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Chemical thermodynamics is of pivotal importance in chemistry, physics, the geosciences, the biosciences and chemical engineering. It is a highly formalised scientific discipline of enormous generality, providing a mathematical framework of equations (and a few inequalities) which yield exact relations between macroscopically observable thermodynamic equilibrium properties and restrict the course of any natural process. This review focuses on internal energy and enthalpy of nonelectrolyte liquids, vapours and gases, either pure or mixed (chemically non-reacting). After presenting the basic postulates, i.e., the first and the second law, the fundamental property relations in the internal energy representation and in the equivalent entropy representation are given. Alternative primary functions, such as enthalpy, Gibbs energy and Massieu function are introduced via Legendre transformations, together with the corresponding alternative forms of the fundamental property relations. Maxwell relations and practically important equations for constant-composition fluids are considered, the focus being on the temperature dependence and the pressure dependence of the internal energy and enthalpy, and relations involving heat capacities as well as the thermodynamic sound speed follow. The concepts of property changes of mixing and of excess properties for liquid multicomponent mixtures are introduced, and a few selected empirical correlations describing the composition dependence of excess molar properties, such as a generalised Kohler equation, are presented.

Life is girt all round with a zodiac of sciences, the contributions of men who have perished to add their point of light to our sky.

Ralph Waldo Emerson, Representative Men. Seven Lectures: I. Uses of Great Men, The Riverside Press, Cambridge, Mass., USA (1883).

This monograph is concerned with internal energy and enthalpy and related properties of fluids, pure and mixed, and their role in the physico-chemical description of systems ranging from pure rare gases to proteins in solution. In this introductory Chapter 1, I shall only consider nonreacting fluid equilibrium systems of uniform temperature T and pressure P (i.e., systems in thermal, mechanical and diffusional equilibrium) characterised by the essential absence of surface effects and extraneous influences, such as electric fields. However, the influence of the earth's gravitational field is omnipresent: though usually ignored, it becomes important near a critical point. Under ordinary conditions, the molar volumes V (or specific volumes V/mm, where mm denotes the molar mass) of homogeneous fluids in equilibrium states are functions of T, P and composition only. Such systems are known as PVT systems or simple systems. However, the generality of thermodynamics makes it applicable to considerably broader types of systems by adding appropriate work terms, i.e., products of conjugate intensive and extensive variables, such as surface tension and area of surface layer. Finally, there is a caveat concerning idealised concepts for systems and processes, such as isolated systems, isothermal and reversible processes, to name but a few. Fortunately, they can be well approximated experimentally, and while classical thermodynamics only treats the corresponding limiting cases, the ensuing restrictions are not severe: values of thermodynamic quantities obtained with different experimental techniques are expected to agree within experimental error. Classical thermodynamics deals only with measurable equilibrium properties of macroscopic systems. It is a formalised phenomenological theory of enormous generality in the following sense:

  • Thermodynamic theory is applicable to all types of macroscopic matter, irrespective of its chemical composition and independent of molecule-based information, i.e., systems are treated as “black boxes” and the concepts used ignore microscopic structure, and indeed do not need it.

  • Classical thermodynamics does not allow ab initio prediction of numerical values for thermodynamic properties. It provides, however, a mathematical network of equations (and a few inequalities) that yields exact relations between measurable equilibrium quantities and restricts the behaviour of any natural process.

The scope of chemical thermodynamics was succinctly summarised by McGlashan:1 

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: augmenting the formal framework of chemical thermodynamics with molecule-based models of material behaviour, i.e., by using concepts from statistical mechanics, experimental thermodynamic data contribute decisively towards a better understanding of molecular interactions, and lead to improved descriptions of macroscopic systems. This field of molecular thermodynamics (the term was coined by Prausnitz2  more than four decades ago) is of great academic fascination, and is indispensable in (bio-)physical chemistry and chemical engineering. Its growth has been stimulated by the increasing need for thermodynamic property data and phase equilibrium data2–26  in the applied sciences, and it has profited from advances in experimental techniques,27–33  from modern formulations of chemical thermodynamics,34–40  from advances in the theory of fluids in general41–48  and from advances in computer simulations of model systems.49–52  In Subsection 1.2 I shall present thermodynamic fundamentals of relevance for the book's topic, while 1.3 is devoted to derived thermodynamic properties and relations of relevance for many of the book chapters. A few comments on nomenclature, a brief outlook and concluding remarks will be given in Subsection 1.4.

Experiments, molecule-based theory and computer simulations are the three pillars of science.53  Experiments provide the basis for inductive reasoning, known informally as bottom-up reasoning, which, after amplifying, logically ordering and generalising our experimental observations, leads to hypotheses and then theories, and thus to new knowledge. In contradistinction, deduction, informally known as top-down reasoning, orders and explicates already existing knowledge, thereby leading to predictions which may be corroborated experimentally, or falsified:54  a theory has no value in science unless it is possible to test it experimentally. As pointed out by Freeman Dyson,55 

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

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. While this is the preferred approach, the model with the fewest assumptions may turn out to be wrong. More elaborate versions of Occam's razor have been introduced by modern scientists, and for in-depth philosophical discussions see Mach,56  Popper,54  Katz57  and Sober.58 

Thermodynamics is a physical science concerned with energy and its transformations attending physical and chemical processes. Historically, it was developed to improve the understanding of steam engines, the focus being on the convertibility of heat into useful work. The concepts of work, heat and energy were developed over centuries,59  and the formulation of the principle of conservation of energy has been one of the most important achievements in physics. 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.

A convenient way to present the fundamentals of the phenomenological theory of thermodynamics is the postulatory approach.34–40  A small number of postulates inspired by observation are assumed to be valid without admitting the existence of more fundamental relations from which the postulates could be deduced. The ultimate justification of this approach rests solely on its usefulness. Consider a homogeneous fluid in a closed PVT system, that is a system with a boundary that restricts only the transfer of matter (constant mass system). For such a system, the existence of a form of energy called total internal energy Ut is postulated, which is an extensive material property (for a definition see below) and a function of T, P and mass m or amount of substance n=m/mm. This designation distinguishes it from kinetic energy Ek and potential energy Ep which the system may possess (external energy).

Next, the first law of thermodynamics is introduced as a generalisation and abstraction of experimental results concerning energy conservation. It applies to a system and its surroundings: the law states that energy may be transferred from a system to its surroundings and vice versa, and it may be converted from one form into another, yet the overall quantity of energy is constant. Thus, for a closed PVT system, 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 (no changes in kinetic and/or potential energy), eqn (1.1) is a statement of the first law:

Equation 1.1

For a differential change it is written

Equation 1.2

Mathematically, dUt is an exact differential of the state function Ut: the change in the value of this extensive property,

Equation 1.3

depends only on the two states. On the other hand, δq and δw are inexact differentials, i.e., they represent infinitesimal amounts of heat and work: q and w are path functions. When integrated, δq and δw give finite amounts q and w, respectively. The notation used in the first law, eqn (1.1), asserts that the sum of the two path functions q and w always yields an extensive state function change ΔUt between two equilibrium state points, independent of the choice of path 1→2.

The exact (total) differential of a function f of n independent variables Xi,

Equation 1.4

is defined by

Equation 1.5
Equation 1.6

and the partial derivative ci and its corresponding variable Xi are known as being conjugate to each other. For a non-pathological function f the order of differentiation in mixed second derivatives is immaterial, thus yielding the Euler reciprocity relation

Equation 1.7

for any two pairs (ci, Xi) and (cj, Xj) of the conjugate quantities. Eqn (1.7) serves as a necessary and sufficient criterion of exactness. For n independent variables, the number of conditions to satisfy is n(n−1)/2. As shown later on, this is the number of Maxwell relations. If df(X1, X2, …, Xn) is exact, we have

Equation 1.8

independent of the integration path; in thermodynamics such a function f is called a state function.

As recommended by the International Union of Pure and Applied Chemistry,60  the sign convention for heat and work is that the internal energy increases when heat “flows” into the system, i.e., q>0, and work is done on the system, i.e., w>0. Eqn (1.1) does not provide an explicit definition of internal energy. In fact, there is no known way to measure absolute values of Ut: the internal energy of a system is an extensive conceptual property. Fortunately, only changes in internal energy are of interest, and these changes can be measured. Consider now a series of experiments performed essentially reversibly on a homogeneous constant-composition fluid (closed system) along different paths from an initial equilibrium state at (T1, P1) to a final equilibrium state at (T2, P2). All measurements show that the experimentally determined sum qrev+wrev is constant, independent of the path selected, as it must be provided the postulate of internal energy being a material property is valid. Thus, one has

Equation 1.9

as a special case of eqn (1.1), which for closed systems is generally applicable for reversible as well as irreversible processes between equilibrium states. In differential form this equation reads

Equation 1.10

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

Equation 1.11

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, though the possibility of falsification, of course, remains.54  No further “proof” exists beyond the experimental evidence.

Internal energy is a macroscopic equilibrium property, and the adjective internal derives from the fact that for homogeneous fluids in closed PVT systems, Ut is determined by the state of the system characterised by the independent variables T, P and m or n. Digressing from formal thermodynamics – where the existence of molecules is never invoked – it is instructive to briefly indicate the molecular interpretation of Ut. On a microscopic level, internal energy is associated with molecular matter, and for nonreacting systems at common T and P contributions to ΔUt essentially derive from changes of the molecular kinetic energy, the configurational energy and the intramolecular energy associated with rotational and vibrational modes.

Eqn (1.1) may be expanded to incorporate external energy changes resulting from changes of the closed system's macroscopic motion and/or position. Thus, when changes in kinetic energy and potential energy are to be considered, the first law becomes

Equation 1.12

Work done on the system with mass m by accelerating it from initial speed v1 to final speed v2 is

Equation 1.13

and the work done on the system by raising it from an initial height h1 to a final height h2 is

Equation 1.14

where g is the acceleration of gravity. Since the zero of potential energy can be chosen arbitrarily, only differences in potential energy are meaningful. A similar comment applies to kinetic energy.

Thermodynamic properties may be classified as being intensive or extensive. Properties that are independent of the system's extent are called intensive; examples are temperature, pressure and composition, i.e., the principal thermodynamic coordinates for homogeneous fluids. A property that is additive for independent, noninteracting subsystems is called extensive; examples are mass and amount of substance. The value of an extensive property is either proportional to the total mass m or to the total amount of substance n, and the proportionality factor is known as a specific property or as a molar property, respectively. Thus, for the extensive total internal energy we have

Equation 1.15

where u denotes the specific internal energy, an intensive property, or alternatively we have

Equation 1.16

where U denotes the intensive molar internal energy. Evidently, the quotient of any two extensive properties is an intensive property. Hence 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 when dividing by the total volume

Equation 1.17

As pointed out, classical thermodynamics is concerned with macroscopic properties and with relations among them. No assumptions are made 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. However, since we do have reliable theories involving molecular properties and interactions, using appropriate molecule-based models statistical mechanics allows the calculation of macroscopic properties. Essentially because of this connection, molar properties are used together with the appropriate composition variable, the mole fraction xi of component i of a homogeneous system, i.e., of a phase:

Equation 1.18

where is the total amount of substance in the phase, and ni is the amount of component i; , and for a pure fluid xi=1. Of course, use is limited to systems of known molecular composition. Thus, if Mt is taken to represent an extensive total property, such as Ut, of a homogeneous system, the corresponding intensive molar property M is defined by

Equation 1.19

The overall molar property of a closed multiphase system, with p equilibrium phases α, β, …, is

Equation 1.20

Here, nαMαMt,α is a total property of phase α, etc., is the total amount of substance, is the total amount of phase p, and nip is the amount of i in phase p. The composition of, say, phase α, is characterised by the set of mole fractions {xiα},

Equation 1.21

, and xiα=1 for a pure fluid i. If the use of masses is preferred, replace niα by miα and nα by mα to obtain the analogous equation defining mass fractions wiα in phase α,

Equation 1.22

with , and wiα=1 for pure i. The total mass of phase α is , where miα=niαmm,i is the mass of component i with molar mass mm,i. Thus,

Equation 1.23

The thermodynamic state of a homogeneous equilibrium fluid (closed PVT system) is specified by the thermodynamic coordinates T, P and the set of compositional variables {xi}. For closed heterogeneous equilibrium systems consisting of several phases α, β, …, each being a PVT system, the intensive thermodynamic state is specified by T, P and {xiα}, {xiβ},…. Total extensive property values of such a system are the sum of the total extensive property values associated with each one of the p constituent phases. Hence, for any equilibrium state of a closed multiphase system, according to eqn (1.20) the overall total internal energy is

Equation 1.24

For a change between two equilibrium states of such a closed multiphase system, experimental results for the change of the overall internal energy are independent of the process path, analogous to the results reported for single-phase fluids in closed equilibrium systems.

Two additional property changes are revealed by systematic experiments on closed PVT systems.

One is already known, i.e. ΔVt, but could be also demonstrated as follows. The work caused by a reversible volume change is given by

Equation 1.25

where δwrev is an inexact differential. Multiplication by 1/P serving as an integrating factor yields an exact differential,

Equation 1.26

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

Equation 1.27

Similarly, careful evaluation of systematic experiments reveals that while qrev is a path function, and δqrev is an inexact differential, multiplication by 1/T serving as an integrating factor yields an exact differential and identifies the so defined total entropy St as a state function:

Equation 1.28
Equation 1.29

These results may be summarised by an additional postulate asserting the existence of an extensive state function St called the total entropy, which for any closed equilibrium PVT system is a function of T, P and the amounts (or masses) of the constituent phases with composition {xiα}, {xiβ},…. The entropy change between two equilibrium states therefore depends solely on the difference between the values of St in these states and is independent of the path, regardless of whether the process is reversible or irreversible. However, in order to calculate the difference ΔSt a reversible path connecting the two equilibrium states must be selected.

In the special case of an adiabatic process q=0 and eqn (1.1) becomes

Equation 1.30

The adiabatic work wad is path independent and depends only on the initial and final equilibrium states. Measurements of wad are measurements of ΔUt, and calorimetric experiments confirm it, thereby providing the primary evidence that Ut is indeed a state function.

Analogous to the statement associated with the total internal energy, i.e. the first law eqn (1.1), eqn (1.29) 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, only entropy differences can be measured: the entropy St is also an extensive conceptual property. With the postulated existence of entropy, experimental results have led to the formulation of another restriction, besides energy conservation, applying to all processes: considering the system and its surroundings, the entire entropy change ΔSentiret associated with any process is given by

Equation 1.31

Eqn (1.31) is known as the second law of thermodynamics. It affirms that every natural (spontaneous) process proceeds in such a direction that the entire entropy change is positive, the limiting value zero being approached by processes approaching reversibility. This postulate completes the postulatory basis upon which classical equilibrium thermodynamics rests.

The first law 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 (1.10). With eqn (1.25) and (1.28) the basic differential equation for closed systems reads

dUt=TdStPdVt, or d(nU)=Td(nS)−Pd(nV).
Equation 1.32

where U, S, and V denote, respectively, the molar internal energy, the molar entropy, and the molar volume. Since Ut=nU, St=nS and Vt=nV are extensive state functions, eqn (1.32) is not restricted to reversible processes, though it was derived for the special case of such a process. It applies to any process in a closed multiphase PVT system causing a differential change from one equilibrium state to another, including mass transfer between phases or/and composition changes due to chemical reactions. However, for irreversible processes TdSt is not the heat transferred nor is −PdVt the work done: δqirrev<δqrev and δwirrev>δwrev, but, of course, dUt=TdStPdVt=δqirrev+δwirrev, since Ut is a state function. Application of eqn (1.32) to a closed single-phase system without chemical reactions indicates

Equation 1.33
Equation 1.34

where {ni} denotes constant amounts of components. With nS as the dependent property, the alternative basic differential equation for closed single-phase systems without chemical reactions is

Equation 1.35
Equation 1.36
Equation 1.37

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

nU=Ut=Ut (nS, nV, n1, n2, …),
Equation 1.38

or, equivalently, a fundamental equation in the entropy representation

nS=St=St (nU, nV, n1, n2, …).
Equation 1.39

Taking into account eqn (1.33), (1.34), (1.36) and (1.37), the corresponding differential forms of the fundamental equations, also known as the fundamental equations for a change of the state of a phase, or the fundamental property relations, or the Gibbs equations, are

Equation 1.40

in the internal energy representation, and, equivalently, in the entropy representation,

Equation 1.41

Eqn (1.40) and (1.41) apply to single-phase multicomponent PVT systems, either open or closed, where the ni vary because of interchange of matter with the surroundings, or because of chemical reactions within the systems, or both. The intensive parameter furnished by the partial derivative of nU with respect to ni is called the chemical potential of component i in the mixture:

Equation 1.42

It is an intensive conceptual state function. From eqn (1.41) we have

Equation 1.43

Hence the fundamental property relation eqn (1.40) can be written in a more compact form,

Equation 1.44

while the fundamental property relation eqn (1.41) becomes

Equation 1.45

Eqn (1.44) and (1.45) are fundamental because they specify all changes that can take place in PVT systems, and they form the basis of extremum principles predicting equilibrium states. The corresponding fundamental equations for single-phase multicomponent PVT systems in which the ni vary either because of interchange of matter with the surroundings, or because of chemical reactions within the system, or both, read in the energy representation

Equation 1.46

and in the entropy representation

Equation 1.47

However, I reiterate that in this introductory chapter only nonreacting simple fluid equilibrium systems will be considered.

Eqn (1.46) and (1.47) are also known as the integrated forms of the fundamental equations for a change of the state of a phase, or as primary functions, or as cardinal functions, or as thermodynamic potentials. They are obtained by integrating eqn (1.44) and (1.45), respectively, over the change in the amount of substance at constant values of the intensive quantities {T,−P,μi} or , respectively. Alternatively, eqn (1.46) and (1.47) can be regarded as a consequence of Euler's theorem which asserts that if f(z1, z2, …) is a homogeneous function of degree k in the variables z1, z2, …, i.e., if it satisfies for any value of the scaling parameter λ

f(λz1, λz2, …)=λkf(z1, z2, …),
Equation 1.48

it must also satisfy

Equation 1.49

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

Ut(λSt, λVt, λn1, λn2, …)=λUt (St, Vt, n1, n2, …),
Equation 1.50
St(λUt, λVt, λn1, λn2, …)=λSt (Ut, Vt, n1, n2, …),
Equation 1.51

use of eqn (1.49) with k=1 yields eqn (1.46) and (1.47), respectively. The corresponding variable sets, i.e., {nS, nV, n1, n2, …} for the energy representation and {nU, nV, n1, n2, …} for the entropy representation, are called the canonical or natural variables. All thermodynamic equilibrium properties of simple systems can be derived from these functions, and for this reason they are called primary functions or fundamental functions or cardinal functions. As indicated by eqn (1.33), (1.34) and (1.42), T, −P and μi are partial derivatives of Ut(nS, nV, n1, n2, …) appearing in the fundamental property relation in the energy representation, and are thus also functions of {nS, nV, n1, n2, …}. These homogeneous zeroth-order equations expressing intensive parameters in terms of independent extensive parameters, that is,

T=T(nS, nV, n1, n2, …),
Equation 1.52
P=P(nS, nV, n1, n2, …),
Equation 1.53
μi=μi(nS, nV, n1, n2, …),
Equation 1.54

are called general equations of state. A single equation of state does not contain complete information on the thermodynamic properties of the system. However, the complete set of these three equations of state is equivalent to the fundamental equation and contains all thermodynamic information. If two equations of state are known, the Gibbs–Duhem equation (see below) can be integrated to yield the third, which will contain, however, an undetermined integration constant. Analogous comments apply to the fundamental property relation in the entropy representation, see eqn (1.36), (1.37) and (1.43), yielding the corresponding general equations of state

Equation 1.55
Equation 1.56
Equation 1.57

For constant-composition fluids, and thus also for pure fluids, T=T (nU, nV, n1, n2, ), or explicitly resolved for the internal energy,

nU=Ut (T, nV, n1, n2, …).
Equation 1.58

This type of equation is known as the caloric equation of state. Clearly, by using eqn (1.56) and (1.58) we obtain either a pressure-explicit thermal equation of state

P=P (T, nV, n1, n2, …),
Equation 1.59

or a volume-explicit thermal equation of state

nV=Vt (T, P, n1, n2, …).
Equation 1.60

A well-known example of a volume-explicit thermal equation of state is the virial equation in pressure, and a well-known example of a pressure-explicit thermal equation of state is the van der Waals equation. Most equations of state in practical use are pressure-explicit.

In the fundamental property relations for an open single-phase PVT system in both the internal energy representation and the entropy representation, the extensive properties are the mathematically independent variables, while the intensive parameters are derived, which does not reflect experimental reality. The choice of nS and nV as independent extensive variables in eqn (1.44), and of nU and nV as independent extensive variables in eqn (1.45), is not convenient. In contradistinction, the conjugate intensive parameters are easily measured and controlled. Hence, in order to describe 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 by their conjugate intensive parameter(s) without loss of information. The appropriate generating method is the Legendre transformation.61–63  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.64 

Consider the exact differential expression (see eqn (1.5))

df(0)=c1dX1+c2dX2+c3dX3+⋯ +cndXn
Equation 1.61

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

f(0)=f(0) (X1, X2, X3, …, Xn),
Equation 1.62


Equation 1.63

Consider now the function obtained by subtracting the product of X1 with its conjugate partial derivative c1 from the base function f(0), eqn (1.62):

Equation 1.64

The total differential reads

Equation 1.65

and with eqn (1.61) one obtains

Equation 1.66

Comparison of eqn (1.61) with eqn (1.66) shows that the original variable X1 and its conjugate c1 have interchanged their roles. For such an interchange it suffices to subtract c1X1 from the base function to yield the first-order partial Legendre transform,

f(1)=f(1) (c1, X2, X3, …, Xn)=f(0)[c1],
Equation 1.67

which is frequently identifed by a bracket notation. This Legendre transform represents a new function with the independent variables {c1, X2, X3, } being the canonical (or natural) variables. Analogously, the second-order partial Legendre transform f(0)[c1, c2] is obtained via

Equation 1.68

forming the total differential,

Equation 1.69

and using eqn (1.61):

Equation 1.70


f(2)=f(2)(c1, c2, X3, …, Xn)=f(0)[c1, c2].
Equation 1.71

Partial Legendre-transformed functions f(p) of order p,

Equation 1.72

have a special property: since f(p) is known as a function of its n independent canonical variables {c1, …, cp, Xp+1, …, Xn}, the n quantities {X1, …, Xp, cp+1, …, cn} remaining from the original set of variables {X1, X2, X3, …, Xn} and their conjugates {c1, c2, c3, …, cn} in the exact differential expression (1.61) are obtained as appropriate partial derivatives of f(p). Specifically,

Equation 1.73
Equation 1.74

and thus

Equation 1.75

Eqn (1.46) suggests the definition of useful alternative energy-based primary functions related to nU and with total differentials consistent with eqn (1.44), but with canonical variables different from {nS, nV, {ni}}, while eqn (1.47) suggests the definition of useful alternative entropy-based primary functions related to nS and with total differentials consistent with eqn (1.45), but with canonical variables different from {nU, nV, {ni}}. The most popular alternative equivalent primary functions are the total enthalpy

Equation 1.76

the total Helmholtz energy

Equation 1.77

and the total Gibbs energy, a double Legendre transform,

nGnU[T, −P]=nUT(nS)+P(nV)=nHT(nS),
Equation 1.78

where U, H, F and G (and V) designate molar quantities. The positive sign of the P(nV) term of eqn (1.76) results from −P being the intensive parameter associated with nV, and not P, see eqn (1.34). The same comment applies to eqn (1.78). The alternative energy-based fundamental property relations for the enthalpy, the Helmholtz energy and the Gibbs energy are thus

Equation 1.79
Equation 1.80
Equation 1.81

with the associated canonical variables {nS, P, {ni}}, {T, nV, {ni}} and {T, P, {ni}}, respectively. Eqn (1.81) is of central importance in solution thermodynamics. Integration over the changes in the amount of substance in the fundamental property relations eqn (1.79) through (1.81), yields the integrated forms known as the alternative fundamental equations, or alternative primary functions, or alternative cardinal functions, or alternative thermodynamic potentials:

Equation 1.82
Equation 1.83
Equation 1.84

These alternative groupings may also be obtained from eqn (1.76)(1.78), respectively, by substituting for nU according to eqn (1.46).

Since eqn (1.79)–(1.81) are equivalent to eqn (1.44), we have

Equation 1.85

Division of eqn (1.46), (1.82)(1.84) by the total amount of substance n yields the corresponding molar functions:

Equation 1.86
Equation 1.87
Equation 1.88
Equation 1.89

For the special case of 1 mol of mixture, we have

Equation 1.90

which is less general than the fundamental property relation eqn (1.44) in an important aspect: while the ni are independent, mole fractions xi are constrained by , and hence by , thus precluding some mathematical operations which are acceptable for eqn (1.44). Analogous comments apply to the other fundamental property relations. The fundamental property relations/primary functions presented so far are equivalent, though each is associated with a different set of canonical variables. The selection of any primary thermodynamic function/fundamental property relation depends on deciding which independent variables simplify the problem to be solved. In physical chemistry and chemical engineering the most useful variables are {T, P, {ni}} and {T, nV, {ni}}, since they are easily measured and controlled. Hence, the total Gibbs energy Gt(T, P, {ni}) and the total Helmholtz energy Ft(T, nV, {ni}) are important. Of the alternative expressions for the chemical potential, eqn (1.85), the preferred one is

Equation 1.91

Partial derivatives of a total property with respect to ni at constant T, P and nj≠i are ubiquitous in solution thermodynamics, hence a survey of relevant definitions and relations is presented below. Denoting an intensive molar mixture property by M(T, P, {xi}), the corresponding extensive total property of the solution phase is

Mt(T, P, n1, n2, …)=nM(T, P, {xi}),
Equation 1.92

where n is the total amount of substance contained in the phase, either closed or open. The total differential of any extensive property of a homogeneous fluid is given by

Equation 1.93

where the subscript {ni} indicates that all amounts of components i and thus the composition {xi} are/is held constant. The summation term of eqn (1.93) is important for the thermodynamic description of mixtures of variable composition and extent. The dervatives are response functions known as partial molar properties Mi and defined by

Equation 1.94

Partial molar properties are intensive state functions, and depending on M they are either measurable or conceptual quantities. With eqn (1.94), the exact differential eqn (1.93) can be written in a more compact form,

Equation 1.95

The last term gives the differential variation of nM caused by amount-of-substance transfer across phase boundaries, or by chemical reactions, or both. nM of a phase is homogeneous of the first degree in the amounts of substance, hence Euler's theorem, eqn (1.49), yields

Equation 1.96

Division by the total amount of substance gives the molar property

Equation 1.97

Eqn (1.96) and (1.97) are known as summability relations. Since M(T, P, {xi}) is an intensive property, the partial molar property Mi(T, P, {xi}) is also intensive. Denoting a molar property of pure i by , in general,

Equation 1.98

However, from eqn (1.97),

Equation 1.99

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

Equation 1.100

From eqn (1.96) the total differential of Mt=nM of a homogeneous PVT fluid is

Equation 1.101

while eqn (1.95) provides an alternative expression for d(nM). Thus, it follows that

Equation 1.102

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

Equation 1.103

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

Equation 1.104

which shows the constraints on composition changes. It is important to note that a partial molar property Mi is an intensive property referring to the entire mixture: it must be evaluated for each mixture at each composition of interest. However, a partial molar property defined by eqn (1.94) can always be used to provide a systematic formal subdivision of the extensive property nM into a sum of contributions of the individual species i constrained by eqn (1.96), or a systematic formal subdivision of the intensive property M into a sum of contributions of the individual species i constrained by eqn (1.97). 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.

To summarise: the following general system of notation is used throughout this chapter:

  • a total property of a single-phase multicomponent solution, such as the volume, is represented by the symbol Mt, or alternatively by the product nM, with ;

  • a molar property of a single-phase multicomponent solution is represented by the symbol M;

  • pure-substance properties are characterised by a superscript asterisk (*) and identified by a subscript, i.e., is a molar property of pure component i = 1, 2, … ;

  • partial molar properties referring to a component i in solution are identified by a subscript alone, i.e., Mi, i = 1, 2, … .

Additional aspects may be indicated by appropriate superscripts/subscripts attached by definition.

After this excursion into partial molar properties, I introduce the remaining Legendre transforms. A primary function which arises naturally in statistical mechanics is the grand canonical potential. It is the double Legendre transform of nU where simultaneously the total entropy is replaced by its conjugate intensive variable, the temperature, and the extensive amount of substance by its conjugate intensive variable, the chemical potential:

Equation 1.105

The corresponding fundamental property relation is

Equation 1.106

with canonical variables {T, nV, {μi}}. The integrated, alternative form is

Equation 1.107

The remaining two primary functions

Equation 1.108
Equation 1.109

are rarely used and, to the best of my knowledge, have not received generally accepted separate symbols or names. The corresponding fundamental property relations are

Equation 1.110
Equation 1.111

with canonical variables {nS, nV, {μi}} and {nS, P, {μi}}. The integrated, alternative forms are

Equation 1.112
Equation 1.113

The complete Legendre transform, i.e., the transform of order p=n, vanishes identically, which follows directly from the definition. The complete transform of the internal energy replaces all extensive canonical variables by their conjugate intensive variables, thus yielding the null-function

Equation 1.114

and correspondingly

Equation 1.115

with canonical variables {T, P, {μi}}. Eqn (1.115) is a form of the Gibbs–Duhem equation.

For an exact differential df(0) with n independent variables Xi and n conjugate partial derivatives ci, see eqn (1.61) and (1.63), respectively, partial Legendre transforms of order p with 1≤pn−1, involve p conjugate pairs {ci, Xi}, and the number of such transforms is given by the number of combinations without repetition:

Equation 1.116

The total number of partial Legendre transforms, i.e. the total number of alternatives, is given by

Equation 1.117

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

Equation 1.118

Treating in the energy representation eqn (1.46) as a single term (thus n=3), the entire number Nt of equivalent thermodynamic potentials, i.e., nU, nH, nF, nG, nJ, nX, nY, and therefore the number of corresponding equivalent fundamental property relations for PVT systems in the energy representation, is seven:

Equation 1.119

With the fundamental property relation corresponding to the null-function, i.e., the Gibbs–Duhem equation eqn (1.115), we have a total of eight equivalent fundamental property relations,

Equation 1.120

that is, eqn (1.44) plus eqn (1.79) through (1.81), (1.106), (1.110), (1.111) and (1.115).

Partial Legendre transformations of the fundamental equation in the entropy representation, nS=St (nU, nV, {ni}), eqn (1.47), resulting in the replacement of one or more extensive variables by the corresponding conjugate intensive variable(s) 1/T, P/T and μi/T, respectively, yield primary functions known as Massieu–Planck functions, whose total differentials are compatible with eqn (1.45). Interestingly, such a Legendre transform of the entropy was already reported by Massieu65  in 1869, and thus predates the Legendre transforms of the internal energy reported by Gibbs in 1875 (see Callen38 ). Again, treating in eqn (1.47) as a single term (thus, n=3), with eqn (1.119) we have seven equivalent primary functions (including nS)) plus the null-function, and therefore eight equivalent fundamental property relations for PVT systems in the entropy representation: eqn (1.45) plus seven alternatives, including the relation presented below, the entropy-based Gibbs–Duhem equation. A first-order transform with respect to P/T, is an unnamed Massieu-Planck function

Equation 1.121

and the corresponding entropy-based fundamental property relation is

Equation 1.122

with canonical variables . The integrated, alternative form reads

Equation 1.123

The Massieu function is defined by

Equation 1.124

and the corresponding entropy-based fundamental property relation is

Equation 1.125

with canonical variables . The integrated, alternative form reads

Equation 1.126

The Planck function is a second-order Legendre transform,

Equation 1.127

and the corresponding entropy-based fundamental property relation reads

Equation 1.128

with canonical variables . The integrated, alternative form is

Equation 1.129

Another second-order Legendre transform is the Kramer function

Equation 1.130

with the corresponding entropy-based fundamental property relation

Equation 1.131

and canonical variables . The integrated, alternative form is

Equation 1.132

The first-order Legendre transform

Equation 1.133

is unnamed, and the corresponding entropy-based fundamental property relation reads

Equation 1.134

with canonical variables . The integrated, alternative form is

Equation 1.135

Finally, we have the unnamed second-order Legendre transform

Equation 1.136

the corresponding entropy-based fundamental property relation,

Equation 1.137

with canonical variables , and its integrated alternative form

Equation 1.138

Though not always immediately recognised, the (molar) Massieu-Planck functions are simply related to the (molar) thermodynamic potentials:

Equation 1.139
Equation 1.140
Equation 1.141
Equation 1.142
Equation 1.143
Equation 1.144

The complete Legendre transform is identically zero, thus yielding the null-function

Equation 1.145

in the entropy representation, and correspondingly, the fundamental property relation

Equation 1.146

with canonical variables . Division by n yields

Equation 1.147

which might be called a form of the entropy-based Gibbs–Duhem equation.

At constant composition, the fundamental property relations corresponding to Legendre transforms excluding the chemical potentials are readily obtained, and for one mole of a homogeneous constant composition fluid the following four energy-based property relations apply:

Equation 1.148
Equation 1.149
Equation 1.150
Equation 1.151

They are exact differentials, hence

Equation 1.152
Equation 1.153
Equation 1.154
Equation 1.155

These relations establish the link between the independent variables S, V, P, T and the energy-based functions U, H, F, G. For simplicity's sake the subscript {xi} has been omitted.

Frequently we are interested in characterising the response of properties of homogeneous constant-composition fluids to changes in the respective canonical variables. Clearly, besides first-order partial derivatives, second-order partial derivatives will be important. In general, for a property f=f(X1, X2, …, Xn) with n independent variables Xi, the exact differential

Equation 1.156

has n first-order partial derivatives (d=1) with n corresponding operators . The number of direct second-order partial derivatives (d=2), i.e., of type , is given by Nds=n, and the number Nms of mixed second-order partial derivatives , ij, each with operators and , is given by the number of variations without repetition, Vdn:

Equation 1.157

The total number Ns=Nds+Nms of second-order partial derivatives without restricting indices is given by the number of variations with repetition, dn:

Equation 1.158

According to the Euler reciprocity relation eqn (1.7),

Equation 1.159

which provides the basis for the important class of thermodynamic equations known as Maxwell relations discussed below. The total number NMw of Maxwell relations is given by the number of combinations without repetition, Cdn: how many ways exist for picking d=2 different operators ∂/∂Xi, ∂/∂Xj out of n operators ∂/∂X1, ∂/∂X2, …, ∂/∂Xn, when order is not important:

Equation 1.160

Of course, NMw=Nms/2.

For a closed constant-composition phase, the fundamental property relation eqn (1.44) becomes (see also eqn (1.32))

Equation 1.161

The first-order partial derivatives of nU with respect to nS and nV are given by eqn (1.33) and (1.34), respectively, and the Nds=2 corresponding direct second-order partial derivatives are

Equation 1.162
Equation 1.163

The mixed second-order partial derivatives, see eqn (1.157), are

Equation 1.164
Equation 1.165

and applying the Euler reciprocity relation, eqn (1.159), we have NMw=1 Maxwell relation, see eqn (1.160) or NMw=Nms/2=V22/2 = 1:

Equation 1.166

In the second-order derivatives used above, the variables kept constant are extensive quantities. The derivatives of eqn (1.162) through (1.164) are usually presented via their reciprocals. Since all apply to closed constant-composition phases, we may drop the subscript {ni} and, by dividing by n, use them in terms of intensive molar properties M instead of extensive total properties nM:

Equation 1.167
Equation 1.168
Equation 1.169

Here, CV denotes the molar heat capacity at constant volume (the molar isochoric heat capacity),66 

Equation 1.170

the isentropic compressibility βS is defined by67 

Equation 1.171

and the isentropic expansivity αS67  is defined by

Equation 1.172

with the mass density ρ of the phase being given by

ρ(T, P, {xi})≡m/[nV(T, P, {xi})].
Equation 1.173

The Maxwell relation eqn (1.166) now becomes

Equation 1.174

This set {CV, βS, αS} of second-order partial derivatives associated with the fundamental property relation in the energy representation, eqn (1.44), where the canonical variables are the extensive quantities nS, nV and {ni}, may be designated the fundamental set for homogeneous PVT fluids of constant composition.

However, with respect to applicability, the second-order derivatives associated with the alternative energy-based fundamental property relation eqn (1.81) are experimentally more useful descriptors of material properties. For a closed constant-composition phase we have

Equation 1.175

with the intensive canonical variables T and P as independent parameters. The first-order partial derivatives of nG with respect to T and P are

Equation 1.176

respectively. The 2 direct second-order partial derivatives thus become

Equation 1.177
Equation 1.178

and the 2 mixed second-order partial derivatives are

Equation 1.179
Equation 1.180

Applying the Euler reciprocity relation, eqn (1.159), we obtain the Maxwell relation

Equation 1.181

Dropping again the subscripts {ni} and using intensive molar quantities, we have

Equation 1.182
Equation 1.183
Equation 1.184


Equation 1.185

represents the molar heat capacity at constant pressure (the molar isobaric heat capacity),66 

βT≡−V−1(∂V/∂P)T−1 (∂ρ/∂P)T
Equation 1.186

denotes the isothermal compressibility,67  and the isobaric expansivity67  is defined by

αPV−1(∂V/∂T)P=−ρ−1 (∂ρ/∂T)P.
Equation 1.187

The Maxwell relation eqn (1.181) now becomes, in analogy to eqn (1.174),

Equation 1.188

This set {CP, βT, αP} of second-order partial derivatives associated with the alternative energy-based eqn (1.81) involving the intensive variables T and P, may be designated the alternative set for homogeneous PVT fluids of constant composition. Incidentally, this set has been suggested by Callen38  for use in his procedure for the “reduction of derivatives” in single-component systems. Though straightforward in principle, in practice this method can become intricate.

Additional second-order partial derivatives may be obtained via the alternative energy-based fundamental property relations eqn (1.79) and (1.80), respectively. For a homogeneous closed constant composition fluid, the alternative fundamental property relation involving the enthalpy is

Equation 1.189

where the canonical variables are the extensive variable nS and the intensive variable P. The first-order partial derivatives of nH with respect to nS and P are, respectively,

Equation 1.190

The 2 direct second-order partial derivatives thus become

Equation 1.191
Equation 1.192

and the 2 mixed second-order partial derivatives are

Equation 1.193
Equation 1.194

With the Euler reciprocity relation, eqn (1.159), we obtain the Maxwell relation

Equation 1.195

Evidently, onlyeqn (1.193) provides a new coefficient. Using again its reciprocal, dropping the subscripts {ni} and using intensive molar quantities, the Maxwell relation reads

Equation 1.196

Here, γS denotes the isentropic thermal pressure coefficient;66,67  it belongs to an alternative set of second-order partial derivatives, i.e., to {CP,1/βS,γS}.

For a closed constant-composition phase, the alternative fundamental property relation in terms of the Helmholtz energy, with canonical variables T (intensive) and nV (extensive), reads

Equation 1.197

The first-order partial derivatives of nF with respect to T and nV are, respectively,

Equation 1.198

The 2 second-order partial derivatives thus become

Equation 1.199
Equation 1.200

and the 2 mixed second-order partial derivatives are

Equation 1.201
Equation 1.202

Applying the Euler reciprocity relation, eqn (1.159), we obtain the Maxwell relation

Equation 1.203

Evidently, onlyeqn (1.201) provides a new coefficient. Dropping the subscripts {ni} and using intensive molar quantities, the Maxwell relation reads, in analogy to eqn (1.196),

Equation 1.204

Here, γV denotes the isochoric thermal pressure coefficient;66,67  it belongs to an alternative set of second-order partial derivatives, i.e., to {CV,1/βT,γV}.

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

Equation 1.205

the three mechanical coefficients are related as follows:

Equation 1.206
Equation 1.207

Additional useful relations for a constant composition phase may now be established systematically between members of the fundamental set and of the alternative sets of second-order partial derivatives. However, here I adopt another approach by placing the emphasis on discussing the responses of U, H, F, G, etc., to changes in T and P, or T and V, respectively, and introducing appropriate relations between second-order partial derivatives en route, whenever convenient. In view of the definitions of F and G, and eqn (1.155), the Gibbs–Helmholtz equations

Equation 1.208
Equation 1.209

are obtained. Simple mathematical transformations lead to the alternative forms

Equation 1.210
Equation 1.211
Equation 1.212
Equation 1.213

Eqn (1.212) suggests an alternative to the fundamental property relation eqn (1.81)via the dimensionless property G/RT, where R=8.3144598 J K−1 mol−1 is the molar gas constant:68 

Equation 1.214

and thus

Equation 1.215

Eqn (1.214) is of considerable utility. All terms have the dimension of amount-of-substance, and in contradistinction to eqn (1.81), the enthalpy rather than the entropy appears in the first term of the right-hand side of this exact differential, with benefits for discussing experimental results. An analogous equation may be derived involving the Helmholtz energy. Introducing the dimensionless property F/RT, the alternative to eqn (1.80) reads

Equation 1.216

and thus

Equation 1.217

In contradistinction to eqn (1.80), the internal energy rather than the entropy appears in the first term of the right-hand side of eqn (1.216).

The parallelism between equations involving molar quantities of constant-composition phases and equations involving corresponding partial molar quantities facilitates the formulation of new relations by analogy. This approach is valid whenever the properties appearing in any equation are linearly related. Consider, for instance, the alternative fundamental property relation eqn (1.81). The number of Maxwell relations for a solution characterised by n independent variables is given by eqn (1.160) with d=2, and it increases rapidly with the number of components. For a ternary solution there are n=5 independent variables {T, P, n1, n2, n3}, and

Equation 1.218

By inspection of the right-hand side of eqn (1.81) we find the Maxwell relation eqn (1.181), three Maxwell relations introducing the partial molar entropy Si (i=1, 2, 3), viz.

Equation 1.219

three Maxwell relations introducing the partial molar volume Vi, viz.

Equation 1.220

and finally three Maxwell relations relating chemical potentials, viz.

Equation 1.221

The partial property analogue to eqn (1.151) is

Equation 1.222

and the analogue to eqn (1.78) is

Equation 1.223


Equation 1.224

denotes the partial molar enthalpy. The molar heat capacity at constant pressure is defined by eqn (1.185), hence for the partial molar heat capacity it follows that

Equation 1.225

A Helmholtz-type equation analogous to eqn (1.212) involving partial molar properties reads

Equation 1.226

etc., etc. Euler's theorem, eqn (1.49), provides additional relations involving μj(T, P, {ni}):

Equation 1.227

(see also eqn (1.221)). Eqn (1.227) is known as the Duhem–Margules relation. Analogous equations apply to any partial molar property defined by eqn (1.94).

Maxwell equations often allow replacement of a difficult to measure derivative by a derivative which is easier to measure,1  preferably involving as experimental parameters T and P, or V, or amount density ρn≡1/V. Eqn (1.188) and (1.204) are particularly useful in EOS research, since they allow the determination of changes of entropy (a conceptual property) in terms of derivatives involving measurables. Maxwell relations form also part of the thermodynamic basis of the relatively new experimental technique known as scanning transitiometry.69,70 

For homogeneous constant-composition fluids, the volume dependence of U and the pressure dependence of H are conveniently derived viaeqn (1.208) and (1.209):

Equation 1.228
Equation 1.229

Both equations can be contracted to yield

Equation 1.230
Equation 1.231

(∂U/∂V)T is a useful property in solution chemistry and has been given the symbol

Π(T, P, {xi})≡(∂U/∂V)T,
Equation 1.232

and a special name, internal pressure. It may be determined at pressure P viaeqn (1.228) by measuring γV, or by using eqn (1.206). Π is related to the solubility parameter, and two chapters of this book are dedicated to these topics.

(∂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

Equation 1.233

It is related to the isenthalpic Joule–Thomson coefficient

Equation 1.234


Equation 1.235

The three quantities φ, μJT and CP of gases/vapours71–73  may be measured by flow calorimetry.

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

Equation 1.236

B is the second virial coefficient of the amount-density series, C is the third virial coefficient, and 〈P〉 is the mean experimental pressure. The zero-pressure value of φ is thus

Equation 1.237

and integration between a suitable reference temperature Tref and T yields74 

Equation 1.238

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 recently measured by McGlashan and Wormald72  in the temperature range 313 K to 413 K, and derived values of φ0 were compared with results from the 1984 NBS/NRC steam tables,75  with data of Hill and MacMillan,76  and with values derived from the IAPWS-95 formulation for the thermodynamic properties of water.7 

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

Equation 1.239

is obtained viaeqn (1.228) and the chain rule, and eqn (1.229) plus the chain rule yields

Equation 1.240

Turning now to the temperature derivatives of U and H, i.e., to the heat capacities of constant-composition fluids, I first recall that

Equation 1.241

and from eqn (1.228),

Equation 1.242

The molar heat capacity at constant pressure is defined by

Equation 1.243

and from eqn (1.229),

Equation 1.244

Finally we note that the isothermal compressibility may also be expressed as

Equation 1.245

and the isentropic compressibility as

Equation 1.246

In high-pressure research,66,67,69,70,77–84 eqn (1.242) and (1.244) are particularly interesting: the pressure dependence of CP of a constant-composition fluid may be determined either from PVT data alone, or by high-pressure calorimetry, or by transitiometry,69,70  or by measuring the speed of ultrasound v0 as a function of P and T,77,79,80,82–90  and the consistency of the experimental results can be ascertained in various ways.

I now present the functional dependence of U and S of a constant-composition fluid on T and V, and of H and S of such a fluid on T and P. Starting with

Equation 1.247
Equation 1.248

and replacing the derivatives viaeqn (1.170) and (1.228), and (1.167) and (1.204) yields

Equation 1.249
Equation 1.250

If T and P are selected as the independent variables, an entirely analogous procedure using eqn (1.185) and (1.229), and (1.182) and (1.188) gives

Equation 1.251
Equation 1.252


Equation 1.253
Equation 1.254
Equation 1.255

thus complementing eqn (1.174) and (1.196), and (1.188) and (1.204). We note that the difference between CP and CV depends on volumetric properties only, i.e., from eqn (1.248),

Equation 1.256

which yields, with eqn (1.167), (1.182), (1.204) and (1.206), the alternative relations

Equation 1.257
Equation 1.258
Equation 1.259

Since the compression factor Z is defined by

Equation 1.260

alternatively66,67,85  the difference is given by

Equation 1.261

The ratio of the molar heat capacities, κCP/CV, is accessible viaeqn (1.167) and (1.182) in conjunction with the chain rule:

Equation 1.262

According to the triple product rule

Equation 1.263

Thus, for homogeneous constant-composition fluids, we obtain the important relation

Equation 1.264

thereby establishing the ultrasonics connection.27,91–96  Using eqn (1.258) together with

Equation 1.265

where v0=v0(T, P, {xi}) is the low-frequency speed of ultrasound, leads to

Equation 1.266

which is one of the most important equations in thermophysics. At low frequencies and small amplitudes, to an excellent approximation (i.e., neglecting dissipative processes, such as those due to shear and bulk viscosity and thermal conductivity) v0 may be treated as an intensive thermodynamic equilibrium property27,77,79,80,82–94  related to βSviaeqn (1.265). Alternatively, by using the relations provided by eqn (1.153) we have

Equation 1.267
Equation 1.268

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

Equation 1.269

While sufficiently small amplitudes of sound waves are readily realised, sufficiently low frequencies f constitute a more delicate problem. Here, I mention only a few aspects in order to alert potential users that not all sound speed data reported in the literature are true thermodynamic data which can be used, say, with eqn (1.265) and (1.266). When sound waves propagate through molecular liquids, several mechanisms help dissipate the acoustic energy. Besides the classical mechanisms causing absorption, i.e., those due to shear viscosity and heat conduction (Kirchhoff-Stokes equation), 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 sound speed v(f) is larger than v0 (for details consult the monograph of Herzfeld and Litovitz91 ). At higher frequencies many liquids show sound speed dispersion,27,85,87,88,91–96  but particular care must be exercised when investigating liquids with molecules exhibiting rotational isomerism, where ultrasonic absorption experiments indicate rather low relaxation frequencies.

At temperatures well below the critical temperature,97–100 γV of liquids is large and the direct calorimetric determination of CV is not easy. It requires sophisticated instrumentation, as evidenced by the careful work of Magee at NIST,101,102  though it becomes more practicable near the critical point where γV is much smaller.

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

Equation 1.270
Equation 1.271

We note that heat capacities may be determined by measuring expansivities and compressibilities. Combining eqn (1.270), (1.271) and (1.258) yields

Equation 1.272
Equation 1.273

Eqn (1.272) establishes a link with Rayleigh–Brillouin light scattering.87,88,95,96,103  For liquid rare gases, the ratio of the integrated intensity of the central, unshifted Rayleigh peak, IR, and of the two Brillouin peaks, 2IB, is given by the Landau–Placzek ratio,

Equation 1.274

From eqn (1.266) the difference between βT and βS may be expressed as

Equation 1.275

while for the difference of the reciprocals we have

Equation 1.276

Isentropic changes on the PVT surface are described in terms of the isentropic compressibility βS, eqn (1.171), the isentropic expansivity αS, eqn (1.172), and the isentropic thermal pressure coefficient γS, eqn (1.196). Useful relations with more conventional second-order derivatives are given below:

Equation 1.277
Equation 1.278
Equation 1.279

For the isentropic thermal pressure coefficient we have

Equation 1.280
Equation 1.281
Equation 1.282

According to eqn (1.280), the rate of an isentropic change of T with P, i.e., (∂T/∂P)S=1/γS, has the same sign as the isobaric expansivity. The three isentropic coefficients are related by

Equation 1.283

Burlew's piezo-thermometric method104  for determining CP is based on eqn (1.280), i.e., on measuring (∂T/∂P)S and (∂V/∂T)P=P.

As pointed out by Rowlinson and Swinton,43  the mechanical coefficients αP, βT, γV are determined, to a high degree of accuracy, solely by intermolecular forces, while the isentropic coefficients αS, βS, γS, with which they are related through the thermal coefficients, i.e., the heat capacities, and the heat capacities themselves depend also on internal molecular properties.

The last topics presented here are property changes of mixing and excess quantities of nonelectrolyte PVT mixtures, in particular liquid mixtures.2,3,34,35,37,39–41,105  Instead of considering total properties Mt=nM(T, P, {xi}), it is advantageous to discuss them in relation to the properties of the pure constituents at the same T, P and {xi}, i.e., to focus on difference measures. Discussion is thus based on a new class of thermodynamic functions known as property changes of mixing, designated by the symbol Δ and, on a molar basis, defined by

Equation 1.284

The corresponding new class of partial molar property changes of mixing is defined by

Equation 1.285

With the summability relation eqn (1.96) we have

Equation 1.286

and in analogy to eqn (1.95), the exact differential of the extensive property (ΔM)t=nΔM is

Equation 1.287

From eqn (1.286) a differential change in nΔM is given by

Equation 1.288

Hence, through comparison with eqn (1.287), and after division by n,

Equation 1.289

This is still another form of the general Gibbs–Duhem equation, eqn (1.103). Here, the focus will be on M=G, S, V, H. Because of direct measurability, ΔV and ΔH are the molar property changes of mixing of special interest.

Alternatively, discussion of real-solution behaviour may be based on deviations from ideal-solution behaviour, i.e., on the differences between property values of real solutions and property values calculated for an ideal-mixture model known as the Lewis–Randall (LR) ideal-solution model at the same T, P and {xi}. This type of ideal solution behaviour is based on the definition

Equation 1.290

for the partial molar Gibbs energy of component i, and model properties will be indicated by a superscript id (alternative ideal-solution models are possible, and are indeed used). Eqn (1.290) serves as a generating function for other partial molar properties of an LR-ideal solution. For instance, the temperature derivative and the pressure derivative yield the partial molar entropy and the partial molar volume, respectively,

Equation 1.291
Equation 1.292

while the Gibbs–Helmholtz eqn (1.226) yields the LR-ideal partial molar enthalpy

Equation 1.293

The LR-ideal molar properties corresponding to the partial molar properties of eqn (1.290) through (1.293) are obtained with the summability relation:

Equation 1.294
Equation 1.295
Equation 1.296
Equation 1.297

The molar property changes of mixing for LR-ideal solutions, ΔMid, may be obtained as a special case from the general defining eqn (1.284):

Equation 1.298
Equation 1.299

That is, by substituting either the corresponding expression for Mid, eqn (1.294) through (1.297), into eqn (1.298), or the corresponding expressions for Miid, eqn (1.290) through (1.293), into eqn (1.299) we obtain

Equation 1.300
Equation 1.301
Equation 1.302
Equation 1.303

The general property ΔMiid(T, P, {xi}) of eqn (1.299) denotes a partial molar property change of mixing for LR-ideal solutions, such as those appearing in eqn (1.300) through (1.303):

Equation 1.304

Quantities measuring deviations of real solution properties M(T, P, {xi}) from LR-ideal solution properties Mid(T, P, {xi}) at the same T, P and {xi} (see eqn (1.294) through (1.297)), constitute another useful new class of functions called excess molar properties. They are designated by a superscript E and defined by

ME(T, P, {xi})≡M(T, P, {xi})−Mid(T, P, {xi}).
Equation 1.305

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

Equation 1.306

and with the summability relation

Equation 1.307

The excess molar Gibbs energy GE as a generating function is of particular interest. As a matter of convenience, eqn (1.290) may be generalised in such a manner that an expression for the partial molar Gibbs energy Gi is obtained which is valid for any real mixture. That is, we may write

Equation 1.308

where γi(T, P, {xi}) is known as the Lewis-Randall (LR) activity coefficient of species i in solution. With the definition eqn (1.306), the excess partial molar Gibbs energy is thus given by

Equation 1.309

In view of eqn (1.307), the excess molar Gibbs energy reads

Equation 1.310

Since Si=−(∂Gi/∂T)P, {xi}, for the excess molar entropy we have

Equation 1.311

since Vi=(∂Gi/∂P)T,{xi}, the excess molar volume is given by

Equation 1.312

and finally, with the Gibbs–Helmholtz equation we obtain for the excess molar enthalpy

Equation 1.313

Of course, GiE=HiETSiE and GE=HETSE, etc. The definition of an excess property is not restricted to any phase, though excess properties are predominantly used for liquid mixtures.

Excess properties and property changes of mixing are closely related and one may readily calculate ME from ΔM and vice versa. By combining the definitionseqn (1.284) and (1.305), in conjunction with ΔMid defined by eqn (1.298), the important relation

ME(T, P, {xi})=ΔM(T, P, {xi})−ΔMid(T, P, {xi})
Equation 1.314

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

MiE(T, P, {xi})=ΔMi(T, P, {xi})−ΔMiid(T, P, {xi}).
Equation 1.315

In eqn (1.314), the difference ΔMMEMid is zero except for the second-law properties M=G, F and S, and similarly for the partial properties in eqn (1.315), the difference ΔMiMiEMiid is zero except for the second-law properties Mi=Gi, Fi and Si. Further, from eqn (1.314) we see immediately that since an excess molar property represents also the difference between the real change of property of mixing and the LR-ideal-solution change of property of mixing, we may identify it alternatively as an excess molar property change of mixing

Equation 1.316

Analogously, from eqn (1.315) we may identify alternatively an excess partial molar property as an excess partial molar property change of mixing

Equation 1.317

Evidently, the terms excess molar property and excess molar property change of mixing may be used interchangeably, and both are indeed found in the literature. If the focus is on properties of mixtures, then ME and MiE are preferred, while for mixing processes the notations ΔME and ΔMiE may be regarded as more appropriate. For the four quantities selected, for a more detailed discussion we have the following equalities:

Equation 1.318
Equation 1.319
VEV, and ViEVi,
Equation 1.320
HEH, and HiEHi.
Equation 1.321

Depending on the point of view, HEH is called either the excess molar enthalpy or the molar enthalpy change of mixing, and VEV is known as either the excess molar volume or the molar volume change of mixing. The relations summarised by eqn (1.318) through (1.321) are reformulations of eqn (1.310) through (1.313).

In analogy to eqn (1.93), (1.95) and (1.287), the exact differential of the extensive property nME(T, P, {xi}) is given by

Equation 1.322

while eqn (1.307) yields for a differential change in nME caused by changes of T, P or ni

Equation 1.323

Comparison with eqn (1.322) and division by n results in

Equation 1.324

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

For convenience, instead of GiE the non-dimensional group GiE/RT is frequently used, which is related to the LR-based dimensionless state function ln γi(T,P,{xi}) by

GiE/RTμiE/RT=ln γi.
Equation 1.325

Using the summability relation, we have

Equation 1.326

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

Equation 1.327


Equation 1.328
Equation 1.329
(∂(nGE/RT)/∂ni)T, P, nj≠i=ln γi.
Equation 1.330

The Gibbs–Duhem equation reads,

Equation 1.331

and at constant T and P

Equation 1.332

The fundamental excess-property relation eqn (1.327) in terms of the canonical variables T, P and {xi} supplies complete information on excess properties. It is of central importance in solution chemistry because HE, VE and ln γi are experimentally accessible quantities: excess enthalpies and excess volumes may be obtained from mixing experiments, while LR activity coefficients are obtained from vapour-liquid (VLE) equilibrium measurements (or solid-liquid equilibrium measurements). For 1 mol of a constant-composition mixture

Equation 1.333

and for the corresponding excess partial molar properties

Equation 1.334

Hence the partial molar analogues of eqn (1.328) and (1.329), respectively, are

Equation 1.335
Equation 1.336

Analogous to eqn (1.243) we have

Equation 1.337

and analogous to eqn (1.229),

Equation 1.338

Modern flow calorimeters allow reliable measurements of HE at elevated T and P, and the results have to be consistent with experimental CPE's and volumetric properties, as indicated by eqn (1.337) and (1.338). Outside the critical region the pressure influence on excess properties is small.

Focussing on the useful excess property GE/(x1x2RT), for a binary mixture, we find

Equation 1.339

These relations are of considerable practical utility when a graphical (visual) evaluation of experimental GEs of binary mixtures is intended. In general, for binary mixtures, extrapolation of ME/x1x2 to x1=0 and x2=0, respectively, is the most convenient and reliable graphical method for determining the infinite-dilution excess partial molar properties M1E,∞ and M2E,∞.

Unfortunately, no general theory exists that satisfactorily describes the composition dependence of excess properties, and relations commonly used are semiempirical at best. Focusing on binary mixtures, perhaps the most popular empirical relation is due to Redlich and Kister,106,107 

Equation 1.340

where the excess partial molar property values at infinite dilution, , are given by

Equation 1.341

For highly skewed data, using more than four terms may cause spurious oscillations of MiEs, and may yield unreliable MiE,∞s. The flexibility to fit strongly unsymmetrical curves is provided by Padé approximants108,109  of order [a/b], where the denominater must never become zero:

Equation 1.342

As alternatives, expressions based on orthogonal polynomials have been suggested,110–112 e.g., expansions based on Legendre polynomials111,112  in z12x1x2:

Equation 1.343

with L0(z12)=1, L1(z12)=z12, L2(z12)=(3z122−1)/2, L3(z12)=(5z123−3z12)/2 and so forth. The summation limit np is selected as required to fit the available experimental data. If HE data are available at several temperatures, the temperature dependence of ap has to be incorporated via, say,

Equation 1.344

For a recent suggestion of an exponential temperature dependence, see Kaptay.113 

Used with necessarily discrete experimental data, Legendre polynomial expansions have the merit that increasing the number of terms to improve the fit will only slightly influence the values of lower-order terms. As pointed out by Pelton and Bale,111,112  using Legendre expansions in terms of Lp(z12) instead in terms of Lp(x1) has certain advantages. Conversion formulae to calculate Legendre coefficients from Redlich-Kister coefficients (or from power series coefficients) have been given by Pelton and Bale,112  Howald and Eliezer,114  and Tomiska.115 

When the number of components increases to three and beyond, experimental work to determine excess properties increases sharply, thus explaining the scarcity of data on multicomponent mixtures. The situation is aggravated by less reliable empirical/semiempirical correlating functions describing their composition dependence. Predictions of multicomponent properties from results on the constituent binaries alone, without ternary (or higher) terms, are always approximate, the most successful correlation of this type being Kohler's equation:116  it relates the excess molar Gibbs energy GE,123 of a ternary liquid mixture with mole fractions {x1, x2, x3}, to the excess molar Gibbs energies GE,ij of the three binary subsystems with composition ,where the mole fractions identified by a superscript prime are defined by

Equation 1.345

Based on the reasonable approximation that pairwise interactions ij remain constant along lines representing mixtures having a constant composition ratio xi/xj, the binary quantities GE,ij are assumed to depend only on , and

Equation 1.346

Kohler's equation treats the binary subsystems equally, and the model does not impose any restrictions on the functional form of the expressions selected to represent the composition dependence of binary GE,ij data. Similar comments apply to HE,123 and CPE,123. Kohler's equation can be generalised to correlate/predict the composition dependence of excess molar properties of multicomponent systems with four or more components. Assuming again that pairwise interactions ij remain constant at conditions imposing a constant composition ratio , such a generalised equation for the excess molar enthalpy HE,12⋯n of an n-component system reads

Equation 1.347

For the composition dependence of the excess molar enthalpies of the binary subsystems, any function, say, Redlich-Kister, Padé or Legendre polynomial, may be used.

For the excess molar heat capacity at constant pressure of a binary subsystem we have with eqn (1.343) and (1.344), and ,

Equation 1.348

Inserting this quantity into

Equation 1.349

yields a Kohler-type equation describing the composition dependence and the temperature dependence of the excess molar isobaric heat capacity CPE,12…n of a liquid n-component mixture:

Equation 1.350

Traditionally, the thermodynamic description of real liquid solutions is based on the excess-property formalism presented above. GE, HE, CPE and VE are measurable properties, and large numbers of (critically) evaluated experimental HE data are available in systematic data collections, such as Landolt-Börnstein,20–24  or in data banks, such as the Dortmund Data Bank.18  Based on this formalism, well-honed semi-empirical models, such as UNIFAC,117–119  DISQUAC120–122  and the new MOQUAC model123  (in which the effect of molecular orientation on interaction is explicitly taken into account), have been developed for correlating, extrapolating and predicting, in particular GE and HE, over reasonably large temperature ranges. Estimated infinite-dilution properties, aqueous solubilities of hydrocarbons, and CPE of liquid mixtures are frequently not satisfactory.124  Similar comments apply to COSMO-RS and related models.125–127 

For the global thermodynamic description of liquid nonelectrolyte mixtures, CPEs are pivotal properties, and taking advantage of the exact relations of eqn (1.337), considerable economy in experimental effort may be attained. Given HE and GE at one suitably selected temperature Tref, and CPE as a function of T, integration over T at constant P and {xi} of the relevant differential equations yields HE, SE and GE over the temperature range of the heat capacity measurements. Well below the vapour-liquid critical region, CPE of a constant-composition mixture at constant pressure frequently shows a simple dependence on temperature, i.e., on τTref/T:53 

Equation 1.351

Starting from eqn (1.337), integration over T yields53 

HE/RT=a3+(a2a3+a5)τa5τ2a4τ ln τ,
Equation 1.352
SE/R=a1+a4+a5/2−a4τa5τ2/2−a3 ln τ,
Equation 1.353
GE/RT=−a1+a3a4a5/2+(a2a3 +a4+a5)τa5τ2/2+a3 ln τa4τ ln τ.
Equation 1.354

The dimensionless coefficients aj=aj(P,{xi}) are related to the excess molar quantities at {Tref, P, {xi}}: CPE(Tref)/R=a3+a4+a5, HE(Tref)/RTref=a2, SE(Tref)/R=a1, and GE(Tref)/RTref=−a1+a2. Global studies of this kind are, however, quite rare, with some of the most careful investigations being those of Ziegler and colleagues.128 

As far as nomenclature/symbols are concerned, in almost all cases I have adhered to the suggestions of IUPAC.60  Deviations are due to my desire to present a concise, unequivocal and logically consistent notation in compliance with usage preferred by the scientific community interested in this review's topics. Such an approach is in accord with the spirit of the Green Book expressed on p. XII, i.e., with the principle of “good practice of scientific language”. The quantities I would like to comment on once again are the mechanical coefficients. For the isothermal compressibility, Rowlinson and Swinton,43  amongst many others, use the symbol βT. Together with the isobaric expansivity αP and the isochoric thermal pressure coefficient γV, a mnemonic triple αP/βT=γV is formed, eqn (1.206); indicating via subscript what quantity to hold constant is advantageous in general, and in particular when discussing related isentropic quantities (subscript S) and saturation quantities (subscript σ).66,67,85  Some symbols may be modified further by adding appropriate subscripts and/or superscripts. For instance, the capital superscript letters L (liquid) and V (vapour) are used because (i) they are easy to read, (ii) they are frequently used in the chemical engineering literature,2,3  including volumes published under the auspices of IUPAC,30–33  and (iii) vapour–liquid equilibrium is usually abbreviated by VLE, and not by vle.

Calorimetry, PVT measurements and phase equilibrium determinations are the oldest and most fundamental experimental areas in physical chemistry. They provide quantitative information on thermodynamic properties to be used for theoretical advances and to improve on applications of science, i.e., chemical engineering. Enormous effort and ingenuity has gone into designing the vast array of apparatus now at our disposal for the determination of caloric properties,28–32,69,70,129,130  of PVT-properties,29–31,33,69,70  and of ultrasonic and hypersonic properties27,29,32,33,87,93,94  of pure and mixed fluids over large ranges of temperature and pressure. During the last decades, the penetration of calorimetry into (traditionally) neighbouring areas has more and more frequently taken place: instruments and experimental data have become indispensable in materials science, but also in biophysics, in drug design and in the medical sciences.

In this introductory chapter, I did not cover any experimental details – the reader is referred to the relevant sections of this book and to pertinent articles and monographs quoted as references. Continuing advances in instrumentation (including automation and miniaturisation) leading to increased precision, accuracy and speed of measurement, as well as the ever widening ranges of application and improved methods of data management, data storage and data transfer provide the impetus for calorimetry on fluid systems to remain an active, developing discipline. Caloric properties are of pivotal importance for physics, chemistry and chemical engineering, and cross-fertilization, notably with bio-oriented fields, will increase. Without doubt, highly interesting research is to be expected, as indicated by the selection of recent articles, reviews and monographs I present, such as ref. 131–147.

Thermodynamics is a vast subject of immense practical as well as fundamental value and beauty.

Combination with molecular theory and statistical mechanics promotes molecule-based insight into macroscopic phenomena, and thus opens the door to advances in chemical engineering. I hope that the topics treated in this book under the “umbrella” internal energy and enthalpy provide a feeling for the scope of the field, for its contributions to the development of thermal physics and chemistry, for its current position in science, and most important, for its future potential. In this connection, it is again my pleasure to acknowledge the many years of fruitful scientific collaboration with more than 80 colleagues, post-doctoral fellows and students from 17 countries. Without them, many projects would have been difficult to carry out, or would have, perhaps, never been started.

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