Gibbs Energy and Helmholtz Energy: Liquids, Solutions and Vapours
 1.1 Introduction
 1.2 Thermodynamic Fundamentals
 1.3 More Thermodynamics and Selected Applications
 1.3.1 Real Fluids: Fundamentals
 1.3.2 Residual Properties, Fugacities and Fugacity Coefficients
 1.3.3 Empirical (Thermal) Equations of State and More: Selected Comments
 1.3.4 Property Changes on Mixing and Excess Properties
 1.4 Concluding Remarks, Future Directions and Acknowledgements
 Appendix
 References
CHAPTER 1: Gibbs Energy and Helmholtz Energy: Introduction, Concepts and Selected Applications

Published:08 Sep 2021

Special Collection: 2021 ebook collection
E. Wilhelm, in Gibbs Energy and Helmholtz Energy: Liquids, Solutions and Vapours, ed. E. Wilhelm and T. M. Letcher, The Royal Society of Chemistry, 2021, pp. 1120.
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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 moleculebased 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 nonelectrolyte 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.
1.1 Introduction
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:
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 moleculebased 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 moleculebased theory; yet regardless of molecular complexity, thermodynamics may still be applied and the results obtained remain exact.
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, moleculebased 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 statisticalmechanical modelling is applied to macroscopic systems, the results have to be consistent with thermodynamics: its discriminative power is an extremely valuable tool for evaluating moleculebased 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 topdown reasoning. It orders and explicates already existing knowledge, thereby leading to predictions that may be corroborated experimentally (or, in principle, falsified^{3 }): 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, moleculebased theory and computer simulations represent the three pillars of science,^{5 }augmenting the formal framework of thermodynamic equations with moleculebased 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 bottomup reasoning, leads to hypotheses and then theories and thus to truly new knowledge. This field of molecular thermodynamics (the term was coined by Prausnitz^{6 } 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 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 indepth philosophical discussions see Mach,^{103 } Popper,^{3 } Katz^{104 } and Sober.^{105 }
1.2 Thermodynamic Fundamentals
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 Abbott^{55 } 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 EditorinChief 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 edition^{115 } 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, nonreactive 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 moleculebased 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_{i}m_{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 nearcritical 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.
System, boundaries .  Interactions 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, boundaries .  Interactions 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 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 Nbody 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 constantcomposition fluid (closed system) along different paths from an initial equilibrium state 1 at (T_{1}, P_{1}) to a final equilibrium state 2 at (T_{2}, P_{2}). 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 reasoning^{3 }).
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 singlephase 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 singlephase 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 singlephase 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 singlephase 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 singlephase, 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, singlephase, multicomponent PVT systems reads
and in the entropy representation we have
To reiterate: in this introductory chapter only nonreacting 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 (z_{1}, z_{2},…) is a homogeneous function of degree k in the variables z_{1}, z_{2},…, 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 firstorder 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, n_{1}, n_{2},⋯} for the internal energy representation and {nU, nV, n_{1}, n_{2},⋯} 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, n_{1}, n_{2},⋯). 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 zerothorder 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, n_{1}, n_{2},⋯), see eqn (1.30), (1.31) and (1.37), yielding the corresponding EOSs:
For constantcomposition fluids and thus also for pure fluids, T = T(nU, nV, n_{1}, n_{2},⋯) 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 pressureexplicit thermal equation of state:
or, after rearrangement, a volumeexplicit thermal equation of state:
Frequently, for the sake of brevity, the adjective thermal (having historical roots) is omitted. A wellknown example of a volumeexplicit (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). Wellknown examples of pressureexplicit (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 densityseries, C(T, { y_{i}}) is the third virial coefficient of the amount densityseries 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.
1.3 More Thermodynamics and Selected Applications
1.3.1 Real Fluids: Fundamentals
In the fundamental property relations for an open singlephase 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. Experimentbased 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 energybased 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 entropybased 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 X_{1} with its conjugate partial derivative c_{1} 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 X_{1} and its conjugate c_{1} have interchanged their roles (and changed sign). For such an interchange, it suffices to subtract from the base function f^{(0)} to obtain the firstorder partial Legendre transform:
which is frequently identified by a bracket notation as indicated. This Legendre transform represents a new function with independent variables {c_{1}, X_{2}, X_{3},⋯, 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 {c_{1}, c_{2},…, 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, singlephase, 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, singlephase, 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 entropybased 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)}(X_{1}, X_{2},…, 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$Le\u2009t$ 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 energybased Gibbs–Duhem equation.^{134 }





nU  nU  Internal energy  (1.40) 
nU + P(nV)  nH  Enthalpy  (1.78) 
nU − T(nS)  nF  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) 





nU  nU  Internal energy  (1.40) 
nU + P(nV)  nH  Enthalpy  (1.78) 
nU − T(nS)  nF  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) 
Alternative fundamental property relations .  Canonical variables . 

nS, nV, {n_{i}} (1.38)  
nS, P, {n_{i}} (1.84)  
T, nV, {n_{i}} (1.85)  
T, P, {n_{i}} (1.86)  
nS, nV, { µ_{i}} (1.87)  
nS, P, { µ_{i}} (1.88)  
T, nV, { µ_{i}} (1.89)  
T, P, { µ_{i}} (1.71) 
Alternative fundamental property relations .  Canonical variables . 

nS, nV, {n_{i}} (1.38)  
nS, P, {n_{i}} (1.84)  
T, nV, {n_{i}} (1.85)  
T, P, {n_{i}} (1.86)  
nS, nV, { µ_{i}} (1.87)  
nS, P, { µ_{i}} (1.88)  
T, nV, { µ_{i}} (1.89)  
T, P, { µ_{i}} (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 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 entropybased 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 Callen^{56 }).
Primary function .  Symbol .  Name .  Alternative expression . 

nS  nS  Entropy  (1.41) 
nΘ  Not named  (1.92)  
nΨ  Massieu function  (1.93)  
nΦ  Planck function  (1.94)  
nΛ  Not named  (1.95)  
nΓ  Not named  (1.96)  
nΩ  Kramers function  (1.97)  
—  Null function  0 (1.72) 
Primary function .  Symbol .  Name .  Alternative expression . 

nS  nS  Entropy  (1.41) 
nΘ  Not named  (1.92)  
nΨ  Massieu function  (1.93)  
nΦ  Planck function  (1.94)  
nΛ  Not named  (1.95)  
nΓ  Not named  (1.96)  
nΩ  Kramers function  (1.97)  
—  Null function  0 (1.72) 
Alternative fundamental property relations .  Canonical variables . 

nU, nV, {n_{i}} (1.39)  
(1.98)  
(1.99)  
(1.100)  
(1.101)  
(1.102)  
(1.103)  
(1.73) 
Alternative fundamental property relations .  Canonical variables . 

nU, nV, {n_{i}} (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:
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 singlephase 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 singlephase equilibrium PVT system, either open or closed, with the last term giving the differential variation of M^{t} = nM caused by amountofsubstance 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 singlephase puresubstance 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 singlephase 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 singlephase 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}.
Puresubstance 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 energybased 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 energybased 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 constantcomposition 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 constantcomposition 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 firstorder derivatives of the primary functions are thermodynamically important, although it is the secondorder derivatives that are indispensable for an indepth discussion of material properties. For simple, constantcomposition 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 energybased 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 secondorder 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 firstorder derivative in terms of these (easily) measurable properties. As pointed out by McGlashan^{4 } (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 energybased fundamental property relation eqn (1.85), see Table 1.3, which for one mole of a homogeneous constantcomposition fluid is given by eqn (1.128). The corresponding three secondorder 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 secondorder 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 firstorder derivative in terms of these three (easily) measurable properties. This approach is particularly convenient when dealing with properties derived from pressureexplicit 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 secondorder derivatives in (S, V) space.^{134 } Based on H(S, P), for a constantcomposition 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 constantcomposition 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 constantcomposition phase may now be established systematically between members of the fundamental set and of the alternative sets of secondorder 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 secondorder 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 amountofsubstance and, in contradistinction to eqn (1.86), the enthalpy rather than the entropy appears in the first term on the righthand 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 righthand 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 liquidstate 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}(H_{2}O) = 273.16 K and T_{tr}(D_{2}O) = 276.97 K, to temperatures slightly above that of the respective density maximum, i.e. T_{max}(H_{2}O) = 277.13 K and T_{max}(D_{2}O) = 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 nbutane, Π_{max} ≈ 190 MPa at an external pressure of about 150 MPa. For CO_{2}, Π_{max} ≈ 345 MPa at an external pressure of about 235 MPa. Note, however, that the statisticalmechanical 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 liquidstate physical chemistry by Scatchard^{163 } 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 perfectgas state. Expanding van Laar's ideas on nonelectrolyte solubility,^{166 } Hildebrand^{167,168 } and Scatchard^{163 } formulated the basic concepts of regular solution theory^{1}, 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_{vap}H^{*}(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 pressureexplicit 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 threeparameter corresponding states theorem (CST),^{175–178 } Pang and McLaughlin^{179 } 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 (δ^{2}/P_{c})^{(0)}, (δ^{2}/P_{c})^{(1)} and (δ^{2}/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 statisticalmechanical expressions.^{165 } Noting that the perfectgas 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 purefluid number density, L is the Avogadro constant^{2} and u(r) denotes the pairpotential energy (pairwise additivity of the potential energy). The pairdistribution 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 purefluid 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–temperaturecomposition relations that would eventually lead to reliable PVTx EOSs, applicable to both gaseous and liquid phases. These EOSs relate the variables in either a volumeexplicit form, nV = V ^{t}(T, P, n_{1}, n_{2},…), as does the virial equation in pressure, eqn (1.54), or in a pressureexplicit form, P = P(T, nV, n_{1}, n_{2},…), 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 gasphase 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 potentialenergy 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 ffunction. Eqn (1.178) provides access to the fundamentally important pair potentialenergy 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 LennardJones in 1924 and connected with gas viscosities,^{189 } the EOSs of real gases,^{190 } Xray measurements on crystals^{191 } and quantum mechanics.^{192 } The most common form of the LennardJones (12,6) function is^{192 }
where σ = 2^{−1/6}r_{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 realgas 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 zeropressure value of φ is thus given by
and integration between a suitable reference temperature T_{ref} and T yields^{196 }
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 Wormald^{194 } in the temperature range 313–413 K and derived values of φ_{0} were compared with results from the 1984 NBS/NRC steam tables,^{197 } with data of Hill and MacMillan^{198 } and with values derived from the IAPWS95 formulation for the thermodynamic properties of water.^{12 }
The isothermal pressure dependence of U of a constantcomposition 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 constantcomposition 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 highpressure research,^{136–140,199–206 }eqn 189 and (1.190) are particularly interesting. For instance, the pressure dependence of C_{P} of a constantcomposition fluid may be determined either from PVT data alone or by highpressure 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 constantcomposition 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
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 constantcomposition 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 v_{0} = v_{0}(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., v_{0} may be treated as an intensive thermodynamic equilibrium property^{32,138,207–215 } related to β_{S}viaeqn (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 v_{0} (for details, consult the classical monograph of Herzfeld and Litovitz^{212 }). 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 constantcomposition 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 secondorder 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 coworkers,^{226,227 } Harrison and MoelwynHughes^{228 } and Nývlt and Erdös.^{229 }
Burlew's piezothermometric method^{230 } 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.
1.3.2 Residual Properties, Fugacities and Fugacity Coefficients
The perfect gas (ideal gas) is a hypothetical substance with the following three defining characteristics:
Its constituent molecules exert no forces on each other (noninteracting molecules), hence the total intermolecular potential energy is independent of the positions of the molecules and may be taken to be zero.
Because of their unimpeded movement the molecules possess kinetic energy and there is temperature.
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 perfectgas 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 realgas 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 realfluid behaviour. Note, that in the limit of vanishingly small pressure, realgas behaviour in many respects approaches perfectgas 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 realfluid properties with perfectgas 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 zeropressure 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 nonzero 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 pairpotential 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 singlephase pure fluid or constantcomposition 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 perfectgas 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 perfectgas state is hypothetical except in the zeropressure limit, where the perfectgas EOS eqn (1.227) is valid (that is, for this real perfectgas 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 zeropressure 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 constantpressure 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's^{3}
threeparameter correspondingstates 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 threeparameter CST correlation
Z^{(0)} represents the simplefluid 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 twoparameter CST correlation suffices). Z^{(1)} represents the nonsimplefluid 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 Pitzertype 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 correspondingstates scaling parameters of the mixture, that is, by essentially empirically averaging purecomponent parameters T_{c,i}, P_{c,i} and ω_{i} to obtain pseudocritical temperatures T_{pc}, pseudocritical pressures P_{pc} and pseudoacentric factors ω_{p} referring to the mixture. This is accomplished by using recipes known as mixing rules. Thus, a threeparameter CST correlation for the compression factor Z of the mixture, in the onefluid approximation, may be written as
where the pseudoreduced temperature T_{pr} and the pseudoreduced pressure P_{pr} are defined by
respectively.
The simplest set of mixing rules for pseudocritical parameters are those of Kay.^{236 } They are defined as mole fractionweighted sums of the purecomponent values and so is ω_{p}, i.e.
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):
where T_{c,ii} and T_{c, jj} denote the critical temperatures of the pure components i and j, respectively, P_{c,ii} and P_{c,jj} are their critical pressures, respectively, and for the evaluation of the cross parameters T_{c,ij} and P_{c,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 vdWtype equations, are pressure explicit.^{6,8,55,63,68,237–241 } With T and V (and for mixtures, of course, also the composition {x_{i}}) being the natural independent (canonical) variables, the focus is on the Helmholtz energy as generating function. Thus, for a singlephase pure fluid or constantcomposition mixture one may also define molar residual properties in (T, V, {x_{i}}) space, i.e. molar isochoric (isometric) residual properties:
To distinguish them from isobaric residual properties, they are indicated throughout by a superscript lowercase r. Again, the Ms denote molar properties of any extensive thermodynamic property nM(T, nV, {x_{i}}) of the fluid, for instance U, H, S, G or F, and M ^{pg}(T, V, {x_{i}}) is the corresponding molar property of the fluid in its hypothetical perfectgas 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 realfluid state and the perfectgas 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:
With the availability of a pressureexplicit EOS, molar isochoric residual properties are the properties of direct interest. In complete analogy with eqn (1.238), we have
Again, experimental evidence indicates that for the pivotal properties internal energy and entropy, the infinitevolume terms (zeropressure terms) can be set equal to zero (at constant T and {x_{i}}), that is,
Thus, for the molar isochoric residual internal energy U ^{r}(T, V, {x_{i}}) we obtain, in conjunction with , and eqn (1.255),
and for the molar isochoric residual enthalpy H^{r}(T, V, {x_{i}}) we obtain, with , , and eqn (1.240),
For the molar isochoric residual entropy S^{r}(T, V, {x_{i}}), we obtain, in conjunction with , and eqn (1.255),
Since C_{V} of a real gas approaches C$Vpg$ for V → ∞ (i.e. for P → 0), together with and , see eqn (1.189), the molar isochoric residual constantvolume heat capacity C$Vr$(T, V, {x_{i}}) may be calculated via
In addition, we have
Equivalent, albeit slightly different, expressions are presented in the Appendix. For the computation of values M of any thermodynamic fluid property,
is used.
The two types of residual functions, i.e. M ^{R} (T, P, {x_{i}}) and M ^{r} (T, V, {x_{i}}), are rigorously related:^{164 }
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 {x_{i}} and they are therefore not related by the perfectgas law: the lower integral limit denotes the gas pressure P = RT/V for which the molar volume of the perfectgas mixture has the same value V as that of the real mixture at T and {x_{i}}. Alternatively, we have
Note that V ^{r}(T, V, {x_{i}}) and P ^{R}(T, P, {x_{i}}) are identically zero.
Since at constant composition the perfectgas properties U ^{pg}, H ^{pg}, C$Ppg$ and C$Vpg$ are all functions of temperature only, i.e. the firstlaw properties are independent of pressure and of volume, the equality
holds for M = U, H, C_{P} and C_{V}. In contradistinction, the secondlaw perfectgas properties S^{pg}, G^{pg} and F ^{pg} are functions of temperature and they do depend on pressure:
and they do depend on volume:
Hence, by virtue of eqn (1.262) or eqn (1.263), respectively, the following relations between the residual secondlaw properties are obtained:
An important property in solution chemistry is the fugacity coefficient ϕ$i\pi $(T, P, {x$i\pi $}) of component i in solution in phase π.^{242 } It is related to the molar isobaric residual chemical potential, which is obtained by applying the partial molar derivative prescription eqn (1.114) to the expression for the molar isobaric residual Gibbs energy of the mixture in any phase π, taking into account that for the model perfectgas mixture we have
Hence
and the partial molar isobaric residual Gibbs energy/the isobaric residual chemical potential of component i in solution in phase π reads
As shown later, eqn (1.274) provides a rigorous basis for the definition of the fugacity coefficient ϕ$i\pi $ (T, P, {x$i\pi $ }) 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 perfectgas (idealgas) systems, while avoiding the troublesome behaviour of the chemical potential when either P or x$i\pi $ 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:
which becomes for a pure perfect gas with V $ipg,*$ = RT/P
we keep the simple structure of eqn (1.276) and, following Lewis, replace P with a new function f$i*$ that makes, by definition, this differential relation generally valid. Hence, as the first part of the definition, we have
where f$i\pi ,*$ = f$i\pi ,*$ (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
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 f$i\u2009\pi ,*$ (T, P). For an isothermal change of state, botheqn 275 and (1.277) are applicable, hence we may write^{55 }
Integration from P = 0 to the pressure of interest P, in conjunction with eqn 1.278, yields
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 perfectgas state to the state of i in the realfluid state (phase π) at the same pressure, in conjunction with eqn (1.278), yields
where G$iR,\pi ,*$ (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 lefthand side of eqn (1.280) and the righthand side of eqn (1.281) as the argument of the logarithm is a new property and is called the fugacity coefficient, ϕ$i\pi ,*$ (T, P), of pure component i in phase π. By definition
or, perhaps more convenient,
Since G$iR,\pi ,*$ (T, P) is a property (a state function) of pure i, so the fugacity coefficient ϕ$i\pi ,*$ (T, P)/the fugacity f $i\pi ,*$ (T, P) is a property (a state function) of pure i in phase π.
For a pure perfect gas, V $iR,pg,*$ = 0 or, equivalently, G$iR,pg,*$ = 0, for all temperatures and pressures and therefore
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)
where f $i\pi $ = f $i\pi $ (T, P, {x$i\pi $ }) is called the component fugacity of substance i in a real solution phase π with composition {x$i\pi $ }. In the literature, a frequently used special notation identifies fugacities and fugacity coefficients of components in solution by a circumflex (^) to distinguish them from purefluid properties and from partial molar properties. Since the partial molar Gibbs energy G$i\pi $(T, P, {x$i\pi $ }) = µ$i\pi $ (T, P, {x$i\pi $ }) can be regarded as a property (a state function) of dissolved i, so the component fugacity f $i\pi $ 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 perfectgas mixture, we demand that
that is
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 $i\pi $(T, P, {x$i\pi $ }) of component i in solution in phase π. General integration of eqn (1.285) at constant T, P and {x$i\pi $ } from the perfectgas state to the real state of interest, i.e. component i in solution, results in
The difference on the lefthand side of eqn (1.288) is the partial molar residual Gibbs energy G$iR,\pi $ (residual chemical potential µ$iR,\pi $ ) 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 righthand side of eqn (1.288) as the argument of the logarithm is a new property and is called the fugacity coefficient, ϕ$i\pi $ (T, P, {x$i\pi $ }), of component i in solution in phase π:
or, perhaps more convenient,
For a component i in a perfectgas mixture, G$iR,pg$ = µ$iR,pg$ = 0 and therefore ϕ$ipg$ = 1.
Eqn (1.286) guarantees recovery of the expression for a perfectgas 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
Since eqn (1.291) is generally valid, it must also hold for a perfectgas mixture. In conjunction with eqn (1.284) and eqn (1.286), we have
and with the summability relation eqn (1.118)
we obtain
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 fractionweighted sum of the molar purecomponent 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
where M may represent, for instance, G, F, S, H, C_{P} and V, and expressions for some molar property changes pertaining to perfectgas mixtures, in addition to eqn (1.294), are as follows:
The entropy change of eqn (1.297), i.e., is called the ideal molar entropy of mixing; it is always positive.