hence, as the first part of the definition, we have

Equation 1.277

where fiπ,*     = fiπ,*     (T, P) is called the fugacity of the pure real substance i in phase π. To preserve consistency between eqn 277 and (1.276), we demand that

Equation 1.278

Eqn (1.278) is the second part of the definition of the fugacity and, together with eqn (1.277), they constitute the complete definition of fi π,*     (T, P). For an isothermal change of state, botheqn 275 and (1.277) are applicable, hence we may write55 

Equation 1.279

Integration from P = 0 to the pressure of interest P, in conjunction with eqn 1.278, yields

Equation 1.280

This equation connects the fugacity concept with the residual volume and prescribes the way to compute numerical values of the fugacity from experimental PVT data for the pure fluid or from appropriate model EOSs.

General integration of eqn (1.277) at constant temperature from the state of pure component i in the perfect-gas state to the state of i in the real-fluid state (phase π) at the same pressure, in conjunction with eqn (1.278), yields

Equation 1.281

where GiR,π,*         (T, P) denotes the molar isobaric residual Gibbs energy of pure component i in phase π. Eqn (1.281) is in accord with eqn 245 and (1.280). The dimensionless ratio appearing on the left-hand side of eqn (1.280) and the right-hand side of eqn (1.281) as the argument of the logarithm is a new property and is called the fugacity coefficient, ϕiπ,*     (T, P), of pure component i in phase π. By definition

Equation 1.282

or, perhaps more convenient,

Equation 1.283

Since GiR,π,*         (T, P) is a property (a state function) of pure i, so the fugacity coefficient ϕiπ,*     (T, P)/the fugacity f  iπ,*    (T, P) is a property (a state function) of pure i in phase π.

For a pure perfect gas, ViR,pg,*           = 0 or, equivalently, GiR,pg,*           = 0, for all temperatures and pressures and therefore

Equation 1.284

An entirely analogous definition can be introduced for the fugacity of a component i in solution. As the first part of the definition, we now have for the partial molar Gibbs energy (chemical potential)

Equation 1.285

where fiπ  = fiπ  (T, P, {xiπ }) is called the component fugacity of substance i in a real solution phase π with composition {xiπ }. In the literature, a frequently used special notation identifies fugacities and fugacity coefficients of components in solution by a circumflex (^) to distinguish them from pure-fluid properties and from partial molar properties. Since the partial molar Gibbs energy Giπ(T, P, {xiπ }) = µiπ (T, P, {xiπ }) can be regarded as a property (a state function) of dissolved i, so the component fugacity fiπ  is a property of i in solution in phase π, with dimension of pressure. To preserve consistency between eqn (1.285) and the known expression for a perfect-gas mixture, we demand that

Equation 1.286

that is

Equation 1.287

Eqn (1.286) is the second part of the definition and together with eqn (1.285) they constitute the complete definition of the fugacity f  iπ(T, P, {xiπ }) of component i in solution in phase π. General integration of eqn (1.285) at constant T, P and {xiπ } from the perfect-gas state to the real state of interest, i.e. component i in solution, results in

Equation 1.288

The difference on the left-hand side of eqn (1.288) is the partial molar residual Gibbs energy GiR,π    (residual chemical potential µiR,π      ) in (T, P, x) space of component i in solution in phase π, as defined by eqn (1.274) The dimensionless ratio appearing on the right-hand side of eqn (1.288) as the argument of the logarithm is a new property and is called the fugacity coefficient, ϕiπ (T, P, {xiπ }), of component i in solution in phase π:

Equation 1.289

or, perhaps more convenient,

Equation 1.290

For a component i in a perfect-gas mixture, GiR,pg       = µiR,pg        = 0 and therefore ϕipg    = 1.

Eqn (1.286) guarantees recovery of the expression for a perfect-gas mixture.55  Integration of eqn (1.285) at constant temperature and pressure from pure component i to the real state of i in solution in phase π yields

Equation 1.291

Since eqn (1.291) is generally valid, it must also hold for a perfect-gas mixture. In conjunction with eqn (1.284) and eqn (1.286), we have

Equation 1.292

and with the summability relation eqn (1.118)

Equation 1.293

we obtain

Equation 1.294

in accord with eqn (1.272). Here, Δ signifies the change on mixing where the extent of change is measured with respect to the mole fraction-weighted sum of the molar pure-component properties. That is, as the defining equation for a new class of thermodynamic functions known as the molar property changes of mixing, at constant T and P, we introduce

Equation 1.295

where M may represent, for instance, G, F, S, H, CP and V, and expressions for some molar property changes pertaining to perfect-gas mixtures, in addition to eqn (1.294), are as follows:

Equation 1.296
Equation 1.297
Equation 1.298
Equation 1.299
Equation 1.300

The entropy change of eqn (1.297), i.e., is called the ideal molar entropy of mixing; it is always positive.

Finally, for the overall fugacity of the solution in phase π we have

Equation 1.301

where GR,π(T, P, {xiπ }) is the molar isobaric residual Gibbs energy of the solution, and

Equation 1.302

is the overall fugacity coefficient of the solution in phase π or, perhaps more convenient,

Equation 1.303

Evidently, fugacity, having the dimension of pressure, and the dimensionless fugacity coefficient are intensive state functions related to exponentials of (partial) molar isobaric residual Gibbs energies made dimensionless by dividing by RT, as indicated in eqn (1.282), (1.289) and (1.302). Viewing the fugacity as a “corrected pressure” may be misleading and obscure these exact relations. As the pressure goes to zero (at constant temperature and composition), the (partial) molar residual Gibbs energies divided by RT, as introduced above, all approach zero on the basis that the perfect-gas state is approached asymptotically; hence f  iπx iπPPi, f  πi,*     → P and f  πP, respectively. Therefore, in summary,

Equation 1.304
Equation 1.305

and

Equation 1.306

The fugacity coefficients are always positive: for real systems; they may be larger than 1 or smaller than 1 and they are unity for a perfect-gas system.

Since GiR,π      (T, P, {xiπ }) is the partial molar isobaric residual Gibbs energy of i in phase π, i.e.

Equation 1.307

the associated summability relation yields the molar isobaric residual Gibbs energy of the solution:

Equation 1.308

For convenience, the non-dimensional quantity GR,π/RT is frequently used instead of GR,π, cf.eqn (1.160a) and (1.160b), hence analogously

Equation 1.309

We recognise that ln ϕiπ is a partial molar property in relation to ln ϕπ, where ϕπ is the overall fugacity coefficient of the solution in phase π:

Equation 1.310

It is for this reason that in the literature a circumflex is frequently applied to distinguish the properties ϕiπ and f  iπ from pure-fluid properties and to show clearly that they are not partial molar properties (frequently identified by an overbar). These remarks indicate, perhaps, the advantages of the nomenclature used in this chapter [cf. the four-point summary in the paragraph after eqn (1.125) in Section 1.3.1], in particular the designation of a pure-fluid property by an asterisk, which makes the use of an overbar unnecessary.

In analogy with eqn (1.160a), the fundamental residual-property relation (with canonical variables T, P, niπ ) valid for a fluid in any phase π, may now be written as

Equation 1.311

or, alternatively,

Equation 1.312

This equation is valid for any PVTx equilibrium phase π, either open or closed, and describes changes in the extensive property nπ ln ϕπ by changes in temperature, pressure and amounts (of substance) caused by mass transfer across the phase boundary or by chemical reactions or both. The associated Gibbs–Duhem equation is immediately obtained from the most general form eqn (1.124) for M = ln ϕπ:

Equation 1.313

In the equations given above,

Equation 1.314

where the molar isobaric residual enthalpy of the solution in phase π is given by

Equation 1.315

and

Equation 1.316

where the molar isobaric residual volume of the solution in phase π is given by

Equation 1.317

Evidently, the partial molar property analogues of eqn 314 and (1.316) are

Equation 1.318

and

Equation 1.319

respectively, where HiR,π     is the partial molar isobaric residual enthalpy and ViR,π     is the partial molar isobaric residual volume of component i in solution in phase π. The corresponding summability relations read

Equation 1.320

and

Equation 1.321

Evaluation of ln ϕiπ (T, P, {xiπ }), using an EOS in conjunction with eqn (1.319), is now straightforward. Since the molar volume of pure component i in the perfect-gas state is given by Vipg,*        = RT/P and its partial molar volume in phase π by , integration at constant T and constant composition from P = 0 (where ϕiπ = 1) to arbitrary pressure P yields

Equation 1.322

Eqn (1.322) constitutes a generally valid equation for the calculation of ln ϕiπ(T, P, {xiπ}) from any volume-explicit EOS. By way of example, consider a binary vapour-phase mixture (π = V, x1V = 1 − x2V ) adequately described by a two-term virial equation in pressure, i.e.. After some mathematical manipulation, we obtain the compact expression for the fugacity coefficients of components 1 and 2, respectively, in a binary vapour mixture:

Equation 1.323

where B(T, x2V) = x1VB11(T) + x2VB22(T) + x1Vx2Vδ12(T) is the second virial coefficient of the mixture, δ12 ≡ 2B12 − (B11 + B22), B11 and B22 are the second virial coefficients of the pure components and B12 designates a composition-independent interaction virial coefficient (cross-coefficient). When focusing on highly dilute systems, the fugacity coefficient of, say, component 2 at infinite dilution in the vapour phase is thus given by

Equation 1.324

For pure substance i in phase π, we have

Equation 1.325

and

Equation 1.326

To conclude this section, I emphasise that GR,π(T, P, {xiπ })/RT is a convenient generating function for molar isobaric residual properties [see also eqn (1.160a)].

In principle, the exact classical thermodynamic method of using isobaric residual functions for the calculation of property changes of single-phase constant-composition PVT fluids for any arbitrary equilibrium change of state is a general and powerful tool; and applications profit greatly from contributions based on the corresponding states theorem, such as the popular Lee–Kesler tables.174  For single-phase constant-composition fluids, in conjunction with perfect-gas heat capacities, these generalised correlations, say for HR and SR, allow the calculation of property changes associated with any change of state from (T1, P1) → (T2, P2).

From PVT EOSs, volumetric properties and also residual functions characterising deviations from perfect-gas (ideal-gas) behaviour can be calculated (see Section 1.3.2). The PVT relation may be a volume-explicit EOS, such as CST-type equations, or a pressure-explicit EOS. The simplest, practically useful pressure-explicit polynomial EOS is cubic in molar volume, since it is capable of yielding the perfect-gas limit for V → ∞ and of representing both liquid-like and vapour-like volumes for sufficiently low temperatures. First, the focus will be on pure fluids and for simplicity's sake the asterisk will be omitted. The five-parameter equation55 

Equation 1.327

where the adjustable parameters b, θ, η, δ, ε depend, in general, on temperature (and for mixtures also on composition), can be considered a generalisation238  of the original vdW EOS, eqn (1.56), to which it reduces for η = b, δ = ε = 0 and θ = constant = a. Note that the derived universal critical compression factor

Equation 1.328

is considerably larger than common experimental values. Over the years, many specialisations of eqn (1.327) have been suggested,245  and two of the most popular generalised vdW equations are the Redlich–Kwong (RK) equation (1949):246 

Equation 1.329

where b = bRK, θ = θRK(T) ≡ aRK/T1/2, η = bRK, δ = bRK and ε = 0, and the Peng–Robinson (PR) equation (1976):247 

Equation 1.330

where b = bPR, θ = θPR(T) ≡ aPR(T), η = bPR, δ = 2bPR and ε = −bPR2. Significantly, all modern cubic EOSs have a temperature-dependent parameter θ.

Although the RK equation yields a somewhat better critical compression factor than the original vdW EOS, i.e.

Equation 1.331

and better second virial coefficients,248  it is still not very accurate in predicting vapour pressures and liquid densities. It was Soave (S) in 1972 249,250  who generalised the temperature dependence of the attractive parameter in the RK equation by writing, with θ = θS(T) ≡ aS(T),

Equation 1.332

Making use of the mathematical requirements for the occurrence of an inflection point on the critical isotherm, that is, and at the critical point (P = Pc, V = Vc), experience shows that the preferred expressions for evaluating aS(Tc) ≡ aS,c and bS are in terms of Tc and Pc, that is,

Equation 1.333

The critical compression factor is the same as for the Redlich–Kwong EOS, i.e. Zc,S = 1/3.

At temperatures TTc, the attraction parameter aS(T) may now be expressed as the product of its value at the critical point and a dimensionless temperature-dependent α-function:

Equation 1.334

hence the Soave EOS, eqn (1.332), may now be written as

Equation 1.335

Clearly, for the limiting value of the α-function as TTc (i.e. Tr → 1) we require

Equation 1.336

To help better fit vapour pressure data of hydrocarbons, Soave suggested as a generally useful empirical function:

Equation 1.337

whose limiting value for TTc is one, in accord with eqn (1.336). With a substantially expanded data set of vapour pressures becoming available (a then new compilation from the American Petroleum Institute), the αS(T, ω) relation was improved by Graboski and Daubert251–253  with

Equation 1.338

For a new expression that allows a more reliable extrapolation of heavy hydrocarbon vapour pressures below the normal boiling point, see ref. 254.

When applying the classical critical constraints to the Peng–Robinson EOS, eqn (1.330), we obtain

Equation 1.339

and the PR equation now reads

Equation 1.340

At temperatures TTc, the attraction parameter θPR(T) = aPR(T) is expressed as the product of its value at the critical point and a dimensionless temperature-dependent α-function:

Equation 1.341

again with the constraint

Equation 1.342

Optimising the correlation of experimental vapour-pressure data with the Peng–Robinson EOS yields

Equation 1.343

Note that the critical compression factor is

Equation 1.344

which value is nearer the common experimental values (particularly for non-polar compounds). This partially explains the fact that the Peng–Robinson EOS + eqn (1.343) predicts pure liquid amount densities more accurately than the Soave EOS + eqn (1.337), as evidenced by a comparison due to Privat et al.255  of Peng–Robinson and Soave EOSs using an extensive data base (about 1400 compounds) of the Design Institute for Physical Properties (DIPPR).256  This study indicated similar performances of the two EOSs in predicting vapour pressures, enthalpies of vaporisation and saturated (orthobaric) liquid heat capacities and showed that second virial coefficients cannot be satisfactorily reproduced.

Since it has been recognised that adequately formulated α-functions substantially improve predictions of vapour pressures, enthalpies of vaporisation, etc., over the years a large number of α-functions of increasing complexity have been suggested. On the occasion of the 40th anniversary of the publication of the Peng–Robinson EOS,247  Lopez-Echeverry et al.257  presented a comprehensive review of work done on this cubic EOS, including application to pure fluids as well as mixtures; they listed more than 70 α-functions published since 1976!

For quite some time, reasonable temperature dependences of α-functions were established for subcritical conditions only, and theory-based guidelines for developing adequate functions in general, and for supercritical conditions in particular, have been lacking until recently. Based on a careful thermodynamic/mathematical analysis of the problem, Le Guennec et al.258,259  established the requirements for formulating thermodynamically consistent α-functions for cubic EOSs, applicable under both subcritical and supercritical conditions. Here, consistent means that a proper α-function, i.e.

Equation 1.345

must satisfy all the following constraints [cf.eqn 334 and (1.336) and eqn 341 and (1.342)]:

Equation 1.346

That is, the dimensionless temperature-dependent α-function becomes unity at the critical temperature and the constraints summarised in eqn (1.346) guarantee that it decreases monotonously with increasing temperature and that it is convex with respect to the temperature coordinate. The negativity constraint on the third derivative helps avoid non-physical changes of CP with temperature, and allows for a good prediction of subcritical properties used in data regression and correct extrapolation into the supercritical region. The results presented in ref. 258 and 259 support and justify a posteriori the extensive use of Soave-type α-functions by Mahmoodi and Sedigh.260–262 

Inaccurate liquid density predictions constitute a significant deficiency of two-parameter cubic EOSs. To improve significantly the description of volumetric fluid properties, in 1982 Péneloux et al. introduced the successful and widely applied empirical concept of volume translation.263  Geometrically, the Péneloux-type volume correction corresponds to a shift of the entire volume–pressure isotherm along the volume axis by an increment −ct, i.e. towards smaller volumes. We recognise that such a volume translation implies that also the covolume b is subjected to the same shift. Hence the volume-translated EOS is obtained by simply replacing V and b in the original (non-translated) vdW-type equation according to

Equation 1.347

For instance, the original vdW EOS becomes the following volume-translated equation:

Equation 1.348

We note that the mathematical structure of the repulsive term has remained unaltered, but the attractive term has changed to the Clausius form.264  The translated Soave EOS reads (for the sake of simplicity subscripts t have been removed)

Equation 1.349

with

Equation 1.350

For the volume-translation parameter, Péneloux et al. recommended

Equation 1.351

where ZRA is the Rackett compression factor.265  If a value of ZRA is not available, it may be estimated using266 

Equation 1.352

We recognise that Zc in eqn (1.350) is no longer a universal constant (ZS,c = 1/3), but is now a substance-specific property closer to the experimental compression factor. The volume translation also modifies the covolume, but leaves the cohesive parameter aS,c unaltered.

A careful investigation of the impact of Péneloux-type volume translations (temperature independent versus temperature dependent) on physical properties evaluated from cubic EOSs was presented by Jaubert et al.267  This is important for deciding which kind of experimental data should be used in optimal data fitting for volume translation.

Translated [cf.eqn (1.347)] and consistent [cf.eqn 345 and (1.346)] versions of the Redlich–Kwong (tc-RK) and the Peng–Robinson (tc-PR) cubic EOSs were recently developed by the Nancy group.268  The adjective translated specifies here that a temperature-independent volume translation was carried out to reproduce exactly the experimental saturated molar liquid volume at Tr = 0.8. As α-function, an equation suggested by Twu et al.269  was generalised, yielding, for instance, for the tc-PR EOS and for 0 ≤ ω ≤ 1.5

Equation 1.353

In the subcritical temperature range it performs slightly better than the Soave α-function and can safely be extrapolated into the supercritical temperature range.

While improving volumetric predictions, application of a volume translation leaves the quality of calculated vapour pressures unaffected. Focusing on a generalised cubic EOS:

Equation 1.354

where the parameters u and w are either universal or substance-specific, Piña-Martinez et al.270  analysed systematically the influence of these two parameters on the accuracy of predicted volumetric and vapour–liquid equilibrium properties for translated and untranslated cubic EOSs. Selecting as optimised values u = 2.16 and w = −0.86, the most accurate untranslated and translated cubic EOSs were simultaneously obtained. Using a temperature-independent volume translation to reproduce exactly the saturated liquid molar volume at Tr = 0.8 in conjunction with the flexible α-function of Twu et al.269  yielded the most accurate cubic EOS (tc-OptiM) ever published. However, the results were only slightly better than those obtained with the tc-PR EOS,268  simply because the PR values for u and w (2 and −1, respectively) are very close to the optimised values.

The basic physical idea of the vdW model is the additive separation of the pressure into two contributions caused by repulsive and attractive interactions, respectively. The structure of simple dense liquids is chiefly determined by repulsive forces, that is, by the way in which molecular hard cores are packed, and the cohesive energy of a simple liquid has only a small effect on its structure.271  Note that for the vdW model fluid, eqn (1.56), the attractive interactional contribution to the compression factor, i.e.avdW/RTV, gives rise to a molar residual internal energy

Equation 1.355

as obtained with eqn (1.256). I reiterate that since Upg,* does not depend on V or P, we have UvdWr,*(T, V) = UvdWR,*(T, P).

With increasing density, the repulsive part of the vdW compression factor

Equation 1.356

compares poorly272  with the exact expression for a fluid of hard spheres (with diameter σhs):273 

Equation 1.357

and, even worse, the maximum attainable packing density corresponds to ξ = 0.25, where the denominator of ZvdW (repulsive), eqn (1.356), has a pole. Here, the compactness274,275  (packing density) reflects the finite volume of the molecules envisaged as hard spheres, each with volume vmolecule = σhs3π/6, hence . Note that ξ for common liquids274,275  at ambient temperature and pressure is about 0.5 (for the densest packings of hard spheres, i.e. the face-centred cubic lattice and the hexagonal closest packing, we have ). In contradistinction, the virial expansion of the Carnahan–Starling EOS for hard spheres:276 

Equation 1.358

reads

Equation 1.359

in excellent agreement277  with the exact result, eqn (1.357).

Most of the EOSs developed since van der Waals have adopted his approach and in a centennial article in 1973 devoted to his legacy, Rowlinson278  illustrated the extent of vdW-concepts still contained in current theories of fluids.

By proposing a classical (although much more elaborate) alternative to the conventional method of evaluating the pure substance parameter b and contributions in addition to the vdW term RT/(Vb) in the EOS, Martin and Hou279  developed a considerably more useful EOS in the van der Waals spirit. Their approach included (besides other properties/conditions) all three critical pure-substance data Tc, Pc, Vc; they also considered the Boyle temperature TB, where

Equation 1.360

and

Equation 1.361

which helps in establishing the empirical relation

Equation 1.362

and they utilised the value of one experimental vapour pressure point (T, Pσ), thereby obtaining empirically at the critical point. Application of a total of 10 conditions (of which I have indicated four) yielded a rather complex vdW-type EOS that gives precise results up to about 1.5ρc:

Equation 1.363

The eight constants are evaluated from the substance-specific input parameters. With a few improvements added, Martin280  subsequently improved the performance of the Martin–Hou EOS to provide high-precision results up to about 2.3ρc.

Equations of state play an important role in many branches of chemical engineering since they can be used without any conceptual difficulties for calculating vapour–liquid equilibria, liquid–liquid equilibria and supercritical fluid-phase equilibria. Because of their importance, EOS research is an attractive and highly active field and numerous excellent reviews have covered various aspects, although frequently focusing on cubic, vdW-type equations; the literature references presented here, i.e.ref. 92, 178, 237–241, 245, 248, 250 and 281–290, are necessarily just an idiosyncratic selection. Therefrom we note, however, that despite the inherent empiricism and new developments, both the Soave and the Peng–Robinson EOSs (and closely related equations) have remained mainstays for calculating the thermodynamic properties of fluids and vapour–liquid equilibria; nearly 150 years after van der Waals' dissertation,291 generalised vdW EOSs are still a hot topic in chemical engineering.

An entirely different approach was adopted by Dieterici in 1899.292  In contradistinction to the additive vdW Ansatz, exemplified by eqn (1.56), he introduced a multiplicative relation between repulsive and attractive terms:

Equation 1.364

Similarly to the vdW quantities, the Dieterici parameter aD is an attractive interaction energy parameter and the repulsive Dieterici parameter bD is related to molecular size. Using standard calculation procedures, the universal critical constants can be derived, yielding for the critical compression factor

Equation 1.365

which value is remarkably close to the experimental result Zc ≃ 0.29 for the heavier rare gases. Despite this advantage over the vdW EOS, the Dieterici EOS has not contributed significantly to the field and relevant research activities have slowly disappeared,293,294  although some years ago there was a revival of interest.295–299  In particular, modifications of the vdW and the Dieterici EOSs obtained by replacing the classical vdW repulsive term with the Carnahan–Starling hard-sphere term, eqn (1.358), were investigated. A careful analysis by Polishuk et al.298  using the global phase diagram methodology revealed several shortcomings of the Dieterici EOS, perhaps the most important being that it cannot describe liquid–liquid immiscibility in real fluid mixtures at low temperatures.

The greatest utility of cubic EOSs is for vapour–liquid phase equilibrium calculations involving mixtures. Extension of vdW-type model equations to multicomponent mixtures usually rests upon the inherent assumption that the same EOS used for the pure fluid components can be used for the c-component mixture, provided that adequate prescriptions for the unlike (cross) mixture parameters are available. The vdW one-fluid approximation provides such recipes for the composition dependence of the mixture interaction energy parameter a, say aS or aPR, and of the mixture molecular size parameter b, say bS or bPR. These mixing rules are quadratic in mole fraction:

Equation 1.366

where xi and xj are the mole fractions of components i and j, respectively. The attractive vdW interaction parameters of the pure components are denoted by aii and ajj, while aij measures the strength of the attractive interaction between unlike molecules. Similarly, the composition dependence of the mixture parameter b (the mixture covolume) may also be approximated by a quadratic mixing rule:

Equation 1.367

where bii and bjj denote the vdW size parameters (covolumes) of the pure components and bij characterises the repulsive interaction between unlike molecules.

For three-parameter cubic EOSs, such as the Patel–Teja (PT) equation:300 

Equation 1.368

where the parameters in the general vdW equation, eqn (1.327), are now given by b = bPT, θ = θPT(T) ≡ aPT(T), η = bPT, δ = bPT + cPT and ε = −bPTcPT, a similar mixing rule for the third parameter, here for cPT, is usually assumed:

Equation 1.369

I emphasise that these commonly used mixing rules are semiempirical approximations and alternative recipes could be used and indeed have been suggested. However, to actually apply eqn (1.366), (1.367) and (1.369), the cross-interaction parameters aij, bij and cij for unlike interactions (ij) have to be known; evaluation of these quantities in terms of pure-substance parameters is one of the key problems in molecular thermodynamics.6,8,80,184,237,287,301  The most common choices for combining rules are the geometric-mean rule for aij suggested by Galitzine in 1890,302  and later by Berthelot in 1898,303,304  and the arithmetic-mean rule for bij, although in engineering calculations both are routinely modified empirically:

Equation 1.370
Equation 1.371

The parameters kij and lij are known as binary interaction parameters. For three-parameter cubic EOSs, cij is typically approximated by

Equation 1.372

mij being another empirical binary interaction parameter.

The vdW-type cubic EOSs account for vapour–liquid equilibria with a critical point, although like all classical analytical equations, in the critical region they become qualitatively incorrect. For instance, cubic EOSs for pure liquids predict for the curve of coexisting vapour densities and liquid densities a square-root dependence on the temperature difference from the critical temperature, i.e. the classical (cl) critical exponent is βcl = 1/2, whereas the non-classical (ncl) way in which the orthobaric (subscript σ) amount densities ρσL,*   and ρσV,*   approach the critical amount density ρc is much flatter;305  it is described by the power law (simple scaling)

Equation 1.373

where B is the corresponding dimensionless, fluid-specific critical amplitude and βncl is the non-classical critical exponent.

Cubic EOSs are unable to reproduce the shape of the critical isotherm, that is, they predict a critical dependence on the third power of the density difference from the critical density, i.e. δcl = 3, whereas for real pure fluids we have non-classically

Equation 1.374

with D being the corresponding dimensionless, fluid-specific critical amplitude.

The non-classical divergence of the isothermal compressibility is expressed by the power law

Equation 1.375

while γcl = 1, and Γ+ is the corresponding dimensionless, fluid-specific critical amplitude.

In addition, classical EOSs are unable to yield a weakly divergent isochoric heat capacity:

Equation 1.376

with A+ being the corresponding dimensionless, fluid-specific critical amplitude; classically, CV* is non-divergent, that is, αcl = 0.

Eqn (1.373)–(1.376) represent power laws with universal critical exponents:117–119,126  a positive exponent relates to properties that remain finite at the critical point, whereas a negative sign preceding the exponent relates to properties that diverge. Using critical parameters, the thermodynamic properties have been made dimensionless, hence the critical amplitudes become numbers. The critical exponents have the same values for all three-dimensional Ising-type systems and are interrelated:306 

Equation 1.377

Theory implies also universal relations between the non-universal amplitudes of all asymptotic thermodynamic power laws, that is,

Equation 1.378

Hence, based on eqn 377 and (1.378), only two of the four universal exponents and only two of the four non-universal amplitudes are independent: all other exponents and amplitudes on any path for all fluids and fluid mixtures can be expressed in terms of just the respective two selected. This principle is known as two-scale-factor universality.118,306  A theoretical approach to correct classical cubic EOSs for the effects of critical density fluctuations was suggested by Wyczalkowska, Sengers and Anisimov.307 

So far, every cubic EOS that has been proposed has some limitations (besides classical critical behaviour), with respect to either the range of operating conditions or types of fluids to which it could be applied. Further progress in developing pressure-explicit multiparameter EOSs240  was initially greatly stimulated by work on the virial equation in the amount (of substance) density ρ ≡ 1/V, which for a pure gas/vapour at not too high densities reads

Equation 1.379

This equation has a sound theoretical foundation.182,286  Here, B(T) is the second virial coefficient, C(T) is the third virial coefficient and so forth. The celebrated EOS of Benedict, Webb and Rubin (BWR)308  of 1940:

Equation 1.380

was an important step in the right direction and represented the volumetric properties of industrially important fluids (hydrocarbons) reasonably well. However, the BWR EOS was found to give unsatisfactory results in low-temperature applications, at high fluid densities and in the critical region. To alleviate these deficiencies, many modifications of the BWR equation, with many more parameters, have been proposed174,309–312  and are still widely used.

As already pointed out, fundamental equations have a great inherent advantage over volumetric PVTx EOSs (thermal EOSs): they contain complete information on the thermodynamic system. Hence, once a judiciously selected empirical relation has been developed for one of the fundamental equations in terms of the appropriate canonical variables, say for nU = Ut(nS, nV, {ni}), nH = Ht(nS, P, {ni}), nF = Ft(T, nV, {ni}) or nG = Gt(T, P, {ni}), in principle all other thermodynamic fluid properties can be calculated by combinations of suitably selected derivatives. Since the independent extensive variable entropy nS is not directly measurable, neither the fundamental equation formulated in terms of internal energy nor that formulated in terms of enthalpy is used for developing empirical correlational equations. The advantage of using nG = Gt(T, P, {ni}) is that the intensive canonical variables T and P are easily measured, monitored and controlled; this makes the Gibbs energy so important in both physical chemistry and chemical engineering. However, owing to the discontinuity in slope of the Gibbs energy surface at the liquid (L)/vapour (V) phase boundary, i.e.

Equation 1.381

the fundamental equation nG = Gt(T, P, {ni}) can only be used to represent the liquid part of the Gibbs energy surface or the vapour part separately, thus precluding a closed mathematical description of the entire fluid range (P = Pσ denotes the vapour pressure). In contradistinction, formulations based on Helmholtz-energy-based fundamental equations, using canonical variables T and V or T and ρ, are suitable for describing the entire fluid region. They are valid for liquid and vapour states and for equilibria between them and the description of supercritical states is included. Their validity range terminates at the melting curve, thus allowing the calculation of liquid-phase properties, but not of properties of the coexisting solid phase. Modern fundamental equations are usually based on the Helmholtz energy.240,313–315  However, in practical applications the dimensionless property F(T,V)/RT, see eqn (1.161), is usually replaced with F(T, ρ)/RT, which is split into a residual part and a perfect-gas (ideal-gas) part, and both are empirically expressed by dimensionless functions of the inverse reduced temperature τTc/T = 1/Tr and the reduced amount density, i.e. the inverse reduced molar volume, δρ/ρc = Vc/V = 1/Vr:

Equation 1.382

The commonly used functional form of the residual term is that found in modified BWR equations; for details, see ref. 240 and 313–315. Note that the dimensionless property F(T,V)/RT is related to the molar Massieu function divided by R [cf.eqn (1.107)]:

Equation 1.383

In recent decades, generalised models for calculating thermodynamic mixture properties using Helmholtz energy-based fundamental equations have been developed.316–319  For instance, the GERG-2008 equation of Kunz and Wagner320  is based on data of 21 natural gas components. Over the entire composition range, it covers the liquid phase, the gas phase, the supercritical region and VLE within the range 90–450 K and up to 35 MPa. However, when applied to significantly asymmetric binary mixtures, say, methane + pentane, it predicts critical curves321  with physically unreasonable temperature maxima;322  hence further work is indicated.

In the high-accuracy mixture models currently available, that is, in the multi-fluid Helmholtz energy-explicit (Massieu function-explicit) formulations, binary interaction parameters used in the combination rules must conventionally be obtained via various semiempirical estimation methods. Fairly recently, Bell and Lemmon323  developed a novel, entirely automatic evolutionary optimisation algorithm to optimally fit the most important interaction parameters for more than 1100 binary mixtures. The source code for the fitter and the entire set of optimised binary interaction parameters were provided as supplemental material.

As shown in Section 1.3.2, the residual Gibbs energy GR/RT and the fugacity coefficient ln ϕ are directly related to experimentally accessible PVTx data and may therefore be discussed, in principle, in terms of judiciously selected EOSs. However, for dense liquid mixtures the focus of the physical chemist and the chemical engineer is frequently on the composition and temperature dependences of thermodynamic mixture/solution properties rather than pressure or density dependences that are usually rather small. Experimental determination, correlation and prediction of mixture properties are topics of central importance in chemical thermodynamics and data on binary liquid mixtures are of particular interest. Primarily, they are useful for testing and guiding theories that attempt to predict thermodynamic mixture properties from the properties of the constituent pure components, and the experimental results provide information on parameters that characterise interactions between unlike species. In turn, these data constitute the very foundation for the development of predictive methods for the properties of liquid multicomponent mixtures which, on the application side, are indispensable for the calculation of phase equilibria. At present, no generally satisfactory theory exists that provides a reliable basis for the prediction/correlation of thermodynamic data for binary liquid mixture and hence, a fortiori, for ternary liquid mixtures, quaternary liquid mixtures and so forth, i.e. for multicomponent liquid mixtures. In fact, liquid mixtures/solutions are frequently much more profitably dealt with through determining/discussing properties that measure the departure from a conveniently selected behaviour reasonably close to reality. Fortunately, large numbers of (critically) evaluated experimental data on various mixture properties are available in systematic data collections, listed here as ref. 9–31 and 256.

In the simplest approach to describe real mixture properties, instead of considering total properties Mt = nM(T, P, {xi}), it is always helpful to discuss the corresponding molar mixture properties in relation to the properties of the c pure liquid constituents at the same T, P and {xi}, i.e. to focus on difference measures with respect to a mole fraction-weighted sum of pure-component properties. Discussion is therefore based on a new class of thermodynamic functions known as molar property changes of mixing, commonly designated by the symbol Δ, which was already introduced in Section 1.3.2, and, for the sake of convenience, is presented again here. On a molar basis, the definition reads

Equation 1.295

and M can represent, for instance, G, F, S, H, V, CP, ln ϕ, etc. The corresponding new class of partial molar property changes of mixing is defined by

Equation 1.384

With the corresponding summability relations at constant T and P, cf.eqn (1.117) or (1.118), we have

Equation 1.385

or

Equation 1.386

and the exact differential of the extensive property nΔM is

Equation 1.387

From eqn (1.385), a differential change in nΔM is given by

Equation 1.388

hence via comparison with eqn (1.387) and after division by n,

Equation 1.389

is obtained; this is another form of the general Gibbs–Duhem equation. In this section, by way of example, the focus will be on M = G, F, S, V, H and because of direct measurability, Δ H(via calorimetry) and ΔV (via dilatometry) are the molar property changes of mixing of special interest; indeed, several chapters in the second42  and third43  volumes of this series of books have been devoted to these topics.

The application of residual-function approaches or EOS methods to liquid mixtures, frequently consisting of chemically quite complex components, is often unsatisfactory. Hence for many applications, one way to proceed is to select in place of the perfect gas (ideal gas) a reference basis that is more appropriate for condensed phases; the most commonly selected basis is the ideal solution (indicated by a superscript id). In fact, this approach constitutes the classic method in solution thermodynamics for presenting liquid mixture properties. Compared with the perfect-gas basis, we recognise that usually changes in composition affect liquid mixture properties much more strongly than changes in pressure. Although no real mixture is truly ideal, the ideal solution model is simple, yet realistic enough to approximate grosso modo the real-fluid behaviour of non-electrolyte solutions. Deviations of real-mixture values from ideal-mixture values (so-called “excesses”) provide the best basis for a molecule-based discussion using physically intuitive concepts as suggested by Wilhelm324–328  (see Figure 1.1). Consider a liquid equilibrium phase with composition {xi} ≡ {xiL } at uniform temperature and pressure. Using the idealised composition dependence of the component fugacity as represented by the Lewis–Randall (LR) rule,55,61,68,242,244  we have by definition

Equation 1.390

which implies, with eqn 282 and (1.289),

Equation 1.391

the fugacity coefficient of component i in a Lewis–Randall ideal solution is equal to the fugacity coefficient of pure component i in the same physical state as the solution and at the same T and P. Also, for the partial molar Gibbs energy/chemical potential we have

Equation 1.392

Any discussion of real-solution behaviour can now be based on deviations from LR ideal-solution behaviour, i.e. on the differences between property values of real solutions and property values calculated for the LR ideal-solution model at the same T, P and {xi}, based on eqn (1.392). That is, the partial molar Gibbs energy Giid   of component i serves as a generating function for other partial molar properties of an LR ideal solution. Alternative ideal-solution models are possible and are indeed used, although the LR ideal solution is the conventional reference model for characterising mixture behaviour of liquid systems. Note, however, that its use is not restricted to the liquid phase – it could also be used for gaseous mixtures. The most important alternative ideal-solution model is that based on Henry's law (HL); this topic is discussed in detail by Wilhelm and Battino in Chapter 2.

Figure 1.1

Heuristic summary of the most important physical phenomena to be considered in discussing the thermodynamic properties of pure liquids and/or liquid mixtures/solutions at the molecular level and at the bulk level (after Wilhelm324–328 ). Adapted from ref. 328 with permission from the Royal Society of Chemistry.

Figure 1.1

Heuristic summary of the most important physical phenomena to be considered in discussing the thermodynamic properties of pure liquids and/or liquid mixtures/solutions at the molecular level and at the bulk level (after Wilhelm324–328 ). Adapted from ref. 328 with permission from the Royal Society of Chemistry.

Close modal

As stated above, all thermodynamic properties of the model ideal solution follow from eqn (1.392)via temperature and pressure derivatives. For instance, differentiating with respect to T and P yields the partial molar entropy and the partial molar volume, respectively:

Equation 1.393
Equation 1.394

while the Gibbs–Helmholtz equation yields the LR ideal partial molar enthalpy:

Equation 1.395

The partial molar results for first-law properties do not depend on composition: they are the pure component values. In contradistinction, the partial molar results for second-law properties do depend on composition via the entropy of mixing terms; we also have

Equation 1.396

Note that each partial molar second-law property diverges at infinite dilution, that is, for xi → 0.

The LR ideal molar properties corresponding to the partial molar properties of eqn (1.392)–(1.396) are obtained with the summability relation:

Equation 1.397
Equation 1.398
Equation 1.399
Equation 1.400
Equation 1.401

The molar property changes of mixing for LR ideal solutions, Δ Mid, may be obtained as special cases from the general defining equation, eqn (1.295):

Equation 1.402a
Equation 1.402b

By substituting the corresponding expression for Mid, eqn (1.397)–(1.401), into eqn (1.402a) or by inserting the corresponding expressions for Miid   , eqn (1.392)–(1.396), into eqn (1.402b), we obtain

Equation 1.403
Equation 1.404
Equation 1.405
Equation 1.406
Equation 1.407

Evidently, when an LR ideal solution is formed from the pure components at constant temperature and pressure, the changes of first-law properties on mixing are all zero. However, changes of second-law properties on mixing are not zero – they amount to an entropy-of-mixing term, that is, ΔSid is always positive, whereas ΔGid and ΔFid are always negative. The general property ΔMiid   (T, P, {xi}) of eqn (1.402b) denotes a partial molar property change of mixing for LR ideal solutions, such as those appearing in eqn (1.403–1.407):

Equation 1.408

Having introduced the concept of an ideal LR solution, we now consider properties that measure deviations of real solution properties M(T, P, {xi}) from LR ideal solution properties Mid(T, P, {xi}) at the same T, P and {xi}, such as those indicated by eqn (1.397)–(1.401). These difference measures constitute another highly useful new class of functions called molar excess properties; they are designated by a superscript E and are defined by

Equation 1.409

Clearly, the prescription is analogous to that used for obtaining molar residual properties, although with id properties replacing pg properties.

The corresponding partial molar excess properties for component i in solution are defined by

Equation 1.410

and with the summability relation we have

Equation 1.411

The definition of an excess property is not restricted to any phase, though excess properties are predominantly used for liquid mixtures.

Substituting the LR ideal-solution expressions eqn (1.397)–(1.401) into eqn (1.409) and taking into account eqn (1.295), we obtain

Equation 1.412
Equation 1.413
Equation 1.414
Equation 1.415
Equation 1.416

The relations presented above clearly show that excess properties and property changes of mixing are closely related and one may readily calculate ME from ΔM and vice versa. More formally, by combining the definitions in eqn 295 and (1.411), in conjunction with ΔMid defined by eqn (1.402a), the important relation

Equation 1.417

is obtained, with a similar one holding for the corresponding partial molar quantities:

Equation 1.418

In eqn (1.417), the difference ΔMME = ΔMid is zero except for the second-law properties M = G, F = GPV and S, see eqn (1.403)–(1.407), and similarly for the partial properties in eqn (1.418), the difference ΔMiMiE = ΔMiid    is zero except for the second-law properties Mi = Gi, Fi = GiPVi and Si. Further, from eqn (1.417), we see immediately that since a molar excess property represents also the difference between the real change of a property on mixing and the LR ideal-solution change of a property on mixing, we may identify it, alternatively to the defining equation eqn (1.409), as a molar excess property change on mixing:

Equation 1.419

Analogously, relating to eqn (1.419), we may identify, alternatively to the defining equation eqn (1.410), a partial molar excess property as a partial molar excess property change on mixing:

Equation 1.420

Evidently, the terms molar excess property and molar excess property change on mixing may be used interchangeably and both are indeed found in the literature. If the focus is on properties of mixtures, then ME and MiE are preferred, whereas for mixing processes the notations (ΔM)E = ΔME and (ΔMi)E = ΔMiE may be regarded as being more appropriate.

For the five quantities selected for more detailed discussion, eqn 417 and (1.418) yield for the second-law properties

Equation 1.421
Equation 1.422
Equation 1.423

and for the first-law properties we have

Equation 1.424
Equation 1.425

in accord with eqn (1.412)–(1.416). First-law excess properties are identical with the property changes of mixing. Depending on the point of view, VE = ΔV is known as either the molar excess volume or the molar volume change of mixing and HE = ΔH is called either the molar excess enthalpy or the molar enthalpy change of mixing.

In analogy with eqn (1.387), the exact differential of the extensive property nME(T, P, {xi}) is given by

Equation 1.426

and eqn (1.411) yields for a differential change in nME

Equation 1.427

Comparison with eqn (1.426) and division by n results in

Equation 1.428

which is still another form of the general Gibbs–Duhem equation.

The fundamental excess property relation for a single-phase fluid system of variable mass and variable composition, in analogy with eqn (1.160a) and (1.311), reads

Equation 1.429

where GiE denotes the partial molar excess Gibbs energy (= µiE, the excess chemical potential). Since the dimensionless property GE(T, P, {xi})/RT is expressed in terms of its canonical variables, it serves as a generating function for all other excess properties and thus contains complete excess property information. The pivotal role of excess properties in solution thermodynamics (experiment and theory) is emphasised. A discussion from a somewhat different angle is presented below.

There is a formal analogy between the definition of a molar isobaric residual property as given by eqn (1.235) and a molar excess property as given by eqn (1.409), hence the following simple relation is easily established:

Equation 1.430

or, in more compact notation,

Equation 1.431

Traditionally, excess properties are directly measured experimental quantities, say via calorimetry (HE, CPE)) or dilatometry (VE), but because of their close relation to residual properties they can also be obtained from PVTx measurements and thus from EOSs, as indicated by eqn (1.431). As discussed above, during the last two decades, research in this area has greatly expanded.

Evidently, as a generating function, the molar excess Gibbs energy GE and the dimensionless property GE/RT are of central interest. As a matter of convenience, eqn (1.392) may be generalised in such a manner that an expression for the partial molar Gibbs energy Gi is obtained that is valid not only for ideal systems but also for any real mixture per definition. Hence we may write

Equation 1.432

where γiLR    (T, P, {xi}) is known as the Lewis–Randall (LR) activity coefficient of component i in solution. For an LR ideal system, γiLR    (T, P, {xi}) = 1. With the prescription eqn (1.114), the partial molar excess Gibbs energy is given by

Equation 1.433

In view of eqn (1.411), the molar excess Gibbs energy reads

Equation 1.434

Since , we have for the molar excess entropy

Equation 1.435

With , the molar excess volume is given by

Equation 1.436

and finally, with the Gibbs–Helmholtz equation or via HE = GE + TSE, we obtain for the molar excess enthalpy

Equation 1.437

The ideal-solution equations for Mid (M = G, S, V, H) which follow from the Lewis–Randall rule are given by eqn (1.397)–(1.401). I reiterate that eqn (1.434)–(1.437) apply only when the Lewis–Randall model for the ideal solution is used.

As already pointed out, central to the development of useful relations for liquid mixtures is the fundamental excess-property relation for GE (or GE/RT, see below), which may now be written as

Equation 1.438

and, of course,

Equation 1.439

and so forth.

For convenience, instead of GiE, the non-dimensional group GiE/RT is frequently used, see eqn (1.429); this quantity is directly related to the LR-based dimensionless state function ln γiLR     (T, P, {xi}):

Equation 1.440

Using the summability relation, we have

Equation 1.441

The corresponding alternative fundamental excess-property relation for a single-phase system, in which the amounts ni may vary either through interchange of matter with its surroundings (open phase) or because of chemical reactions within the system, or both, now reads

Equation 1.442

in complete analogy with eqn (1.160a). By inspection, we find

Equation 1.443
Equation 1.444
Equation 1.445

and from eqn (1.438)

Equation 1.446

The corresponding Gibbs–Duhem equation reads

Equation 1.447

which at constant T and P reduces to

Equation 1.448

Rewriting eqn (1.448) for a binary mixture, we have

Equation 1.449

We recognise that if in a binary mixture γ1LR increases (or decreases) with increasing x1, then γ2LRmust simultaneously decrease (or increase) with increasing x1. In addition, when x2 → 0 and thus x1 → 1, the slope of the curve ln γ1LR    versus x1 is zero and, vice versa, when x1 → 0 and thus x2 → 1, the slope of the curve ln γ2LR    versus x1 is zero: each ln γiLR    curve (i = 1, 2) terminates at xiLR     = 1 with zero value and zero slope.

The usefulness of the fundamental excess-property relation eqn (1.442) originates from the fact that HE and its temperature dependence, the molar excess heat capacity CPE at constant pressure (molar excess isobaric heat capacity),33,41,43,138,139,326,327,329,330 

Equation 1.450

and also VE and γiLR are all experimentally accessible, the latter being obtained from vapour–liquid equilibrium (VLE) data as briefly discussed below. Excess volumes may be obtained more or less directly via dilatometry or pycnometry or by vibrating-tube densimetry or buoyancy methods (exploiting the Archimedes principle).42  The excess Gibbs energy, in principle the key excess property, is a conceptual property that cannot be measured directly (nor can the excess entropy), although it can be quantitatively deduced from VLE measurements via determination of LR activity coefficients.36,40,331–335  For mixtures at low to moderate pressure, i.e. well below the critical pressure, the conventional highly effective and refined “phi/gamma” approach to VLE is commonly used (note that isothermal measurements are advantageous55 ). For applications at pressures up to a few bar, to an excellent approximation the equilibrium relation for a binary system reads

Equation 1.451

and

Equation 1.452a
Equation 1.452b

Here, δij is defined by

Equation 1.453

where the identically subscripted second virial coefficients Bii and Bjj refer to the pure vapour of component i or j, respectively, and Bij is known as the interaction virial coefficient or cross-coefficient. Note that we have δij = δji. In eqn (1.452a), ϕi V(T, P, yi) is the vapour-phase fugacity coefficient of component i and ϕiV,*    (T, Pσ,i) is the fugacity coefficient of pure saturated vapour at T and Pσ,i. These phase-equilibrium relations are easily extended to multicomponent mixtures.

The classical data reduction approach uses activity coefficients directly determined using eqn (1.451). Insertion of the so-obtained γi LR    s into eqn (1.434) or (1.441) yields values of the dimensionless property that are then fitted to an appropriate analytical correlating equation representing the composition dependence of GE/RT. For more recent methods, see Van Ness and Abbott.55  Combination with calorimetrically measured excess enthalpies yields the molar excess entropy:

Equation 1.454

For one mole of a constant-composition mixture,

Equation 1.455

and for the corresponding partial molar excess properties, see eqn (1.440) and (1.442),

Equation 1.456

is obtained. Hence the partial molar analogues of eqn 443 and (1.444) are

Equation 1.457

and

Equation 1.458

respectively. Hence the partial molar excess entropy is given by

Equation 1.459

Finally, I list the useful relations

Equation 1.460

and, important for calorimetric work at high pressure,

Equation 1.461

and

Equation 1.462

Modern calorimeters allow reliable measurements of HE and CP (and thus of CPE) at elevated T and P and the results have to be consistent with experimentally determined volumetric properties, as indicated by eqn 461 and (1.462), respectively. However, outside the critical region the influence of pressure on excess properties is usually rather small.

Focusing now on the non-dimensional excess property GE/(x1x2RT) for a binary mixture, we find this quantity to be of considerable practical utility, especially when a graphical (visual) evaluation of experimentally determined GEs is intended. Note that

Equation 1.463

where γ1 LR,∞       and γ2 LR,∞       are the LR activity coefficients at infinite dilution. These quantities play an important role in solution chemistry and have found many applications in the characterisation of solution behaviour. In general, for binary mixtures extrapolation of ME/x1x2 to x1 = 0 and x2 = 0 is the most convenient and reliable graphical method for determining the infinite-dilution partial molar excess properties M 1E,∞    and M2 E,∞   , respectively.

Activity coefficients γiLR,∞          at infinite dilution characterise the thermodynamic behaviour of a single solute molecule i completely surrounded by solvent molecules j, hence it usually indicates maximum non-ideality and, in the absence of i,i interactions, it provides direct information on bulk i,j interactions, that is, on solute–solvent interactions. Hence they are key parameters in the discussion of dilute solutions, for instance, of those encountered in environmental studies.336–338  In fact, given the infinite-dilution activity coefficients of each component in the other in a binary mixture, values of parameters in popular two-parameter activity coefficient models can be easily obtained; in turn, these can be used for phase equilibrium predictions over the entire composition range.339  With improved experimental techniques,340–342  precise measurements at low concentrations can be made with less effort compared with conventional VLE measurements and with greater accuracy since extrapolation of activity coefficients obtained at higher mole fractions to infinite dilution is quite demanding.343 

For the prediction of the composition dependence of LR-based GE, many empirical equations have been proposed, and for binary mixtures some of the simpler ones are special cases of one of the following power series expansions in the mole fractions:

Equation 1.464

known as the Redlich–Kister expansion,344,345  or

Equation 1.465

With restriction to two parameters and on rearrangement, i.e. B + C = A21 and BC = A12, eqn (1.464) yields the equivalent two-parameter three-suffix Margules equation:346,347 

Equation 1.466

with the following expressions for the LR activity coefficients:

Equation 1.467
Equation 1.468

With restriction to two parameters and on rearrangement, i.e. 1/(BC) = A12 and 1/(B + C) = A21, eqn (1.465) yields the equivalent two-parameter van Laar-type equation:348,349 

Equation 1.469

with the following expressions for the LR activity coefficients:

Equation 1.470
Equation 1.471

In 1964, Wilson suggested a novel type of equation for GE by introducing the intuitively appealing concept of local composition, i.e. where the local mole fraction of component i in a mixture {i + j} deviates from the bulk mole fraction,350  a concept that has been developed impressively since then. For a binary mixture, the molar excess Gibbs energy is given by

Equation 1.472

and the corresponding activity coefficients are

Equation 1.473

and

Equation 1.474

where the parameter Γ is given by

Equation 1.475

Thus, at infinite dilution we obtain

Equation 1.476

Eliminating one of the parameters, say Λ21, yields

Equation 1.477

and then we obtain

Equation 1.478

An iterative procedure is required to evaluate the adjustable parameters Λ12 and Λ21 (they must always be positive numbers). However, more than one set of parameters is mathematically possible when the infinite-dilution LR activity coefficients are smaller than unity.351,352  In Wilson's derivation they are related to the liquid pure-component molar volumes and to characteristic interaction energy differences, i.e.

Equation 1.479

Numerical values of the parameters λijλii can only be found through reduction of experimental VLE data. The Wilson equation is a very flexible equation with a built-in temperature dependence; it is able to represent mixtures that exhibit strong deviations from ideality and it is easily generalised to describe multicomponent behaviour using only two binary parameters from each of the binary subsystems:

Equation 1.480
Equation 1.481

However, Wilson's equation is unable to predict limited miscibility and should therefore be used only for liquid systems where the components are completely miscible.

Enormous research efforts have been invested in developing the local composition concept, for instance in developing the NRTL equation, the UNIQUAC and UNIFAC formalism and the DISQUAC model.353–364  This topic definitely needs and deserves a review of its own.

Chemical thermodynamics is a vast field of attractive, formal beauty, yet with many connections to real-world problems. Hence scientific advances frequently originate from the work of chemical engineers. Although thermodynamics is based on the observation of and experiments on bulk matter, combination with molecular theory in conjunction with statistical mechanics promotes inductive molecule-based insight into macroscopic phenomena and thus continues to contribute decisively to progress in physics/chemistry and in chemical engineering, but also increasingly in biophysical chemistry, biophysics, drug development and biomedical research. The focus of the present book and of its three companion volumes41–43,335  published so far is on liquids, solutions and vapours. As I have pointed out recently,134  the liquid state of matter houses by far the largest group of unsolved or crudely solved problems in physical chemistry, especially when biophysical chemistry is included. For obvious reasons, research on systems with bio-relevance is an exciting, dynamically evolving field: supported by unabated progress in instrumentation, improved means for data processing and data transfer and increasingly sophisticated computer simulation techniques, new scientific information at the microscopic, mesoscopic and macroscopic levels stimulates the building of connections with a growing number of neighbouring fields. However, advances always rest on reliable, precise and reasonably comprehensive experimental results obtained via classical equilibrium thermodynamic methods, subsequently augmented by appropriate molecule-based statistical-mechanical models. Hence calorimetry, PVTx measurements and phase equilibrium determinations (vapour–liquid, liquid–liquid and solid–liquid equilibria), as the oldest and most fundamental experimental disciplines in chemical thermodynamics/physical chemistry, play a central role in providing new quantitative data on key properties of fluid systems. In turn, these data are used for theoretical advances on the one hand and for improvements/innovations in chemical engineering and in bioscience-related applications on the other. The indicated sequence of classical thermodynamic methods (one might call it an approach based on “integration”) is reflected by the sequence of topics covered by the four books that we (E. Wilhelm and T. M. Letcher) have edited in this series published by The Royal Society of Chemistry; throughout the focus was/is on fluid systems:

  • I. Heat Capacities (2010)

  • II. Volume Properties (2015)

  • III. Enthalpy and Internal Energy (2018)

  • IV. Gibbs Energy and Helmholtz Energy (2021)

The sequence of topics that we adopted is evident: from directly measurable properties, such as heat capacities, to conceptual properties, such as Gibbs and Helmholtz energies.

The main reason for the importance to the chemist of an understanding of liquid solutions is simply the fact that most chemical reactions involve solutions; and we note the continuously growing interest in solutions of environmental relevance (to indicate the topics covered, I have also supplied the respective titles).365–380  With increasing concern for the environmental dynamics of natural and anthropogenic substances, for their biological impact in general and on human health (environmental risk assessment) in particular, physicochemical/thermodynamic work on dilute aqueous solutions has increased significantly. New instruments have been developed381–387  and relevant data compilations have become available.365–373,380  In this introductory chapter, I have not covered any experimental details – the reader is referred to the relevant chapters of this book and to the pertinent review articles and monographs cited as references. Suffice it to say that enormous effort and ingenuity has gone into scientific design to provide the array of apparatus now at our disposal for high-precision calorimetry, for the determination of PVTx properties of pure and mixed fluids over large ranges of temperature and pressure, for sophisticated ultrasonics equipment and for vapour/liquid-phase equilibrium instruments. Undoubtedly, automation of apparatus will continue and the speed of measurement will increase. In fact, the availability of good commercial instruments has resulted in an unprecedented growth of published experimental property data, as documented by ref. 8–33, ref. 256 and the increasing number of data-focused review articles. Through ever-widening ranges of application in conjunction with improved measuring accuracy and advanced methods of data management and data storage, chemical thermodynamics will remain an active fundamental discipline. Cross-fertilisation with bio-oriented fields will continue to increase and, as pointed out above, will lead to fascinating, important and, perhaps, unexpected research results, as indicated by the selection of recent articles, reviews and monographs collected as ref. 388–417.

In addition to a concise overview covering fundamental parts of chemical thermodynamics, in this introductory chapter I hope to have also communicated my admiration for thermodynamics and my conviction that scientific advances inevitably lead to a broadening of the field and to a merging of neighbouring areas of research, and that globally, cross-disciplinary research provides, perhaps, the most potent stimulus in science and in technological innovation. I hope that the topics treated in this book under the “umbrella” Gibbs energy and Helmholtz energy provide a feeling for the scope of the field, for its contributions to the development of (bio-)physical chemistry and chemical engineering, for its current position in science and, most important, for its future potential. In this connection, it is again my pleasure to acknowledge the many years of fruitful scientific collaboration with more than 80 colleagues, post-doctoral fellows and students from 17 countries. Without them, many projects would have been difficult to carry out or would have, perhaps, never been started.

My view of chemical thermodynamics has evolved over the years and I have benefited from collaborations and discussions with many friends and colleagues and some of them have contributed to one or more volumes of this series of books that have been published under the auspices of both IUPAC and IACT. In fact, quite early in my career I was asked to participate in various IUPAC activities and I did indeed devote an appreciable part of my “scientific life” to further the Union's goals, both nationally and internationally. Thus, I was particularly pleased to participate, as an Invited Lecturer, at the 47th IUPAC World Chemistry Congress: Frontiers in Chemistry, celebrating 100 Years with IUPAC, in Paris, France, 7–12 July 2019, where a Special Symposium (Homage to Eduard Hála) was devoted to chemical thermodynamics: this event clearly illustrates the importance of this field of science. In fact, chemical thermodynamics has grown so large and the subtopics so numerous that covering G- and F-related areas exhaustively in a reasonably sized monograph is essentially impossible. However, it is hoped that the various subjects selected and discussed here in the 17 chapters of the book by internationally acknowledged colleagues will provide a feeling for the scope of the field and its position in the development of science and engineering. By necessity, I have limited my introductory chapter to a few core topics, the selection of which was, of course, influenced by my current interests. To summarise: I hope to have.

  • formulated concisely some important aspects of the thermodynamic formalism needed in this area of research;

  • discussed and made transparent some key issues when applying chemical thermodynamics to fluid mixtures;

  • shown how to apply and extend well-known concepts appropriately to perhaps less familiar problems;

  • stimulated some colleagues to enter this fascinating field of research.

Success in any of these points would be most rewarding.

The presentation here is based on that given by Rowlinson418  in Handbuch der Physik, Volume 12, Chapter 1. The logarithmic infinity in S, F and G at zero pressure makes it more convenient to select as reference states the hypothetical perfect gas at the same temperature and either at the same volume or at the same pressure as the real fluid. Thermodynamic properties referred to such perfect-gas states are known as residual properties. We note that in practice a more useful variable than V is the amount (of substance) density, ρ = 1/V. Molar isochoric (isometric) residual properties of pure fluids are thus defined by

Equation 1.A1
Equation 1.A2

When focusing on evaluating molar isochoric residual properties based on ρ as the independent parameter, that is, when using eqn (1.A2), it is advantageous to introduce, by definition, the isochoric residual function Qr(T, V) that remains finite at zero density:

Equation 1.A3

At zero density (or zero pressure), Qr(T, V) is finite at all temperatures:

Equation 1.A4

with the exception of the Boyle temperature, where Qr(T, V) = 0. Qr(T, V) is a very regular function of density and temperature, even at the critical point, and it is nearly a linear function of temperature even at high densities, because

Equation 1.A5

and the curvature of isochores is usually quite small.

From eqn (1.A2), the following equations for the molar isochoric residual properties are obtained in terms of Qr(T, V):

Equation 1.A6
Equation 1.A7
Equation 1.A8
Equation 1.A9
Equation 1.A10
Equation 1.A11
Equation 1.A12

The limiting values of these molar isochoric residual properties at zero density (infinite volume) are zero in each case.

The molar isochoric residual properties are related to each other as follows:

Equation 1.A13
Equation 1.A14

where

Equation 1.A15

is the residual pressure.

An analogous set of molar isobaric residual properties for pure fluids is obtained from the defining equation

Equation 1.A16

When focusing on evaluating molar isobaric residual properties based on P as the independent parameter, that is, when using eqn (1.A16), it is advantageous to define an isobaric residual function that remains finite at zero pressure: in analogy with the definition of the residual function Qr(T, V), we now have

Equation 1.A17

which is the molar isobaric residual volume.

From eqn (1.A16), the following equations for the molar isobaric residual properties are obtained in terms of VR(T, P):

Equation 1.A18
Equation 1.A19
Equation 1.A20
Equation 1.A21
Equation 1.A22
Equation 1.A23
Equation 1.A24

The molar isobaric residual properties are related to each other as follows:

Equation 1.A25
Equation 1.A26
1

A comprehensive biographical memoir for Joel Henry Hildebrand (16 November 1881–30 April 1983) was prepared by Kenneth S. Pitzer.169  As a scientist, Professor Hildebrand contributed to a remarkable diversity of fields, although his research on liquids and non-electrolyte solutions (e.g. the solubility of gases in liquids) is most important; he introduced the solubility parameter and his efforts to improve the regular solution theory continued, in fact, until a final paper in 1979 170  and a historical résumé in 1981.171  After his PhD in 1906, he spent a postdoctoral year in Germany to learn the then new scientific discipline Physical Chemistry (attending lectures by J. H. van ’t Hoff and W. Nernst and doing some research under Nernst), and in 1913 G. N. Lewis invited him to join the young faculty of the Chemistry Department at the University of California, Berkeley, where he remained. After the influential monograph (with R. L. Scott) on The Solubility of Nonelectrolytes in 1950,69  he co-authored books that focused on topics of his particular interests, e.g. Regular and Related Solutions: The Solubility of Gases, Liquids and Solids in 1970 with J. M. Prausnitz and R. L. Scott.74  A comprehensive biographical memoir for George Scatchard (19 March 1892–10 December 1973) was prepared by John T. Edsall and Walter H. Stockmayer.172  Professor Scatchard's scientific contributions focused almost entirely on the physical chemistry of solutions, comprising electrolytes and non-electrolytes, high-precision VLE studies (e.g. with S. E. Wood, J. E. Mochel and W. J. Hamer), amino acids and proteins, although his PhD thesis (Columbia, 1916) was on synthetic organic chemistry. After he was drafted to France in 1918, at the end of the First World War (as First Lieutenant in the Sanitary Corps), he worked with Victor Grignard in Paris (Sorbonne) and returned to the USA by ship on his 27th birthday, first to Amherst College, then to MIT (1923), where he remained for the rest of his life. In his Chemical Reviews paper of 1931,163  entitled Equilibria in Non-electrolyte Solutions in Relation to the Vapor Pressures and Densities of the Components, he gave the simplest successful equation for predicting the enthalpies of mixing by introducing the concept of cohesive energy density. He later considered it as his first important paper. Another important article (1949) discussed the binding of small molecules and ions to proteins,173  and led to a simple equation for plotting binding data to evaluate the number of binding sites and the association constants involved (Scatchard plots).

2

The letter L is used in honour of Josef Loschmidt (1821–1895), pioneering Austrian physicist, who, in 1865, provided the first reasonable estimate for the number of particles N in a given volume of gas under ambient conditions. With the Avogadro constant L and the Boltzmann constant kB now being exactly defined,180,181  that is, L = 6.02214076 × 1023 mol−1 and, kB = 1.380649 × 10−23 J K−1, the (molar) gas constant is R = LkB = 8.314462618 J K−1 mol−1, exactly.

3

A comprehensive biographical memoir for Kenneth Sanborn Pitzer (6 January 1914–26 December 1997) was prepared by Robert F. Curl.232 Professor Pitzer's scientific contributions to physical chemistry cover an impressive diversity of fields, comprising thermodynamics and statistical thermodynamics (e.g. extension of CST), theoretical organic chemistry (C–C hindered rotation in hydrocarbons, strain energies of cyclic hydrocarbons), electrolytes (from dilute solutions to fused salts) and quantum chemistry. In 1936, while still a graduate student, he discovered (with fellow student J. D. Kemp, J. Chem. Phys., 1936, 4, 749) that a barrier V(ϕ) to internal rotation of the methyl groups in ethane (ϕ is the rotational angle) of the form was required to explain its third law entropy, a discovery of great importance for organic chemistry and hydrocarbon thermodynamics. Immediately after receiving his PhD from the University of California, Berkeley, in 1937, he was appointed to the faculty of UCB's Chemistry Department. With several interruptions (World War II, Atomic Energy Commission, President of Rice University, President of Stanford University), from 1971 on he finally stayed in Berkeley, became Professor Emeritus in 1984, although he continued with active and productive research until his death – an impressive 211 of his 405 papers were published during these final years. According to his biographer, a few years before his death Pitzer was puzzled that most of his still living contemporaries had left research, which for him was a source of both pleasure and satisfaction. In 1988, at the 10th ICCT in Prague, he presented the IUPAC Rossini Lecture.

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