CHAPTER 1: Gibbs Energy and Helmholtz Energy: Introduction, Concepts and Selected Applications

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 

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 

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 

The growth of molecular thermodynamics has been stimulated by the continuously increasing need for thermodynamic property data and phase equilibrium data^8–31  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 fluids^6,69–97  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 {w_i}, defined by

where m_i denotes the mass of component i, is the total mass of the phase, and for a pure fluid w_i = 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 m_m,i of each component i is known, the composition of a phase is preferably characterised by a set of mole fractions {x_i}, defined by

where n_i denotes the amount (of substance) of component i, is the total amount in the phase, and for a pure fluid x_i = 1. Since m_i = n_im_m,i is the total mass of component i, we have

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 V ^t of a phase is an extensive property that does depend on the quantity of material and may thus be alternatively expressed either by

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 ρ_m ≡ m/V ^t and in (statistical) thermodynamics, the amount (of substance) density ρ ≡ n/V ^t = 1/V and the number density ρ_N ≡ N/V ^t.

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} 

that is, for a pure fluid with critical pressure P_c, critical molar volume V_c and critical temperature T_c the divergence of β_T is expressed by the power law (simple scaling)^118,119 

where γ = 1.239 ± 0.002 denotes the critical exponent (the classical value is 1), and ρ_c 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),

and to the divergence of the molar isobaric heat capacity,

where C_V 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 theory^123–125  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.

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 U ^t 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 U ^t, 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

where, in analogy with eqn (1.4), U ≡ U ^t/n denotes the molar internal energy of the fluid and u ≡ U ^t/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

Mathematically, dU ^t is an exact differential of the state function U ^t: 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

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 ΔU ^t 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

Measurement of the adiabatic work W_ad 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 U ^t 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

and Q_mech 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 U ^t: 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 Q_rev + W_rev 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

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

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

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. ΔV ^t, but this fact could also be demonstrated as follows. The work caused by a reversible volume change is given by

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

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

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

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

• Postulate 4: There exists a material property called total entropy S^t, 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 S^t 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 ΔS^t, 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 S^t 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:

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 M ^t is the sum of the total property values of the p constituent phases:

Hence the overall molar property is obtained from

where

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

where U, S and V denote the molar internal energy, the molar entropy and the molar volume, respectively. Since U^t = nU, S^t = nS and V ^t = 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 TdS^t is not the heat transferred, nor is −PdV ^t the work done: δQ_irrev < δQ_rev and δW_irrev > δW_rev, but, of course, dU ^t = TdS^t − PdV ^t = δQ_irrev + δW_irrev, since U ^t 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

where the subscript {n_i} 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

To reiterate: since U^t = nU, S^t = nS and V ^t = 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

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 n_i may vary because of interchange of matter with its surroundings or because of chemical reactions within the system or both. Since U^t and S^t are the only conceptual properties, one may have either a fundamental equation in the internal energy representation:

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

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

in the internal energy representation, and, equivalently,

in the entropy representation. eqn 34 and (1.35) apply to single-phase multicomponent PVT systems, either open or closed, where n_i 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 n_i, at constant entropy and volume, is called the chemical potential of component i in the mixture:

From eqn (1.35), we obtain

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

while the fundamental property relation eqn (1.35) becomes

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

and in the entropy representation we have

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, {n_i}) and nS(nU, nV, {n_i}) 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, µ_i} or , respectively. These two primary functions (cardinal functions, thermodynamic potentials) are, of course, related:

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

it must also satisfy

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 U^t and S^t, that is,

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 µ_i are partial derivatives of U^t(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

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 S^t(nU, nV, n₁, n₂,⋯), see eqn (1.30), (1.31) and (1.37), yielding the corresponding EOSs:

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.

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:

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

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:

where Z is known as the compression factor, B′(T, { y_i}) denotes the second virial coefficient of the pressure series, C′(T, { y_i}) is the third virial coefficient of the pressure series and so forth and y_i 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:

and the van der Waals (vdW) equation:

In eqn (1.55), B(T, { y_i}) denotes the second virial coefficient of the amount density-series, C(T, { y_i}) 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,

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/V ². 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, {n_i}} 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, {n_i}} and potentially more practical.

Consider the exact (total) differential

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

where

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):

The total differential reads

and with eqn (1.59) we obtain

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:

which is frequently identified by a bracket notation as indicated. This Legendre transform represents a new function with independent variables {c₁, X₂, X₃,⋯, X_n}, 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,…, c_p} into f⁽⁰⁾ reads

and the associated total differential is

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

The associated differential expression reads

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

and correspondingly

with canonical variables {T, P, { µ_i}}. 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 { µ_i} 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

and correspondingly

with canonical variables {1/T, P/T, { µ_i/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 {µ_i/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,…, X_n), with 1 ≤ p ≤ (n − 1), is obtained via subtraction of p products of X_i 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

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

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

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 N ^t 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:

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 

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

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

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 N ^t 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, {n_i} by the corresponding conjugate intensive variable(s) 1/T, P/T and µ_i/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 ).

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:

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

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, {n_i}} and {T, nV, {n_i}}, since they are easily measured and controlled. Hence the total Gibbs energy G^t(T, P, {n_i}) and the total Helmholtz energy F^t(T, nV, {n_i}) are particularly important.

Partial derivatives of a total property with respect to n_i at constant T, P and n_j≠i 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) {n_i} of all the components present:

where is the total amount contained in the phase and M(T, P, {x_i}) 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

where the subscript {n_i} indicates that the amounts of all components i and thus the composition {x_i} 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 M_i and are defined by

The partial molar property M_i(T, P, {x_i}) is an intensive state functions: it quantifies the change (response) of the total (extensive) property M ^t = nM when an infinitesimal amount (of substance) dn_i of component i is added to the solution at constant temperature and pressure, while keeping the amounts of all the other components, i.e. n_j≠i, constant. Note that when the amounts n_i are replaced with the masses m_i of the components, for instance, because their molar masses m_m,i are unknown (n_i = m_i/m_m,i), and n with , eqn (1.114) yields partial specific properties and the composition of the mixture is then characterised by weight fractions w_i 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. M̄_i.

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

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

From experiments, we know that M ^t is homogeneous of the first degree in the amounts of substance, hence Euler's theorem, eqn (1.44), yields

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

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

eqn 117 and (1.118) are known as summability relations. To reiterate: since M(T, P, {x_i}) is an intensive property, the partial molar property M_i(T, P, {x_i}) 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

However, from eqn (1.118),

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

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

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

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

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

which shows the constraints on changes of mixture composition. It is important to note that a partial molar property M_i 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 M^t = 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 n_i is the amount of component i, or alternatively by the symbol M ^t.

• Pure-substance properties are characterised by an asterisk (*) and identified by a subscript, e.g. M$i*$ 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. M_i, 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:

They are exact differentials, hence

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 {n_i} or {x_i}, 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):

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 M ^t(T, P) = nM:

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 related^56,134 ) consists of the molar heat capacity at constant pressure (molar isobaric heat capacity):^{134,138–140} 

the isothermal compressibility:^134,139,140 

and the isobaric expansivity:^134,139,140 

The conventional choice of the set {C_P, β_T, α_P} 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 {C_P, β_T, α_P} 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} 

the isothermal compressibility:^134,139,140 

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

In (T, V) space, the set {C_V, 1/β_T, γ_V} 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:

for the inverse isentropic compressibility^{134,138–140}  and

for the isentropic expansivity^134,139,140 

and {1/C_V, 1/β_S, 1/α_S} 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

and the isentropic thermal pressure coefficient^134,139 

thus providing the set {1/C_P, β_S, 1/γ_S}.¹³⁴ 

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

the three mechanical coefficients are related as follows:

and

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:

Simple mathematical manipulations lead to the following alternative forms:

and

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

and thus

Eqn (1.160a) is of considerable utility: knowledge of nG/RT as a function of the canonical variables T, P and n_i 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

and thus

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

Note that both equations can be contracted:

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

and a special name, internal pressure. It may be directly determined at any pressure viaeqn (1.162) by measuring γ_V ≡ (∂P/∂T)_V with a piezometer or by using experimental results for the isobaric expansivity α_P and the isothermal compressibility β_T 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

At ambient temperatures and pressures, usually P ≪ Tγ$VL,*$, notable exceptions being liquid water and liquid heavy water in the temperature range from the respective triple point, T_tr(H₂O) = 273.16 K and T_tr(D₂O) = 276.97 K, to temperatures slightly above that of the respective density maximum, i.e. T_max(H₂O) = 277.13 K and T_max(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, Π_max ≈ 190 MPa at an external pressure of about 150 MPa. For CO₂, Π_max ≈ 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 c^L,*(T, P) was introduced to liquid-state physical chemistry by Scatchard¹⁶³  and is defined by (see Wilhelm^164,165 )

where E$cohL,*$(T, P) denotes the cohesive energy per mole of pure liquid, the molar isobaric residual internal energy is defined by U^R,L,*(T, P) ≡ U^L,*(T, P) − U^pg,*(T), with the superscript R characterising a residual property in (T, P, {x_i}) 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  Hildebrand^167,168  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:

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

where the subscript σ indicates saturation conditions (vapour–liquid equilibrium, VLE), that is, P_σ = P_σ(T) denotes the vapour pressure and Δ_vapH^*(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 > T_c and P > P_c, 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

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 T_r ≡ T/T_c and P_r ≡ P/P_c:

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

where P_σ,r ≡ P_σ/P_c 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 = Tγ$V*$(T, P) − P and the cohesive energy density c^* ≡ −U^R,*/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^* ≡ −U^R,*/V^* measures the total molecular cohesion of the fluid, whereas Π^* ≡ (∂U^*/∂V)_T = (∂U^R,*/∂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

Hence the cohesive energy density is given by

The total pressure is given by

hence the internal pressure is obtained as

where ρ$N*$ ≡ L/V^* = Lρ^* 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, ρ$N*$), has been written explicitly as a function of the interparticle distance r and also of T and the pure-fluid number density ρ$N*$ 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^* ≡ Tγ$V*$ − 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 = V ^t(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 enormously^80,182–186  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

where k_B 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 

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(r_min) = −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 

where σ = 2^−1/6r_min.

(∂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,

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

by

The three quantities φ, µ_JT and C_P of gases/vapours can be measured by flow calorimetry.^193–195  Since for perfect gases Tα_P = 1, φ = 0 and µ_JT = 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:

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

and integration between a suitable reference temperature T_ref and T yields¹⁹⁶ 

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:

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

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

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

In high-pressure research,^{136–140,199–206} eqn 189 and (1.190) are particularly interesting. For instance, the pressure dependence of C_P 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

we obtain

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

and

Note that the difference between C_P and C_V depends on volumetric properties only. The heat capacity difference may therefore also be expressed by^138–140 

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:

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:

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

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

thus complementing eqn 198 and (1.200).

The ratio of the molar heat capacities, κ ≡ C_P/C_V , is accessible via eqn 139 and (1.142) in conjunction with the chain rule, i.e.

and the triple product rule:

Thus, for homogeneous constant-composition fluids we obtain

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

leads to

which is one of the most important equations in fluid phase thermophysics. Here, ρ_m ≡ m_m/V = m_m ρ = m_m ρ_N/L denotes the mass density and v₀ = v₀(T, P, {x_i}) 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 η_s, thermal conductivity λ, bulk viscosity η_v, etc., v₀ may be treated as an intensive thermodynamic equilibrium property^{32,138,207–215}  related to β_Sviaeqn (1.207). Alternatively, by using the relations provided by eqn 131 and (1.132), we have

and

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

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 η_s and thermal conductivity λ, are described by the Kirchhoff–Stokes equation:²¹² 

where α_cl denotes the classical amplitude absorption coefficient. Sound dispersion due to classical absorption is almost always negligible and the product v(  f  )v$02$ in eqn (1.214) is usually replaced with v$03$. 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}  γ_V of liquids is large and the direct calorimetric determination of C_V 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 γ_V is much smaller.

From the equations for the difference C_P − C_V of a constant-composition fluid, it follows that

and

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

and

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, I_R, and of the two Brillouin peaks, 2I_B, is given by the Landau–Placzek ratio:

From eqn (1.208), the difference between β_T and β_S may be expressed as

and for the difference of the reciprocals we have

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

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

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

Most values of β_S 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 C_P is based on eqn (1.225), i.e. on measuring (∂T/∂P)_S and (∂V/∂T)_P = Vα_P, similarly to the approach of Richards and Wallace.²³¹ 

As pointed out by Rowlinson and Swinton,⁷⁹  the mechanical coefficients α_P, β_T and γ_V are determined, to a high degree of accuracy, solely by intermolecular forces, while the isentropic coefficients α_S, β_S and γ_S, 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:

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

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

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, {x_i}} and {T, V, {x_i}}. 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 V ^pg, two obvious choices exist: one may quantify deviations in terms of a ratio measure, here the compression factor:

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

The two functions are, of course, related:

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 V ^R becomes indeterminate, necessitating application of de l’Hôpital's rule:

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

Hence V ^R 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 V^R, molar isobaric residual properties M ^R of a single-phase pure fluid or constant-composition fluid mixture are defined similarly by^55,68,79,165 

where the superscript R identifies a residual function in (T, P, {x_i}) space. The Ms denote molar values of any extensive thermodynamic property nM (T, P, {x_i}), such as U, H, S, V, G or F. M (T, P, {x_i}) is the actual molar property value of the fluid at the temperature, pressure and composition of interest and M ^pg (T, P, {x_i}) is the molar property value for the fluid in its perfect-gas state at the same T, P and {x_i}. 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

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

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

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 {x_i}):

Thus, for the molar isobaric residual enthalpy H^R(T, P, {x_i}) we obtain, in conjunction with and ,

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

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

Since

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

For the remaining molar isobaric residual properties, we have

The two terms of the integrand in the G^R 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, {x_i}} 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

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 T_r ≡ T/T_c, reduced pressure P_r ≡ P/P_c and Pitzer's acentric factor ω, which is defined by eqn (1.173). Specifically, in the key three-parameter CST correlation

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 T_c and P_c, 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⁾(T_r, P_r) and Z⁽¹⁾(T_r, P_r), and also for derived functions for both liquid and vapour phases, covering large temperature and pressure ranges, i.e. 0.30 ≤ T_r ≤ 4.00 and 0.01 ≤ P_r ≤ 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 T_c,i, P_c,i and ω_i to obtain pseudocritical temperatures T_pc, pseudocritical pressures P_pc and pseudo-acentric factors ω_p 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

where the pseudoreduced temperature T_pr and the pseudoreduced pressure P_pr are defined by