Chapter 1: Internal Energy and Enthalpy: Introduction, Concepts and Selected Applications

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/m_m, where m_m 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 

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 data^2–26  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 general^41–48  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 

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 U^t 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/m_m. This designation distinguishes it from kinetic energy E_k and potential energy E_p 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:

For a differential change it is written

Mathematically, dU^t is an exact differential of the state function U^t: the change in the value of this extensive property,

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 ΔU^t 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 X_i,

is defined by

and the partial derivative c_i and its corresponding variable X_i 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

for any two pairs (c_i, X_i) and (c_j, X_j) 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, …, X_n) is exact, we have

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 U^t: 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 q_rev+w_rev 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

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

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, 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, U^t 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 U^t. On a microscopic level, internal energy is associated with molecular matter, and for nonreacting systems at common T and P contributions to ΔU^t 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

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

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

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

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

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

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 x_i of component i of a homogeneous system, i.e., of a phase:

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

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

Here, n^αM^α≡M^t,α is a total property of phase α, etc., is the total amount of substance, is the total amount of phase p, and n$ip$ is the amount of i in phase p. The composition of, say, phase α, is characterised by the set of mole fractions {x$iα$},

, and x$iα$=1 for a pure fluid i. If the use of masses is preferred, replace n$iα$ by m$iα$ and n^α by m^α to obtain the analogous equation defining mass fractions w$iα$ in phase α,

with , and w$iα$=1 for pure i. The total mass of phase α is , where m$iα$=n$iα$m_m,i is the mass of component i with molar mass m_m,i. Thus,

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 {x_i}. 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 {x$iα$}, {x$iβ$},…. 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

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

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

since 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 reveals that while q_rev is a path function, and 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:

These results may be summarised by an additional postulate asserting the existence of an extensive state function S^t 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 {x$iα$}, {x$iβ$},…. The entropy change between two equilibrium states therefore depends solely on the difference between the values of S^t 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 ΔS^t 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

The adiabatic work w_ad is path independent and depends only on the initial and final equilibrium states. Measurements of w_ad are measurements of ΔU^t, and calorimetric experiments confirm it, thereby providing the primary evidence that U^t 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 S^t 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 ΔS$entiret$ associated with any process is given by

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

where U, S, and V denote, respectively, the molar internal energy, the molar entropy, and the molar volume. Since U^t=nU, S^t=nS and V^t=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 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 eqn (1.32) to a closed single-phase system without chemical reactions indicates

where {n_i} 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

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

or, equivalently, a fundamental equation in the entropy representation

Taking into account eqn (1.33), (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

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

Eqn (1.40) and (1.41) apply to single-phase multicomponent PVT systems, either open or closed, where the n_i 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 n_i is called the chemical potential of component i in the mixture:

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

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

while the fundamental property relation eqn (1.41) becomes

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 n_i 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

and in the entropy representation

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,μ_i} 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

it must also satisfy

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,

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 μ_i are partial derivatives of U^t(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,

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

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

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

or a volume-explicit thermal equation of state

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))

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

where

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

The total differential reads

and with eqn (1.61) one obtains

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,

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

forming the total differential,

and using eqn (1.61):

Hence,

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

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

and thus

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, {n_i}}, 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, {n_i}}. The most popular alternative equivalent primary functions are the total enthalpy

the total Helmholtz energy

and the total Gibbs energy, a double Legendre transform,

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

with the associated canonical variables {nS, P, {n_i}}, {T, nV, {n_i}} and {T, P, {n_i}}, 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:

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

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

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

which is less general than the fundamental property relation eqn (1.44) in an important aspect: while the n_i are independent, mole fractions x_i 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, {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 important. Of the alternative expressions for the chemical potential, eqn (1.85), the preferred one is

Partial derivatives of a total property with respect to n_i at constant T, P and n_j≠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, {x_i}), the corresponding extensive total property of the solution phase is

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

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

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,

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

Division by the total amount of substance gives the molar property

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

However, from eqn (1.97),

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

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

while eqn (1.95) 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. This equation is of central importance in chemical thermodynamics. For changes at constant T and P it simplifies to

which shows the constraints on composition changes. It is important to note that a partial molar property M_i 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 M^t, 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., M_i, 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:

The corresponding fundamental property relation is

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

The remaining two primary functions

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

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

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

and correspondingly

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

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

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

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

Treating in the energy representation eqn (1.46) as a single term (thus n=3), the entire number N_t 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:

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,

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=S^t (nU, nV, {n_i}), 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 μ_i/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

and the corresponding entropy-based fundamental property relation is

with canonical variables . The integrated, alternative form reads

The Massieu function is defined by

and the corresponding entropy-based fundamental property relation is

with canonical variables . The integrated, alternative form reads

The Planck function is a second-order Legendre transform,

and the corresponding entropy-based fundamental property relation reads

with canonical variables . The integrated, alternative form is

Another second-order Legendre transform is the Kramer function

with the corresponding entropy-based fundamental property relation

and canonical variables . The integrated, alternative form is

The first-order Legendre transform

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

with canonical variables . The integrated, alternative form is

Finally, we have the unnamed second-order Legendre transform

the corresponding entropy-based fundamental property relation,

with canonical variables , and its integrated alternative form

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

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

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

with canonical variables . Division by n yields

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:

They are exact differentials, hence

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 {x_i} 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, …, X_n) with n independent variables X_i, the exact differential

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 N_ds=n, and the number N_ms of mixed second-order partial derivatives , i≠j, each with operators and , is given by the number of variations without repetition, V_dⁿ:

The total number N_s=N_ds+N_ms of second-order partial derivatives without restricting indices is given by the number of variations with repetition, V̄$dn$:

According to the Euler reciprocity relation eqn (1.7),

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

Of course, N_Mw=N_ms/2.

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

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 N_ds=2 corresponding direct second-order partial derivatives are

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

and applying the Euler reciprocity relation, eqn (1.159), we have N_Mw=1 Maxwell relation, see eqn (1.160) or N_Mw=N_ms/2=V$22$/2 = 1:

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 {n_i} and, by dividing by n, use them in terms of intensive molar properties M instead of extensive total properties nM:

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

the isentropic compressibility β_S is defined by⁶⁷ 

and the isentropic expansivity α_S⁶⁷  is defined by

with the mass density ρ of the phase being given by

The Maxwell relation eqn (1.166) now becomes

This set {C_V, β_S, α_S} 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 {n_i}, 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

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

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

and the 2 mixed second-order partial derivatives are

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

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

Here,

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

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

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

This set {C_P, β_T, α_P} 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

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,

The 2 direct second-order partial derivatives thus become

and the 2 mixed second-order partial derivatives are

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

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

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

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

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

The 2 second-order partial derivatives thus become

and the 2 mixed second-order partial derivatives are

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

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

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

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

the three mechanical coefficients are related as follows:

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

are obtained. Simple mathematical transformations lead to the alternative forms

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 

and thus

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

and thus

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

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 S_i (i=1, 2, 3), viz.

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

and finally three Maxwell relations relating chemical potentials, viz.

The partial property analogue to eqn (1.151) is

and the analogue to eqn (1.78) is

where

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

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

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

(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 ρ_n≡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 via eqn (1.208) and (1.209):

Both equations can be contracted to yield

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

and a special name, internal pressure. It may be determined at pressure P viaeqn (1.228) by measuring γ_V, 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

It is related to the isenthalpic Joule–Thomson coefficient

by

The three quantities φ, μ_JT and C_P of gases/vapours^71–73  may be measured by flow calorimetry.

Since for perfect gases Tα_P=1, φ=0 and μ_JT=0, the real-gas values of these coefficients are directly related to molecular 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:

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

and integration between a suitable reference temperature T_ref and T yields⁷⁴ 

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

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

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

and from eqn (1.228),

The molar heat capacity at constant pressure is defined by

and from eqn (1.229),

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

and the isentropic compressibility as

In high-pressure research,^{66,67,69,70,77–84}  eqn (1.242) and (1.244) are particularly interesting: the pressure dependence of C_P of a constant-composition fluid may be determined either from PVT data alone, or by high-pressure calorimetry, or by transitiometry,^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

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

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

whence

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

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

Since the compression factor Z is defined by

alternatively^66,67,85  the difference is given by

The ratio of the molar heat capacities, κ≡C_P/C_V, is accessible via eqn (1.167) and (1.182) in conjunction with the chain rule:

According to the triple product rule

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

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

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

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 property^{27,77,79,80,82–94}  related to β_Sviaeqn (1.265). Alternatively, by using the relations provided by eqn (1.153) we have

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

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  γ_V of liquids is large and the direct calorimetric determination of C_V 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 γ_V is much smaller.

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

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

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

From eqn (1.266) the difference between β_T and β_S may be expressed as

while 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 (1.171), the isentropic expansivity α_S, eqn (1.172), and the isentropic thermal pressure coefficient γ_S, eqn (1.196). Useful relations with more conventional second-order derivatives are given below:

For the isentropic thermal pressure coefficient we have

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

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

As pointed out by Rowlinson and Swinton,⁴³  the mechanical coefficients α_P, β_T, γ_V are determined, to a high degree of accuracy, solely by intermolecular forces, while the isentropic coefficients α_S, β_S, γ_S, with which they are related through the thermal coefficients, i.e., the heat capacities, and the heat capacities themselves depend also on internal molecular properties.

The 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 M^t=nM(T, P, {x_i}), it is advantageous to discuss them in relation to the properties of the pure constituents at the same T, P and {x_i}, 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

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

With the summability relation eqn (1.96) we have

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

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

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

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 {x_i}. This type of ideal solution behaviour is based on the definition

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,

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

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

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

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

The general property ΔM_i^id(T, P, {x_i}) 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):

Quantities measuring deviations of real solution properties M(T, P, {x_i}) from LR-ideal solution properties M^id(T, P, {x_i}) at the same T, P and {x_i} (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

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

and with the summability relation

The excess molar Gibbs energy G^E 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 G_i is obtained which is valid for any real mixture. That is, we may write

where γ_i(T, P, {x_i}) 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

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

Since S_i=−(∂G_i/∂T)_P, {xi}, for the excess molar entropy we have

since V_i=(∂G_i/∂P)_T,{xi}, the excess molar volume is given by

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

Of course, G$iE$=H$iE$−TS$iE$ and G^E=H^E−TS^E, etc. The definition of an excess property is not restricted to any phase, though excess properties are predominantly used for liquid mixtures.