Any sufficiently advanced technology is indistinguishable from magic.
Arthur C. Clarke, Profiles of the Future: An Inquiry into the Limits of the Possible, revised edition. Harper & Row, New York, 1973.
How often have we walked this narrow thoroughfare without having noticed that little shop? Or, if noticed, passed it by? Let us just try the door … it opens … a bell tinkles … What an odd shop … What can it have for sale? Shall we go in?
Avram Davidson, editor, Magic for Sale, Ace Science Fiction, 1983.
Abstract
The liquid state is one of the three principal states of matter and arguably the most important one; and liquid mixtures represent a large research field of profound theoretical and practical interest. This topic is of importance in many areas of the applied sciences, such as in chemical engineering, geochemistry, the environmental sciences, biophysics and biomedical technology. First, I will concisely present a review of important concepts from classical thermodynamics of nonelectrolyte solutions; this will be followed by a survey of (semi)empirical approaches to representing the composition and temperature dependence of selected thermodynamic mixture properties, and finally the focus will be on dilute binary nonelectrolyte solutions where one component, a supercritical solute, is present in much smaller quantity than the other component, called the solvent. Partial molar properties in the limit of infinite dilution (indicated by a superscript ∞) are of particular interest. For instance, activity coefficients (Lewis–Randall (LR) convention) are customarily used to characterize mixing behavior, and infinitedilution values \(\gamma_{i}^{{{\text{LR,}}\infty }}\)provide a convenient route for obtaining binary parameters for several popular solution models. When discussing solute (j)—solvent (i) interactions in solutions where the solute is supercritical, the Henry fugacity \(h_{j,i} \left( {T,P} \right)\), also known as Henry’s law (HL) constant, is a measurable thermodynamic key quantity. Its temperature dependence yields information on the partial molar enthalpy change on solution \(\Delta H_{j}^{\infty } \left( {T,P} \right)\), while its pressure dependence yields information on the partial molar volume \(V_{j}^{{{\text{L,}}\infty }} \left( {T,P} \right)\) of solute j in the liquid phase (superscript L). I will clarify issues frequently overlooked, touch upon solubility data reduction and correlation, report a few recent highprecision experimental results on dilute aqueous solutions of supercritical nonelectrolytes, and show the equivalency of results for caloric quantities (e.g. \(\Delta H_{j}^{\infty }\)) obtained via van ’t Hoff analysis of highprecision solubility data with directly measured calorimetric data.
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1 Introduction
1.1 Preliminaries
Physical chemistry and chemical physics (if they are, indeed, fields apart) of liquids and liquid mixtures or solutions are indispensable parts of many areas of the pure and applied sciences, such as chemistry, physics, biophysics, chemical engineering, and geoscience, among others. In particular, this large and fundamentally important research field forms bridges between topics where work on macromolecules, colloids, glasses, liquid crystals and biological materials constitutes the highly active and rapidly expanding field of softmatter chemical physics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Most processes of interest in chemical engineering are carried out in fluid solution phases, and nonelectrolyte solutions constitute, perhaps, the most important subfield which will be considered in this review. Based on thermodynamics, molecular physics and statistical mechanics, the three main reasons for the enormous effort invested into experimental, theoretical and computerbased research in this field are:

(I)
It is hoped that by studying mixture/solution properties and solubilities we will steadily improve our knowledge on interactions between molecules in bulk liquid phases. Any advance in this area will directly benefit science in general, and separation industry in particular, since the most common processes there are predominantly based on vapor–liquid equilibria (VLE) involving multicomponent systems. A classic book in this area, combining thermodynamic theory with experimental practice, has been authored by Hála et al. [20].

(II)
The appearance of new physical phenomena not found with the pure liquid components is scientifically fascinating as well as challenging. It adds a new dimension to thermophysical research and opens the door for novel ideas and developments.

(III)
Sciencebased chemical engineering is one of the most important segments of a modern society, with fluid phase separation technology and chemical/biochemical product design being cornerstones of the field. Solvents have always been the mainstay of the chemical industry, and associated technological problems must be resolved as fast as possible; that includes restrictions aimed at preserving the quality of human life in general, i.e., environmental (sustainability, toxicology, etc.) [21, 22] and specific health issues. The practitioners involved must deal with them efficiently and pragmatically.
Experiments are the fundament of science: they provide the basis for inductive reasoning (known informally as bottomup reasoning), which after amplifying, logically ordering and generalizing our experimental observations leads to hypotheses and then theories, and thus to new knowledge. In contradistinction, deduction (informally known as topdown reasoning), orders and explicates already existing knowledge, thereby leading to predictions which may be corroborated experimentally, or falsified [23]. In fact, a theory has no value in science unless it is possible to test it experimentally. Though correct in principle, from a practical point of view the following problem exists: the huge number of potentially useful solubility data connected with binary, ternary, quaternary, etc., systems at different temperatures T and pressures P effectively precludes the experimental determination of mixture/solution properties and solubilities for all but a few representative key systems of physical–chemical/technological interest. This is best illustrated by calculating the number of multicomponent solutions containing c components, which can be formed out of, say, p = 500 chemicals (this is a reasonable estimate of the number of solvents generally available to chemists working in academic, governmental and industrial research laboratories). The number of multicomponent solutions is given by the ccombination
Hence, \(C_{2}^{500} = 0.12475 \times 10^{6}\) different binary solutions may be formed, \(C_{3}^{500} = 20.7085 \times 10^{6}\) different ternary solutions, \(C_{4}^{500} = 2.573031125 \times 10^{9}\) different quaternary solutions, and so forth. Thus, reliable and effective prediction methods for thermodynamic properties of fluids, pure and mixed, are an indispensable tool of the trade.
In order to facilitate development and discussion of ideas, I consider it helpful to introduce here a few physical concepts that will be discussed in more detail later on (see Sect. 2.3). The obvious power and utility of an equationofstate (EOS) has stimulated research in this field for about 150 years, that is, since van der Waals (vdW). He was one of the first to utilize molecular theory to describe semiquantitatively thermodynamic properties of liquid mixtures in terms of purecomponent properties [24,25,26,27,28]. It was based on his then newly developed celebrated volumetric equation of state for pure fluids,
or alternatively written,
Here, V is the molar volume, and R denotes the molar gas constant. The parameter b is known as the covolume and allows for the finite hard size of the molecules, and the averaged attractive intermolecular interaction in the real fluid leads to a correction of the pressure amounting to \({a \mathord{\left/ {\vphantom {a {V^{2} }}} \right. \kern\nulldelimiterspace} {V^{2} }}.\)
The vdW equation of state is cubic in V, that is,
and such a cubic expression (this is a minimum requirement) is capable of representing both liquidlike and vaporlike molar volumes at sufficiently low temperatures (via the Maxwell equalarea rule). It yields also a critical point (indicated by a subscript c) with critical temperature \(T_{{\text{c}}}\), critical pressure \(P_{{\text{c}}}\) and critical molar volume \(V_{{\text{c}}}.\) For the first time, properties of both liquid and vapor could be described by an equation of state using molecularbased parameters, which feature is prerequisite for using an EOS in the calculation of vapor–liquid equilibria. A frequently used alternative form of the vdW equation of state is
where Z is known as the compression factor. As best seen with Eq. 5, in the limit as \(V \to \infty\) the vdW equation yields the perfectgas (pg) [29], or idealgas, equation of state, that is,
All pressureexplicit equations of state should satisfy this limit. The term “equation of state” most frequently refers to the equilibrium relation between pressure, volume, temperature and composition of the uniform fluid phase, in the absence of special external force fields.^{Footnote 1} Most conveniently, the composition is characterized by the mole fractions \(x_{i}\) of the components, i.e.,
Here, \(n_{i}\) is the amount of substance of component i in the mixture, \(n \equiv \sum\limits_{i}^{c} {n_{i} }\) is the total amount of substance, and for a pure substance \(x_{i} = 1\).
Based on his equation of state, Eq. 2, van der Waals formulated his corresponding states theorem (CST) for pure fluids. However, the derived universal critical compression factor
is considerably larger than common experimental values. Although first derived for a vdW fluid, it was shown later that CST is much more generally applicable: it is, in fact, valid for any EOS with two adjustable parameters (twoparameter CST), and, as shown by Pitzer [36], it can be derived from statistical mechanics.
Nevertheless, from such a volumetric vdWtype EOS, for any given composition, PVT information may be obtained. Using classical thermodynamics, isobaric residual properties and isochoric (isometric) residual properties can be computed, and vapor–liquid phase equilibria (VLE) can be calculated (for definitions see Sect. 2). However, for many practical applications concerning liquid solutions, the focus is on the direct modeling of the molar excess Gibbs energy \(G^{{\text{E}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) and related quantities, such as activity coefficients (see Sect. 2), though in principle an adequate PVTx equation of state implies a reliable \(G^{{\text{E}}}.\) Note that only PVTx systems comprised of chemically nonreactive components will be considered. With few exceptions, the following general system of notation will be used throughout:

Solutions/mixtures are of prime interest, and their molar singlephase properties, such as the molar volume V, will be represented by the plain symbol M; additional superscripts (such as E, identifying an excess property, or R, indicating a residual property) will be attached as needed. Thus, I have adopted the nomenclature suggested by Van Ness and Abbott [17].

A total property of a singlephase multicomponent solution is either indicated by a superscript t, or represented by the product nM, i.e.,\(M^{{\text{t}}} \equiv nM;\) \(n = \sum\limits_{i} {n_{i} }\) denotes the total amount (of substance) in the phase, and \(n_{i}\) is the amount (of substance) of component i. Again, the suggestion of Van Ness and Abbott [17] is adopted.

Molar singlephase puresubstance properties will be characterized by a superscript asterisk and, if needed, identified by a subscript, i.e., \( M_{i}^{*} i = 1,2, \ldots , \)

Partial molar properties referring to a component i in solution will be identified by a subscript, i.e., \( {\text{M}}_{i} = 1,2, \ldots , \)additional superscripts/subscripts will be attached as needed. Since a superscript asterisk characterizes a puresubstance property, no overbar is needed to indicate a partial molar property.
The notation adopted in this review facilitates reading equations and avoids as much as possible indexcluttering.
1.2 Thermodynamic Fundamentals
Consider a singlephase, multicomponent PVTx system, either open or closed, where the amounts \(n_{i} ,n_{j} ,n_{k} ,\ldots\) of components \(i,j,k,\ldots\) may vary either because of interchange of matter with the surroundings, or because of chemical reactions within the system, or both. The fundamental equation for a change of state of a phase, also known as the fundamental property relation, or the differential form of the fundamental equation in the internal energy representation, reads
and, equivalently,
in the entropy representation [17, 37]. Here, \(nU = U^{{\text{t}}} \left( {nS,nV,n_{1} ,n_{2} ,\ldots} \right)\) denotes the extensive total internal energy (indicated by the superscript t), and U the intensive molar internal energy, \(nS = S^{{\text{t}}} \left( {nU,nV,n_{1} ,n_{2} ,\ldots} \right)\) denotes the extensive total entropy, and S the intensive molar entropy.
The intensive parameter furnished by the firstorder partial derivative of the total internal energy with respect to the amount of substance \(n_{i}\) of component i,
is called the chemical potential of component i in the mixture. It is an intensive conceptual state function. And from Eq. 10 one obtains
Hence the fundamental property relation Eq. 9 can be written in a more compact form,
while the fundamental property relation Eq. 10 becomes
Equations 13 and 14 are fundamental because they completely specify all changes that can take place in simple, singlephase, multicomponent PVTx systems, either open or closed, and they form the basis of extremum principles predicting equilibrium states. However, I reiterate that in this review only simple, nonreacting fluid systems will be covered.
The fundamental property relations for open singlephase systems (either in the internal energy representation, Eq. 13, or in the entropy representation, Eq. 14), that is, the differential forms of the fundamental equations for open systems, can be integrated over the change in the amount of substance at constant values of the intensive quantities \(\left\{ {T,  P,\mu_{i} } \right\}\) or \(\left\{ {{1 \mathord{\left/ {\vphantom {1 T}} \right. \kern\nulldelimiterspace} T},{P \mathord{\left/ {\vphantom {P T}} \right. \kern\nulldelimiterspace} T},  {{\mu_{i} } \mathord{\left/ {\vphantom {{\mu_{i} } T}} \right. \kern\nulldelimiterspace} T}} \right\}\), respectively, and immediately yield the fundamental equations for an open, singlephase, multicomponent PVTx system:
in the internal energy representation, and
in the entropy representation. These equations are also known as the integrated forms of the fundamental equations for a change of the state of a phase; and the state functions \(nU\left( {nS,nV,\left\{ {n_{i} } \right\}} \right)\) and \(nS\left( {nU,nV,\left\{ {n_{i} } \right\}} \right)\) are commonly known as primary functions, or as cardinal functions, or as thermodynamic potentials.
Alternatively, Eqs. 15 and 16 can be regarded as a consequence of Euler’s theorem on homogeneous functions, which asserts the following: if \(f\left( {z_{1} ,z_{2} ,\ldots} \right)\) is a homogeneous function of degree k in the variables \(\left\{ {z_{1} ,z_{2} ,\ldots} \right\}\), that is, if it satisfies for any value of the constant scaling parameter \(\lambda\) the relation
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 firstorder properties of both \(U^{{\text{t}}}\) and \(S^{{\text{t}}}\), that is,
where for a more compact notation, in analogy to \(U^{{\text{t}}}\) and \(S^{{\text{t}}},\) \(nV\) has been replaced by \(V^{{\text{t}}},\) use of Eq. 18 with \(k = 1,\) in conjunction with Eqs. 13 and 14, yields Eqs. 15 and 16, respectively. The corresponding sets of extensive variables, i.e., \(\left\{ {nS,nV,n_{1} ,n_{2} , \ldots } \right\}\) for the internal energy representation, and \(\left\{ {nU,nV,n_{1} ,n_{2} , \ldots } \right\}\) for the entropy representation, are called independent canonical or natural variables. With a satisfactory fundamental equation established, all thermodynamic equilibrium properties of a PVTx phase can be calculated by fairly simple mathematical manipulations, that is, by combinations of appropriate derivatives of the corresponding primary function (thermodynamic potential); it is for this reason that they are called fundamental equations.
As indicated by Eq. 13, the parameters T, \( P\) and \(\mu_{i}\) are partial derivatives of \(nU = U^{{\text{t}}} \left( {nS,nV,n_{1} ,n_{2} , \cdot \cdot \cdot } \right)\) and are thus also functions of \(\left\{ {nS,nV,n_{1} ,n_{2} , \cdot \cdot \cdot } \right\}:\)
T, P and \(\mu_{i}\) are each homogeneous zerothorder functions (intensive functions) in the independent extensive variables nS, nV and \(\left\{ {n_{i} } \right\}\), and any relation expressing an intensive parameter in terms of independent extensive parameters, as indicated by Eqs. 21–23, is called a general equation 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. Analogous comments apply to the fundamental property relation in the entropy representation, Eq. 14, leading to the corresponding general equations of state as indicated below:
For constantcomposition fluids (and thus also for pure fluids), \(T = T\left( {nU,nV,n_{1} ,n_{2} , \cdot \cdot \cdot } \right),\) or explicitly resolved for the extensive total internal energy \(nU = U^{{\text{t}}},\)
This type of equation is known as the caloric equation of state.
Clearly, by using Eqs. 25 and 27 we obtain either a pressureexplicit equation of state
or, when resolved for the extensive total volume \(nV = V^{{\text{t}}}\), a volumeexplicit equation of state
A wellknown example of a volumeexplicit EOS is the virial equation in pressure, and a wellknown example of a pressureexplicit EOS is the van der Waals equation. Note, however, that most equations of state in practical use are explicit in pressure.
In the fundamental property relations for an open singlephase PVTx 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. Clearly, this situation does not conform to experimental practice. The choice of nS and nV as independent extensive variables in Eq. 13, and of nU and nV as independent extensive variables in Eq. 14, is not convenient: experimentbased experience shows that the conjugate intensive parameters \(\left\{ {T,P} \right\}\) and \(\left\{ {{1 \mathord{\left/ {\vphantom {1 T}} \right. \kern\nulldelimiterspace} T},{P \mathord{\left/ {\vphantom {P T}} \right. \kern\nulldelimiterspace} T}} \right\}\), respectively, are much more easily measured and controlled. In order to describe the system behavior when one or more of its intensive parameters are held constant (for instance, in isothermal or isobaric processes), alternative versions of the fundamental equations are necessary in which one or more of the extensive parameters are replaced by the conjugate intensive parameter(s) without loss of information. The appropriate formal mathematical technique for generating equivalent alternative thermodynamic potentials is the Legendre transformation [37,38,39,40].
Equation 15 suggests the definition of useful alternative energybased primary functions related to nU and with total differentials (fundamental property relations) consistent with Eq. 13, but with a set of canonical variables different from {nS, nV, {n_{i}}}, while Eq. 16 suggests the definition of useful alternative entropybased primary functions related to nS and with total differentials (fundamental property relations) consistent with Eq. 14, but with a set of canonical variables different from {nU, nV, {n_{i}}}.
Consider the exact (total) differential
pertaining to the function \(f^{\left( 0 \right)}\) with n independent variables \(X_{i}\),
where
Consider now the function obtained by subtracting the product of \(X_{1}\) with its conjugate partial derivative \(c_{1}\) from the base function \(f^{\left( 0 \right)}\), Eq. 31:
The corresponding total differential reads
and with Eq. 30 one obtains
Comparison of Eq. 30 with Eq. 35 shows that the original variable \(X_{1}\) and its conjugate \(c_{1}\) have interchanged their roles (and a minus sign was introduced): for such an interchange it suffices to subtract \(c_{1} X_{1} = \left( {{{\partial f^{\left( 0 \right)} } \mathord{\left/ {\vphantom {{\partial f^{\left( 0 \right)} } {\partial X_{1} }}} \right. \kern\nulldelimiterspace} {\partial X_{1} }}} \right)_{{X_{j \ne 1} }} X_{1}\) from the base function to yield the firstorder partial Legendre transform,
which is frequently identified by a bracket notation indicated above. This Legendre transform represents a new function with the independent variables \(\left\{ {c_{1} ,X_{2} ,X_{3} , \ldots ,X_{n} } \right\}\) being the canonical (or natural) variables.
The secondorder partial Legendre transform is obtained via
which yields the total differential
Hence
Analogously, the Legendre transformation of higher order p of the base function \(f^{\left( 0 \right)}\) that introduces the partial derivatives \(\left\{ {c_{1} ,c_{2} , \ldots ,c_{p} } \right\}\) into \(f^{\left( 0 \right)}\) reads
and the associated total differential is
The complete Legendre transform, i.e., the transform of order \(p = n\), replaces all variables by the conjugate partial derivatives, and vanishes identically for any system; this follows directly from the definition:
The associated differential expression reads
In thermodynamic theory, the complete Legendre transform of the internal energy nU of an open PVTx phase with c components has all extensive canonical variables replaced by their conjugate intensive variables, thus yielding the null function
and correspondingly
with canonical variables \(\left\{ {T,P,\left\{ {\mu_{i} } \right\}} \right\}.\) This property of the complete Legendre transform gives rise to the Gibbs–Duhem equation, which represents an important relation between the intensive parameters T, P and \(\left\{ {\mu_{i} } \right\}\) of the system and shows that they are not independent of each other.
When focusing on the fundamental equation in the entropy representation for an open, singlephase, multicomponent (c components) PVTx system, Eq. 16, the complete Legendre transform of the total entropy nS (again all extensive canonical variables are replaced by their conjugate intensive variables) is identically zero, thus yielding the null function
and correspondingly,
with canonical variables \(\left\{ {{1 \mathord{\left/ {\vphantom {1 T}} \right. \kern\nulldelimiterspace} T},{P \mathord{\left/ {\vphantom {P T}} \right. \kern\nulldelimiterspace} T},\left\{ {{{\mu_{i} } \mathord{\left/ {\vphantom {{\mu_{i} } T}} \right. \kern\nulldelimiterspace} T}} \right\}} \right\}\). This property of the complete Legendre transform gives rise to the entropybased Gibbs–Duhem equation, which shows that the intensive parameters characterizing the system, i.e., \({1 \mathord{\left/ {\vphantom {1 T}} \right. \kern\nulldelimiterspace} T}\), \({P \mathord{\left/ {\vphantom {P T}} \right. \kern\nulldelimiterspace} T}\) and \({{\mu_{i} } \mathord{\left/ {\vphantom {{\mu_{i} } T}} \right. \kern\nulldelimiterspace} T}\), are not independent of each other.
As shown in Eq. 40, a partial Legendre transform \(f^{\left( p \right)}\) of order p of the base function \(f^{\left( 0 \right)} \left( {X_{1} ,X_{2} ,\ldots,X_{n} } \right)\), with \(1 \le p \le \left( {n  1} \right)\), is obtained via subtraction of p products of \(X_{i}\) with its conjugate partial derivative \(c_{i} = \left( {{{\partial f^{\left( 0 \right)} } \mathord{\left/ {\vphantom {{\partial f^{\left( 0 \right)} } {\partial X_{i} }}} \right. \kern\nulldelimiterspace} {\partial X_{i} }}} \right)_{{X_{j \ne i} }}\), i.e., via subtraction of \(\sum\limits_{i = 1}^{p} {c_{i} X_{i} }\). The number of partial Legendre transforms of order p is therefore given by the number of combinations without repetition, that is, by
The total number \(N_{{{\text{Le,}}p}}\) of partial Legendre transforms, that is, the total number of equivalent alternatives to \(f^{\left( 0 \right)}\), is thus obtained from
Since the total number \(N_{{{\text{Le}}}}^{{\text{t}}}\) of Legendre transforms includes the complete transform, it is given by
The complete Legendre transform, that is, the transform of order p = n, vanishes identically.
Application of the above results to the fundamental equations for an open multicomponent PVTx phase either in the internal energy representation, Eq. 15, or in the entropy representation, Eq. 16, is now straightforward. Provided the summation term \(\sum\limits_{i}^{c} {\mu_{i} n_{i} }\) in the former is treated as a single term (in this case \(n = 3\)), the entire number \(N^{{\text{t}}}\) of equivalent primary functions (equivalent thermodynamic potentials) related to the internal energy, including \(nU\), and therefore the number of the corresponding equivalent fundamental property relations, i.e., of the total differentials of these functions, is seven:
They are presented in Tables 1 and 2, respectively, together with the null function and its associated internal energybased Gibbs–Duhem equation [37].
Since the total differentials of the primary functions presented in Table 2 are equivalent, alternatives to the definition of the chemical potential \(\mu_{i}\) of component \(i\) by Eq. 11 are possible:
The last equality, that is,
is the preferred working definition, because T and P are the most useful experimental thermodynamic coordinates. Recalling the definition of a partial molar property, that is,
where \(nM = M^{{\text{t}}} \left( {T,P,n_{1} ,n_{2} ,\ldots} \right)\) is an extensive mixture thermodynamic property, and \(M = M\left( {T,P,x_{1} ,x_{2} ,\ldots} \right)\) stands for any intensive thermodynamic mixture property (excluding, of course, temperature, pressure and composition), we recognize that \(\mu_{i}\) is the partial molar Gibbs energy \(G_{i}\) of component i in solution. Note that a partial molar property \(M_{i} \left( {T,P,x_{1} ,x_{2} ,\ldots} \right)\) is an intensive state function, a property of the mixture, and in general \(M_{i} \ne M_{i}^{ * }\); \(M_{i}\) s obeys the summability relations:
respectively.
Analogically, when treating the summation term \(\sum\limits_{i}^{c} {\frac{{\mu_{i} }}{T}} n_{i}\) in Eq. 16 (entropy representation) as a single term (in this case again \(n = 3\)), the entire number of equivalent primary functions (equivalent thermodynamic potentials) related to the entropy, including \(nS\), and therefore the number of corresponding equivalent fundamental property relations is also seven. They are summarized in Tables 3 and 4, respectively, together with the appropriate null function and its associated entropybased Gibbs–Duhem equation [37]. The replacement of one or more of the extensive variables \(nU,nV,\left\{ {n_{i} } \right\}\) by the corresponding conjugate intensive variable(s) 1/T, P/T and \(\mu_{i} /T\), respectively, yields primary functions known as Massieu–Planck functions. Interestingly, such a Legendre transform of the entropy was already reported by Massieu in 1869, and thus predates the Legendre transforms of the internal energy reported by Gibbs in 1875 (see Callen [30]).
Though not always immediately recognized, the (molar) Massieu–Planck functions are simply related to the (molar) thermodynamic potentials [37]:
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 fluid at constant composition, the following four energybased fundamental property relations apply:
Focusing now on the molar Helmholtz energy and the molar Gibbs energy, Eqs. 62 and 63 yield
and in view of the definitions of F and G (see Table 1) and Eq. 64, the Gibbs–Helmholtz equations
and
are obtained. For simplicity’s sake the subscript \(\left\{ {x_{i} } \right\}\), indicating constant composition, has been omitted. Simple mathematical manipulations yield the following alternative forms:
and
Equation 68 suggests an alternative to the fundamental property relation involving the Gibbs energy as presented in Table 2 i.e.,
By introducing the dimensionless property \({G \mathord{\left/ {\vphantom {G {RT}}} \right. \kern\nulldelimiterspace} {RT}},\)
is obtained, and
Equation 70 is of considerable utility. All terms have the dimension of amountofsubstance and, in contradistinction to Eq. 69, the enthalpy rather than the entropy appears in the first term of the righthand side of this exact differential, with obvious benefits for the discussion of experimental results.
Analogously, Eq. 67 suggests an alternative to the fundamental property relation involving the Helmholtz energy, F, as presented in Table 2 i.e.,
By introducing the dimensionless property \({F \mathord{\left/ {\vphantom {F {RT}}} \right. \kern\nulldelimiterspace} {RT}}\), the corresponding fundamental property relation reads
with
Note, that in contradistinction to Eq. 72, the first term on the righthand side of Eq. 73 contains the internal energy instead of the entropy.
2 Thermodynamic Properties of Fluids
2.1 Residual Properties
The thermodynamic equations formally introduced in Sect. 1.2 establish exact relations between system properties and judiciously selected variables, the most convenient being the sets \(\left\{ {T,P,\left\{ {x_{i} } \right\}} \right\}\) and \(\left\{ {T,V,\left\{ {x_{i} } \right\}} \right\}\). However, they do not provide numerical values for any thermodynamic property: reliable experimental data and reliable models are necessary to reach this goal. The perfect gas [29] (or ideal gas), identified by the superscript pg, is a hypothetical substance for which the intermolecular potential energy is zero, and which obeys (see Eq. 6) the simple PVT equation of state
While no real fluid conforms to this model, nevertheless the concept is eminently useful, because (a) the associated simple equations may frequently be used as reasonable approximations of realgas behavior at low pressures, and (b) the model serves as the basis for the definition of extremely useful auxiliary functions known as residual functions, which play a central role in the description of realfluid behavior. Note, that in the limit of vanishingly small pressure, realgas behavior in many respects approaches perfectgas behavior, but not in all, a fact which is frequently overlooked. Specifically, for a constant composition gas (this includes a pure gas) at constant temperature,
For comparing realfluid properties to perfectgas properties at the same temperature, same pressure and same composition, say, the actual molar volume V to \(V^{{{\text{pg}}}}\), two obvious choices exist: one may quantify deviations in terms of a ratio measure, here the compression factor
or in terms of a difference measure, here the molar residual volume,
The two functions are, of course, related,
and since the zeropressure limit of \(V^{{\text{R}}}\) becomes indeterminate, that is, both the numerator \(\left( {Z  1} \right)\) and the denominator P vanish as the pressure reaches its limiting value zero, de l’Hôpital’s rule yields
Since experiments show that \(\left( {{{\partial Z} \mathord{\left/ {\vphantom {{\partial Z} {\partial P}}} \right. \kern\nulldelimiterspace} {\partial P}}} \right)_{T}\) remains generally finite in the limit of vanishingly small pressure, \(V^{{\text{R}}}\) remains also generally finite; it is given by the second amount densityseries virial coefficient B,
Hence \(V^{{\text{R}}}\) is nonzero except at the Boyle temperature at which \(B = 0\). Equation 81 identifies an experimentally accessible macroscopic property as a key thermophysical quantity establishing an important link to the intermolecular pairpotential energy function \(u\left( r \right)\) [41,42,43]:
Here, \(N_{{\text{A}}}\) is the Avogadro constant, \(k_{{\text{B}}}\) is the Boltzmann constant,^{Footnote 2} and \(N_{{\text{A}}} k_{{\text{B}}} = R\). Note, however, that Eq. 82 is for pairpotential energies that do not depend on the relative orientation of the molecules.
In analogy to \(V^{{\text{R}}}\), molar isobaric residual properties \(M^{{\text{R}}}\) of a singlephase pure fluid or constantcomposition fluid mixture are defined similarly by [17, 18, 29, 46]
where the Ms denote molar values of any extensive thermodynamic property \(nM\left( {T,P,\left\{ {x_{i} } \right\}} \right)\), such as U, H, S, V, G or F. \(M\left( {T,P,\left\{ {x_{i} } \right\}} \right)\) is the actual molar property value of the fluid at the temperature, pressure and composition of interest, and \(M^{{{\text{pg}}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) is the molar property value for the fluid in its perfectgas state at the same T, P and \(\left\{ {x_{i} } \right\}\). Note that the perfectgas state is hypothetical except in the zeropressure limit, where the perfectgas EOS is valid (that is, for this real perfectgas state \(PV = RT\)). Residual properties are the most direct measures of the effects of the intermolecular forces. The computation of values M of any thermodynamic fluid property is based on
From the defining equation, Eq. 83, we have for a differential change in state at constant T and constant \(\left\{ {x_{i} } \right\}\)
and integration from \(P = 0\) to the pressure of interest P yields
In contradistinction to the observed limiting behavior of the residual volume, see Eq. 81, experimental evidence indicates that for the pivotal properties enthalpy and entropy the zeropressure terms can be set equal to zero (at constant T and \(\left\{ {x_{i} } \right\}\)), that is,
Thus, for the molar isobaric residual enthalpy \(H^{{\text{R}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) we obtain, in conjunction with \(\left( {{{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial P}}} \right. \kern\nulldelimiterspace} {\partial P}}} \right)_{{T,\left\{ {x_{i} } \right\}}} = V  T\left( {{{\partial V} \mathord{\left/ {\vphantom {{\partial V} {\partial T}}} \right. \kern\nulldelimiterspace} {\partial T}}} \right)_{{P,\left\{ {x_{i} } \right\}}} \), \(\left( {{{\partial H^{{{\text{pg}}}} } \mathord{\left/ {\vphantom {{\partial H^{{{\text{pg}}}} } {\partial P}}} \right. \kern\nulldelimiterspace} {\partial P}}} \right)_{{T,\left\{ {x_{i} } \right\}}} = 0\), and Eq. 87
Rand for the molar isobaric residual entropy, we obtain, in conjunction with \(\left( {{{\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial P}}} \right. \kern\nulldelimiterspace} {\partial P}}} \right)_{{T,\left\{ {x_{i} } \right\}}} =  \left( {{{\partial V} \mathord{\left/ {\vphantom {{\partial V} {\partial T}}} \right. \kern\nulldelimiterspace} {\partial T}}} \right)_{{P,\left\{ {x_{i} } \right\}}}\), \(\left( {{{\partial S^{{{\text{pg}}}} } \mathord{\left/ {\vphantom {{\partial S^{{{\text{pg}}}} } {\partial P}}} \right. \kern\nulldelimiterspace} {\partial P}}} \right)_{{T,\left\{ {x_{i} } \right\}}} =  {R \mathord{\left/ {\vphantom {R P}} \right. \kern\nulldelimiterspace} P}\), and Eq. 87,
Since Eq. 87 is also valid for \(M = C_{P}\), and the pressure dependence of the molar isobaric heat capacity \(C_{p} = \left( {{{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial T}}} \right. \kern\nulldelimiterspace} {\partial T}}} \right)_{{P,\left\{ {x_{i} } \right\}}}\) is given by
and \(\left( {{{\partial C_{P}^{{{\text{pg}}}} } \mathord{\left/ {\vphantom {{\partial C_{P}^{{{\text{pg}}}} } {\partial P}}} \right. \kern\nulldelimiterspace} {\partial P}}} \right)_{{T,\left\{ {x_{i} } \right\}}} = 0\), the molar isobaric residual constantpressure heat capacity is obtained from
Clearly,
and all three functions approach zero for \(P \to 0\). The isobaric residual functions are the conventional forms since they are advantageously based on \(\left\{ {T,P,\left\{ {x_{i} } \right\}} \right\}\) as independent variables. They have been most useful in applications to real gases and gas mixtures, though their suitability in dealing with liquid systems has been greatly furthered by the application of the correspondingstates theorem (CST): this theorem is firmly based on statistical mechanics and occupies a leading position in the field of property estimation [1, 16,17,18, 36, 47,48,49,50]. The most popular approaches are based on Pitzer’s threeparameter CST correlation [51,52,53,54,55]. For pure (*) fluids, that is, for gases, vapors and liquids, the compression factor \(Z^{ * } = {{PV^{ * } } \mathord{\left/ {\vphantom {{PV^{ * } } {RT}}} \right. \kern\nulldelimiterspace} {RT}}\) is expressed as a function of reduced temperature \(T_{{\text{r}}} \equiv {T \mathord{\left/ {\vphantom {T {T_{{\text{c}}} }}} \right. \kern\nulldelimiterspace} {T_{{\text{c}}} }}\), reduced pressure \(P_{{\text{r}}} \equiv {P \mathord{\left/ {\vphantom {P {P_{{\text{c}}} }}} \right. \kern\nulldelimiterspace} {P_{{\text{c}}} }}\) and acentric factor \(\omega\), which quantity is defined by
Here, \(P_{{{\upsigma },{\text{r}}}} \equiv {{P_{{\upsigma }} } \mathord{\left/ {\vphantom {{P_{{\upsigma }} } {P_{{\text{c}}} }}} \right. \kern\nulldelimiterspace} {P_{{\text{c}}} }}\) is the reduced vapor pressure \(P_{{\upsigma }} \left( T \right)\) at reduced temperature \(T_{{\text{r}}} = 0.7\). Specifically, in the key threeparameter CST correlation
\(Z^{\left( 0 \right)}\) represents the simplefluid contribution to \(Z^{ * }\) that is based on experimental PVT data of Ar, Kr and Xe for which \(\omega\) is essentially zero (and thus a twoparameter CST correlation suffices); \(Z^{\left( 1 \right)}\) represents the nonsimplefluid contribution to \(Z^{ * }\): it is determined via experimental PVT data of selected fluids with \(\omega \ne 0\) (quantum fluids, strongly polar fluids and fluids with strong hydrogen bonds are excluded). Critically evaluated values of the \(\omega\) s for many fluids, together with values for \(T_{{\text{c}}}\) and \(P_{{\text{c}}}\), are tabulated in Ref. 49. One of the best Pitzertype correlations is that developed by Lee and Kesler [56,57,58]: these authors present tables for the contributions \(Z^{\left( 0 \right)} \left( {T_{{\text{r}}} ,P_{{\text{r}}} } \right)\) and \(Z^{\left( 1 \right)} \left( {T_{{\text{r}}} ,P_{{\text{r}}} } \right)\), as well as for derived functions for both liquid and vapor phases, covering large temperature and pressure ranges, i.e., \(0.30 \le T_{{\text{r}}} \le 4.00\) and \(0.01 \le P_{{\text{r}}} \le 10.00\).
In order to use generalized CST correlations for mixtures, conventional practice is based on the assumption that mixture properties can be represented by the same correlation types developed for pure fluids, though with appropriately defined values for the correspondingstates scaling parameters of the mixture, that is, by essentially empirically averaging purecomponent parameters \(T_{{{\text{c,}}i}}\), \(P_{{{\text{c,}}i}}\) and \(\omega_{i}\) to obtain pseudocritical temperatures \(T_{{{\text{pc}}}}\), pseudocritical pressures \(P_{{{\text{pc}}}}\) and pseudoacentric factors \(\omega_{{\text{p}}}\) referring to the mixture. This is accomplished by using recipes known as mixing rules. Thus, a threeparameter CST correlation for the mixture compression factor Z, in the onefluid approximation, may be written as
where the pseudoreduced temperature \(T_{{{\text{pr}}}}\) and the pseudoreduced pressure \(P_{{{\text{pr}}}}\) are defined by
The simplest set of mixing rules for pseudocritical parameters are those of Kay [59]. They are defined as molefractionweighted sums of the purecomponent values, and so is \(\omega_{{\text{p}}}\), i.e.,
Although simple to apply, for mixtures of molecularly noticeably dissimilar fluids Kay’s rules are often inadequate, and more flexible and thus more elaborate recipes must be introduced [49], such as quadratic mixing rules (reminiscent of those used in the multicomponent vdW model):
where \(T_{{{\text{c,}}ii}}\) and \(T_{{{\text{c,}}jj}}\) denote the critical temperatures of the pure components i and j, respectively, \(P_{{{\text{c,}}ii}}\) and \(P_{{{\text{c,}}jj}}\) are their critical pressures, respectively, and for the evaluation of the cross parameters \(T_{{{\text{c,}}ij}}\) and \(P_{{{\text{c,}}ij}}\) empirical recipes known as combining rules are required.
However, I emphasize that temperature and volume, or alternatively, amount density \(\rho_{n} \equiv {1 \mathord{\left/ {\vphantom {1 V}} \right. \kern\nulldelimiterspace} V}\), or number density, are the commonly used variables in statistical mechanics; and in addition, on the practical side, most PVTx equations of state, such as cubic vdWtype equations, are pressureexplicit [1, 16,17,18, 49, 60,61,62,63,64,65]: hence T and V (and for mixtures, of course, also the composition \(\left\{ {x_{i} } \right\}\)) are the natural (canonical) independent variables. Thus, for a singlephase pure fluid or constantcomposition mixture one may also define molar residual functions in \(\left( {T,V,\left\{ {x_{i} } \right\}} \right)\)space, i.e., molar isochoric (isometric) residual functions
Again, the Ms denote molar properties of any extensive thermodynamic property \(nM\left( {T,nV,\left\{ {x_{i} } \right\}} \right)\) of the fluid, for instance, U, H, S, G or F, and \(M^{{{\text{pg}}}} \left( {T,V,\left\{ {x_{i} } \right\}} \right)\) is the corresponding molar property of the fluid in its hypothetical perfectgas state at the same temperature, the same molar volume and the same composition. Note the important fact that if the temperature and the volume are the same for the real fluid state and the perfect gas state, the pressure is not the same. Equation 99 may also be used for the definition of a residual pressure, an intensive property:
With the availability of a pressureexplicit EOS in the form \(P = P\left( {T,V,\left\{ {x_{i} } \right\}} \right)\), isochoric residual properties are the properties of direct interest. In complete analogy to Eqs. 86 and 87 we have
Again, experimental evidence indicates that for the pivotal properties internal energy and entropy the infinitevolume terms (zeropressure terms) can be set equal to zero (at constant T and \(\left\{ {x_{i} } \right\}\)), that is,
Thus, for the molar isochoric residual internal energy \(U^{{\text{r}}} \left( {T,V,\left\{ {x_{i} } \right\}} \right)\) we obtain, in conjunction with \(\left( {{{\partial U} \mathord{\left/ {\vphantom {{\partial U} {\partial V}}} \right. \kern\nulldelimiterspace} {\partial V}}} \right)_{{T,\left\{ {x_{i} } \right\}}} =  P + T\left( {{{\partial P} \mathord{\left/ {\vphantom {{\partial P} {\partial T}}} \right. \kern\nulldelimiterspace} {\partial T}}} \right)_{{V,\left\{ {x_{i} } \right\}}}\), \(\left( {{{\partial U^{{{\text{pg}}}} } \mathord{\left/ {\vphantom {{\partial U^{{{\text{pg}}}} } {\partial V}}} \right. \kern\nulldelimiterspace} {\partial V}}} \right)_{{T,\left\{ {x_{i} } \right\}}} = 0\), and Eq. 102,
and for the molar isochoric residual enthalpy \(H^{{\text{r}}} \left( {T,V,\left\{ {x_{i} } \right\}} \right)\) we obtain, in conjunction with \(\left( {{{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial V}}} \right. \kern\nulldelimiterspace} {\partial V}}} \right)_{{T,\left\{ {x_{i} } \right\}}} = T\left( {{{\partial P} \mathord{\left/ {\vphantom {{\partial P} {\partial T}}} \right. \kern\nulldelimiterspace} {\partial T}}} \right)_{{V,\left\{ {x_{i} } \right\}}} + V\left( {{{\partial P} \mathord{\left/ {\vphantom {{\partial P} {\partial V}}} \right. \kern\nulldelimiterspace} {\partial V}}} \right)_{{T,\left\{ {x_{i} } \right\}}} ,\left( {{{\partial H^{{{\text{pg}}}} } \mathord{\left/ {\vphantom {{\partial H^{{{\text{pg}}}} } {\partial V}}} \right. \kern\nulldelimiterspace} {\partial V}}} \right)_{{T,\left\{ {x_{i} } \right\}}} = 0\), and Eq. 87, at constant temperature and composition
For the molar isochoric residual entropy \(S^{{\text{r}}} \left( {T,V,\left\{ {x_{i} } \right\}} \right)\) we obtain, in conjunction with \(\left( {{{\partial S} \mathord{\left/ {\vphantom {{\partial S} {\partial V}}} \right. \kern\nulldelimiterspace} {\partial V}}} \right)_{{T,\left\{ {x_{i} } \right\}}} = \left( {{{\partial P} \mathord{\left/ {\vphantom {{\partial P} {\partial T}}} \right. \kern\nulldelimiterspace} {\partial T}}} \right)_{{V,\left\{ {x_{i} } \right\}}}\), \(\left( {{{\partial S^{{{\text{pg}}}} } \mathord{\left/ {\vphantom {{\partial S^{{{\text{pg}}}} } {\partial V}}} \right. \kern\nulldelimiterspace} {\partial V}}} \right)_{{T,\left\{ {x_{i} } \right\}}} = {R \mathord{\left/ {\vphantom {R V}} \right. \kern\nulldelimiterspace} V}\), and Eq. 101, at constant temperature and composition
Since \(C_{V}\) of a real gas approaches \(C_{V}^{{{\text{pg}}}}\) for \(V \to \infty\) (i.e., for \(P \to 0\)), Eqs. 101 and 102 apply, and together with \(\left( {{{\partial C_{V}^{{{\text{pg}}}} } \mathord{\left/ {\vphantom {{\partial C_{V}^{{{\text{pg}}}} } {\partial V}}} \right. \kern\nulldelimiterspace} {\partial V}}} \right)_{{T,\left\{ {x_{i} } \right\}}} = 0\), and
the molar isochoric residual constantvolume heat capacity \(C_{V}^{{\text{r}}} \left( {T,V,\left\{ {x_{i} } \right\}} \right)\) at constant T and \(\left\{ {x_{i} } \right\}\) may be calculated via
Clearly,
and all three functions approach zero for \(V \to \infty\) (i.e., for \(P \to 0\)). For the computation of values M of any thermodynamic fluid property,
is used.
The two types of residual functions, i.e., \(M^{{\text{R}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) and \(M^{{\text{r}}} \left( {T,V,\left\{ {x_{i} } \right\}} \right)\), are rigorously related:
Note that P and V are parameters associated with the state of the real fluid system at temperature T and constant composition \(\left\{ {x_{i} } \right\}\), and are therefore not related by the perfectgas law: the lower integral limit denotes the gaspressure \(P = {{RT} \mathord{\left/ {\vphantom {{RT} V}} \right. \kern\nulldelimiterspace} V}\) for which the molar volume of the perfectgas mixture has the same value V as that of the real mixture at T and \(\left\{ {x_{i} } \right\}\).
Alternatively, we have
Note that \(V^{{\text{r}}} \left( {T,V,\left\{ {x_{i} } \right\}} \right)\) and \(P^{{\text{R}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) are identically zero.
Since at constant composition the perfectgas properties \(U^{{{\text{pg}}}} ,H^{{{\text{pg}}}} ,C_{P}^{{{\text{pg}}}}\) and \(C_{V}^{{{\text{pg}}}}\) are all functions of temperature only, i.e., the firstlaw properties are independent of pressure and of volume, the equality
In contradistinction, the secondlaw perfectgas properties \(S^{{{\text{pg}}}}\), \(G^{{{\text{pg}}}}\) and \(F^{{{\text{pg}}}}\) are functions of temperature and they do depend on pressure:
and they do depend on volume:
Hence, by virtue of Eq. 110 or Eq. 111, respectively, the following relations between the residual secondlaw properties are obtained:
2.2 Isobaric Residual Properties, Fugacities and Fugacity Coefficients
One of the most important functions in solution chemistry is the fugacity coefficient \(\phi_{i}^{{\uppi }} \left( {T,P,\left\{ {x_{i}^{{\uppi }} } \right\}} \right)\) of component i in solution in phase π [66]. It is related to the isobaric residual chemical potential, which may be obtained either by applying the partial molar derivative prescription
to the expression for the molar isobaric residual Gibbs energy of the mixture in phase π,
or by substituting the expression for the chemical potential of component i in a perfectgas mixture
directly into the defining equation for the isobaric residual chemical potential (the two formulations follow from Eq. 52b). Hence the partial molar isobaric residual Gibbs energy/the isobaric residual chemical potential of component i in solution in phase π reads
providing a rigorous basis for the definition of the fugacity coefficient of component i in solution in phase π. The fugacity concept was introduced by Lewis [67]. It serves to maintain the simple formal structure of thermodynamic equations applicable to perfectgas (idealgas) systems, while avoiding the troublesome behavior of the chemical potential when either P or \(x_{i}^{{\uppi }}\) approaches zero. Thus, in analogy to the expression for an isothermal change of the chemical potential of component i in a perfectgas mixture,
where \(P_{i} = x_{i} P\) is the partial pressure of component i, the fugacity \(f_{i}^{{\uppi }} \left( {T,P,\left\{ {x_{i}^{{\uppi }} } \right\}} \right)\) of component i in a real solution phase \({\uppi }\) with composition \(\left\{ {x_{i}^{{\uppi }} } \right\}\) is partially defined by.
For thermodynamic consistency we require as the second part of the definition
that is,
Equations 125 and 126together constitute the complete definition of the fugacity \(f_{i}^{{\uppi }}\) of component i in solution in phase \({\uppi }\). General integration of Eq. 125 at constant temperature [17] from the state of component i in a perfectgas mixture to the state of i in the real mixture (phase \({\uppi }\)) at the same pressure and the same composition yields
where the partial molar residual Gibbs energy in \(\left( {T,P,x} \right)\)space of component i in solution in phase π, \(G_{i}^{{{\text{R,}{\uppi }}}} \left( {T,P,\{ x_{i}^{{\uppi }} \} } \right)\), and the residual chemical potential in \(\left( {T,P,x} \right)\)space of component i in solution in phase π, \(\mu_{i}^{{{\text{R,}{\uppi }}}} \left( {T,P,\{ x_{i}^{{\uppi }} \} } \right)\), are given by Eq. 123. The dimensionless ratio appearing on the righthand side of Eq. 128 as the argument of the logarithm is a new property and is called the fugacity coefficient \(\phi_{i}^{{\uppi }}\) of component i in solution in phase π:
or, perhaps, more convenient,
For a perfectgas system, necessarily \(G_{i}^{{{\text{R,}{\uppi }}}} = \mu_{i}^{{{\text{R,}{\uppi }}}} = 0\) and thus \(\phi_{i}^{{{\text{pg}}}} = 1\). For a real pure substance i we obtain
where \(G_{i}^{{{\text{R,}{\uppi ,}} * }} \left( {T,P} \right)\) denotes the molar isobaric residual Gibbs energy of pure component i in phase π, \(f_{i}^{{{\pi ,} * }} \left( {T,P} \right)\) is its fugacity, and
is the fugacity coefficient \(\phi_{i}^{{{\pi ,} * }}\) of pure i in phase π, or, perhaps, more convenient
For the overall fugacity of the solution in phase π we have
where \(G^{{{\text{R,}{\uppi }}}} \left( {T,P,\{ x_{i}^{{\uppi }} \} } \right)\) is the molar isobaric residual Gibbs energy of the solution, and
is the overall fugacity coefficient of the solution in phase π, or, perhaps, more convenient,
Evidently, fugacity, having the dimension of pressure, and the dimensionless fugacity coefficient are intensive state functions related to exponentials of (partial) molar isobaric residual Gibbs energies (divided by RT) as indicated above by Eqs. 129, 132 and 135. Viewing the fugacity as a “corrected pressure” may be misleading and obscure these exact relations. As the pressure goes to zero (at constant temperature and composition), the (partial) molar residual Gibbs energies divided by RT, as introduced above, all approach zero on the basis that the perfectgas state is approached asymptotically; hence \(f_{i}^{{\uppi }} \to x_{i}^{{\uppi }} P \equiv P_{i}\), \(f_{i}^{{{\pi ,} * }} \to P\) and \(f^{{\uppi }} \to P\), respectively. Thus,
and
The fugacity coefficients are always positive: for real systems, they may be larger than 1 or smaller than 1, and they are unity for a perfectgas system.
Since \(G_{i}^{{{\text{R,}{\uppi }}}} \left( {T,P,\{ x_{i}^{{\uppi }} \} } \right)\) is the partial molar isobaric residual Gibbs energy of component i in phase π, i.e.,
the associated summability relation yields the molar residual Gibbs energy of the solution
For convenience, the nondimensional quantity \({{G^{{{\text{R,}{\uppi }}}} } \mathord{\left/ {\vphantom {{G^{{{{R,\pi }}}} } {RT}}} \right. \kern\nulldelimiterspace} {RT}}\) is frequently used instead of \(G^{{{\text{R,}{\uppi }}}}\), hence
We recognize that \(\ln \phi_{i}^{{\uppi }}\) is a partial molar property in relation to \(\ln \phi^{{\uppi }},\) where \(\phi^{{\uppi }}\) is the overall fugacity coefficient of the solution in phase π:
The fundamental residualproperty relation (canonical variables T, P, \(n_{i}^{{\uppi }}\)) valid for a fluid in any phase π, may now be written as
or, alternatively,
Here,
where \(H^{{{\text{R,}{\uppi }}}}\) is the molar isobaric residual enthalpy of the solution in phase π, that is,
and
where \(V^{{{\text{R,}{\uppi }}}}\) is the molar isobaric residual volume of the solution in phase π, that is,
Evidently, the partial molar property analogues of Eqs. 145 and 147 are
and
respectively. Here, \(H_{i}^{{{\text{R,}{\uppi }}}}\) is the partial molar isobaric residual enthalpy, and \(V_{i}^{{{\text{R,}{\uppi }}}}\) is the partial molar isobaric residual volume of component i in solution in phase π, respectively. The corresponding summability relations read
and
Evaluation of \(\ln \phi_{i}^{{\uppi }} \left( {T,P,\{ x_{i}^{{\uppi }} \} } \right)\), using an equation of state in conjunction with Eq. 150, is now straightforward. Since the molar volume of pure component i in the perfectgas state is given by \(V_{i}^{{{\text{pg,}} * }} = {{RT} \mathord{\left/ {\vphantom {{RT} P}} \right. \kern\nulldelimiterspace} P}\), and its partial molar volume in phase π by \(V_{i}^{{\uppi }} = \frac{RT}{P}\left[ {{{\partial \left( {n^{{\uppi }} Z^{{\uppi }} } \right)} \mathord{\left/ {\vphantom {{\partial \left( {n^{{\uppi }} Z^{{\uppi }} } \right)} {\partial n_{i}^{{\uppi }} }}} \right. \kern\nulldelimiterspace} {\partial n_{i}^{{\uppi }} }}} \right]_{{T,P,\left\{ {n_{j \ne i}^{{\uppi }} } \right\}}}\), integration at constant T and constant composition from P = 0 (where \(\phi_{i}^{{\uppi }} = 1\)) to arbitrary pressure P yields
Equation 153a is a generally valid relation for the determination of \(\ln \phi_{i}^{{\uppi }} \left( {T,P,\{ x_{i}^{{\uppi }} \} } \right)\) from any volumeexplicit EOS. By way of example, consider the binary vaporphase mixture (\({{\uppi =}{\text{V}}}\), \(x_{1}^{{\text{V}}} = 1  x_{2}^{{\text{V}}}\)) adequately described by a twoterm virial equation in pressure, i.e., \(Z^{{\text{V}}} \left( {T,P,x_{2}^{{\text{V}}} } \right) = 1 + \frac{{B\left( {T,x_{2}^{{\text{V}}} } \right)}}{RT}P\). After some mathematical manipulation we obtain the compact expression for the fugacity coefficients of component 1 and 2, respectively, in the binary vapor mixture:
Here, \(B\left( {T,x_{2}^{{\text{V}}} } \right) = x_{1}^{{\text{V}}} B_{11} \left( T \right) + x_{2}^{{\text{V}}} B_{22} \left( T \right) + x_{1}^{{\text{V}}} x_{2}^{{\text{V}}} \delta_{12} \left( T \right)\) is the second virial coefficient of the mixture, \(\delta_{12} \equiv 2B_{12}  \left( {B_{11} + B_{22} } \right)\), B_{11} and B_{22} are the second virial coefficients of the pure components, and B_{12} designates a compositionindependent interaction virial coefficient (crosscoefficient). When focusing on highly dilute systems, the fugacity coefficient of, say, component 2 at infinite dilution in the vapor phase is thus given by
For pure substance i in phase \({\uppi }\) we have
and
To conclude this section, I emphasize that \({{G^{{{\text{R,}{\uppi }}}} \left( {T,P,\{ x_{i}^{{\uppi }} \} } \right)} \mathord{\left/ {\vphantom {{G^{{{{R,\pi }}}} \left( {T,P,\{ x_{i}^{{\uppi }} \} } \right)} {RT}}} \right. \kern\nulldelimiterspace} {RT}}\) is a convenient generating function for molar isobaric residual properties (see Eq. 144a).
In principle, the exact classical thermodynamic method of using isobaric residual functions for the calculation of property changes of singlephase constantcomposition fluids for any arbitrary equilibrium change of state is a quite general and powerful tool; and applications profit greatly from contributions based on the corresponding states theorem, such as the popular Lee–Kesler tables [56,57,58]. However, for a complete description, the perfectgas heat capacities \(C_{P}^{{{\text{pg}}}} \left( {T,\left\{ {x_{i} } \right\}} \right)\) must be available.
To illustrate the convenient use of isobaric residual functions in the calculation of property changes for singlephase, constantcomposition fluids (pure or multicomponent systems), consider an arbitrary change of state, say, from an initial condition characterized by \(\left\{ {T_{1} , \, P_{1} } \right\}\) to a final condition \(\left\{ {T_{2} , \, P_{2} } \right\}\). Such a change of state causes a change
of any molar thermodynamic property \(M\left( {T,P,\left\{ {x_{i} } \right\}} \right)\). When interest focuses on M = H, the change of enthalpy per mole of mixture \(\Delta_{1 \to 2} H\) is given by
Here, \(\left( {{{\partial H^{{{\text{pg}}}} } \mathord{\left/ {\vphantom {{\partial H^{{{\text{pg}}}} } {\partial P}}} \right. \kern\nulldelimiterspace} {\partial P}}} \right)_{{T,\left\{ {x_{i} } \right\}}} = 0\) and \(\left( {{{\partial H^{{{\text{pg}}}} } \mathord{\left/ {\vphantom {{\partial H^{{{\text{pg}}}} } {\partial T}}} \right. \kern\nulldelimiterspace} {\partial T}}} \right)_{{P,\left\{ {x_{i} } \right\}}} = C_{P}^{{{\text{pg}}}}\) were taken into account. For \(M = S\), the change of entropy per mole of mixture \(\Delta_{1 \to 2} S\) is given by
since \({\text{d}}S^{{{\text{pg}}}} = \left( {{{C_{P}^{{{\text{pg}}}} } \mathord{\left/ {\vphantom {{C_{P}^{{{\text{pg}}}} } T}} \right. \kern\nulldelimiterspace} T}} \right){\text{d}}T  \left( {{R \mathord{\left/ {\vphantom {R P}} \right. \kern\nulldelimiterspace} P}} \right){\text{d}}P\). Evaluation of the integrals involving \(C_{P}^{{{\text{pg}}}}\) requires empirical expressions for the temperature dependence of the perfectgas heat capacities involved [68]. For pure fluids, a commonly used empirical equation [49] reads
and extensive data compilations are available [69,70,71]. For fairly simple molecules, \(C_{P}^{{{\text{pg,}} * }} \left( T \right)\) may be calculated using the frequencies of their normal modes of vibration [72,73,74].
The heat capacities of perfectgas mixtures are rigorously obtained by molefraction averaging the puresubstance perfectgas heat capacities, that is,
and their difference is the same as that for pure perfect gases, \(C_{P}^{{{\text{pg}}}} \left( {T,\left\{ {x_{i} } \right\}} \right)  C_{V}^{{{\text{pg}}}} \left( {T,\left\{ {x_{i} } \right\}} \right) = R.\)
Since the perfectgas state is characterized by the absence of intermolecular forces, the correspondingstates theorem is inapplicable, and when experimental data on \(C_{P}^{{{\text{pg,}} * }} \left( T \right)\) are lacking, reliable estimation methods based on molecular structure have to be used [49, 75,76,77].
2.3 Empirical Equations of State: Selected Comments
From PVT equations of state, volumetric properties as well as residual functions characterizing deviations from perfectgas (idealgas) behavior can be calculated (see Sects. 2.1 and 2.2). The PVT relation may be a pressureexplicit EOS or a volumeexplicit (amount densityexplicit) EOS. The simplest, practically useful polynomial EOS are cubic in molar volume (i.e., they are pressureexplicit), since they are capable of yielding the perfectgas limit for \(V \to \infty\), and of representing both liquidlike and vaporlike volumes for sufficiently low temperatures. The fiveparameter equation [17]
where the adjustable parameters \(b,\theta ,\eta ,\delta ,\varepsilon\) depend, in general, on temperature and composition, can be considered a generalization of the vdW equation Eq. 3 [61], to which it reduces for \(\eta = b\), \(\delta = \varepsilon = 0\), and \(\theta = {\text{constant = }}a\). Over the decades, many specializations of Eq. 161 have been suggested [78], two of the most popular being the Redlich–Kwong (RK) Eq. (1949) [79],
where \(b = b_{{{\text{RK}}}}\), \(\theta = \theta_{{{\text{RK}}}} \left( T \right) \equiv {{a_{{{\text{RK}}}} } \mathord{\left/ {\vphantom {{a_{{{\text{RK}}}} } {T^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern\nulldelimiterspace} 2}}} }}} \right. \kern\nulldelimiterspace} {T^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern\nulldelimiterspace} 2}}} }}\), \(\eta = b_{{{\text{RK}}}}\), \(\delta = b_{{{\text{RK}}}}\) and \(\varepsilon = 0\), and the Peng–Robinson (PR) Eq. (1976) [80],
where \(b = b_{{{\text{PR}}}}\), \(\theta = \theta_{{{\text{PR}}}} \left( T \right) \equiv a_{{{\text{PR}}}} \left( T \right)\), \(\eta = b_{{{\text{PR}}}}\), \(\delta = 2b_{{{\text{PR}}}}\), and \(\varepsilon =  b_{{{\text{PR}}}}^{2}\).
Significantly, all modern cubic equations of state have a temperaturedependent \(\theta\). While the RK equation gives a somewhat better critical compression factor than the original van der Waals EOS (see Eq. 8 and associated comments), i.e.,
and better second virial coefficients [81], it is still not very accurate for vapor pressures and liquid densities. It was Soave (S) [82, 83] who generalized, in 1972, the temperature dependence of the attractive parameter in the RK equation by writing, with \(\theta = \theta_{{\text{S}}} \left( T \right) \equiv a_{{\text{S}}} \left( T \right)\)),
In order to obtain the EOS parameters \(a_{{\text{S}}} \left( T \right)\) and \(b_{{\text{S}}}\), the most frequently employed conventional approach makes use of the mathematical requirements for the occurrence of an inflection point on the critical isotherm in the \(\left\{ {P,V} \right\}\)plane, that is, \(\left( {{{\partial P} \mathord{\left/ {\vphantom {{\partial P} {\partial V}}} \right. \kern\nulldelimiterspace} {\partial V}}} \right)_{{T_{{\text{c}}} }} = 0\) and \(\left( {{{\partial^{2} P} \mathord{\left/ {\vphantom {{\partial^{2} P} {\partial V^{2} }}} \right. \kern\nulldelimiterspace} {\partial V^{2} }}} \right)_{{T_{{\text{c}}} }} = 0\) at the critical point (\(P = P_{{\text{c}}} {, }V = V_{{\text{c}}}\)). When applying these classical critical constraints, experience shows that the preferred sets of expressions for evaluating \(a_{{\text{S}}} \left( {T_{{\text{c}}} } \right) \equiv a_{{\text{S,c}}}\) and \(b_{{\text{S}}}\) are in terms of \(T_{{\text{c}}}\) and \(P_{{\text{c}}}\), simply because critical pressures are known for more substances with better precision than critical molar volumes \(V_{{\text{c}}}\)[1, 49]. For the Soave equation of state we obtain
and the same value for the critical compression factor as that of the Redlich–Kwong equation of state, i.e., \(Z_{{\text{c,S}}} = \frac{1}{3}\).
The Soave equation of state, Eq. 165, may now be written as
Thus, at temperatures \(T \le T_{{\text{c}}}\), the attraction parameter \(a_{{\text{S}}} \left( T \right)\) is now expressed as the product of its value at the critical point and a dimensionless temperaturedependent αfunction, that is
Clearly, for the limiting value of the αfunction as \(T \to T_{{\text{c}}}\) (i.e.,\(T_{{\text{r}}} \to 1\)) we require
This empirical function was primarily formulated to help better fit vapor pressure data of hydrocarbons, and Soave suggested as a generally useful form
whose limiting value for \(T \to T_{{\text{c}}}\) is one, in accord with Eq. 169. With a substantially expanded property set of vapor pressure data becoming available (a then new compilation from the American Petroleum Institute), the \(\alpha_{{\text{S}}} \left( {T,\omega } \right)\)relation was refitted by Graboski and Daubert [84,85,86] to yield an improved expression with
When applying the classical critical constraints to the Peng–Robinson equation of state, Eq. 163, we obtain
and the EOS may be rewritten as follows:
Thus, at temperatures \(T \le T_{{\text{c}}}\), the attraction parameter \(\theta_{{{\text{PR}}}} \left( T \right) = a_{{{\text{PR}}}} \left( T \right)\) is now expressed as the product of its value at the critical point and a dimensionless temperaturedependent αfunction, that is,
again with the constraint
Correlating vaporpressure data with the Peng–Robinson equation of state yielded
Over the years, many other αfunctions of increasing complexity have been proposed. The Soavetype αfunction is a generalized, polynomial expression which has been honed over decades, thereby making it so popular. It has been recognized that adequately formulated αfunctions substantially improved predictions of vapor pressure, especially of polar liquids [87]. However, its behavior was established for subcritical conditions only, and theorybased guidelines for developing adequate αfunction in general, and for supercritical conditions in particular, have been lacking until very recently. Based on a careful thermodynamic/mathematical analysis of the problem, researchers from Nancy, France, that is, Le Guennec, Lasala, Privat and Jaubert, have established the requirements for consistent αfunctions to be used in cubic equations of state applied at both subcritical and supercritical conditions [88,89,90].
The method of parameter evaluation for the PR equation of state remains unchanged, and the critical compression factor is \(Z_{{\text{c,PR}}} = 0.307\). This value is nearer the common experimental values (particularly for nonpolar compounds) and explains partially the fact that the PR equation of state predicts liquid densities more accurately than the Soave equation of state.
Despite the inherent empiricism [91], both the Soave and the Peng–Robinson EOS have remained mainstays for calculating thermodynamic properties of fluids and vapor–liquid equilibria: nearly 150 years after van der Waals’ dissertation [24], generalized vdW equations are still a hot topic in chemical engineering.
The greatest utility of cubic equations of state is for vapor–liquid phase equilibrium calculations involving mixtures. Extension of vdWtype model equations to multicomponent mixtures rests upon the inherent assumption that the same EOS used for the pure fluid components can be used for the ccomponent mixture, provided adequate prescriptions for the mixture parameters are available. The van der Waals onefluid approximation provides such recipes for the composition dependence of the mixture interaction energy parameter a, say, \(a_{{\text{S}}}\) or \(a_{{{\text{PR}}}}\), and of the mixture molecular size parameter b, say, \(b_{{\text{S}}}\) or \(b_{{{\text{PR}}}}\). These mixing rules are quadratic in mole fraction:
where \(x_{i}\) and \(x_{j}\) are the mole fractions of components i and j, respectively. The attractive vdW interaction parameters of the pure components are denoted by \(a_{ii}\) and \(a_{jj}\), while \(a_{ij}\) measures the strength of the attractive interaction between unlike molecules. Similarly, the composition dependence of the mixture parameter b (also known as the mixture covolume) is also approximated by a quadratic mixing rule
where \(b_{ii}\) and \(b_{jj}\) denote the vdW size parameters (covolumes) of the pure components, and \(b_{ij}\) characterizes the repulsive interaction between unlike molecules.
For threeparameter cubic equations of state, such as the Patel–Teja (PT) equation [92],
where the parameters in the generalized vdW equation, Eq. 161, are given by \(b = b_{{{\text{PT}}}}\), \(\theta = \theta_{{{\text{PT}}}} \left( T \right) \equiv a_{{{\text{PT}}}} \left( T \right)\), \(\eta = b_{{{\text{PT}}}}\), \(\delta = b_{{{\text{PT}}}} + c_{{{\text{PT}}}}\), and \(\varepsilon =  b_{{{\text{PT}}}} c_{{{\text{PT}}}}\), a similar mixing rule for the third parameter, here for \(c_{{{\text{PT}}}}\), is usually assumed:
I emphasize that these commonly used mixing rules are semiempirical approximations, and alternative recipes could be used and have indeed been suggested. However, to apply Eqs. 177 and 178 (and 180), for unlike interactions \(\left( {i \ne j} \right)\) the crossinteraction parameters \(a_{ij}\) and \(b_{ij}\) (and \(c_{ij}\)) have to be known: evaluation of these quantities in terms of puresubstance parameters is one of the key problems in molecular thermodynamics [1, 41,42,43, 49, 60, 93, 94]. The most common choices for combining rules are the geometricmean rule for \(a_{ij}\) suggested by Galitzine in 1890 [95], and later by Berthelot in 1898 [96, 97], and the arithmeticmean rule for \(b_{ij}\), though in engineering calculations, both are routinely modified on an empirical basis:
The parameters \(k_{ij}\) and \(l_{ij}\) are known as binary interaction parameters. For threeparameter cubic equations of state, \(c_{ij}\) is typically approximated by
\(m_{ij}\) being another empirical binary interaction parameter.
So far, every cubic EOS that has been proposed has some limitations, either with respect to the range of operating conditions or types of fluids it could be applied to. Further progress in developing pressureexplicit multiparameter equations of state, was initially greatly stimulated by work on the virial equation in the amount density \(\rho_{n} = {1 \mathord{\left/ {\vphantom {1 V}} \right. \kern\nulldelimiterspace} V}\), which for a pure gas/vapor at not too high densities reads
This equation has a sound theoretical foundation [98]. Here, \(B\left( T \right)\) is the second virial coefficient, \(C\left( T \right)\) is the third virial coefficient, and so forth. The celebrated EOS of Benedict, Webb and Rubin (BWR) [99,100,101] of 1940,
was an important step in the right direction and represented the volumetric properties of industrially important fluids (hydrocarbons) reasonably well. However, the BWR equation of state was found to give unsatisfactory results in lowtemperature applications, at high fluid densities, and in the critical region. To alleviate these deficiencies, many modifications of the BWR equation, with many more parameters, have been proposed [56,57,58, 102,103,104,105] and are still widely used.
As already pointed out, fundamental equations have a great inherent advantage over volumetric PVTx equations of state: they contain complete information on the thermodynamic system. Thus, once a judiciously selected empirical relation has been developed for one of the fundamental equations, say, for \(nU = U^{{\text{t}}} \left( {nS,nV,\left\{ {n_{i} } \right\}} \right)\), or \(nH = H^{{\text{t}}} \left( {nS,P,\left\{ {n_{i} } \right\}} \right)\), or \(nF = F^{{\text{t}}} \left( {T,nV,\left\{ {n_{i} } \right\}} \right)\), or \(nG = G^{{\text{t}}} \left( {T,P,\left\{ {n_{i} } \right\}} \right)\), in principle, all other thermodynamic fluid properties can be calculated by combinations of appropriate derivatives. Since the independent variable entropy nS is not directly measurable, neither the fundamental equation formulated in terms of internal energy, nor that formulated in terms of enthalpy, are used for developing empirical correlational equations. The advantage of using \(nG = G^{{\text{t}}} \left( {T,P,\left\{ {n_{i} } \right\}} \right)\) is that the intensive canonical variables T and P are easily measured, monitored and controlled; this makes the Gibbs energy so important in Physical Chemistry as well as in Chemical Engineering. However, due to the discontinuity in slope of the Gibbs energy surface at the liquid (L)/vapor (V) phase boundary, i.e.,
\(nG = G^{{\text{t}}} \left( {T,P,\left\{ {n_{i} } \right\}} \right)\) can only be used to represent the liquid part of the Gibbs energy surface or the vapor part separately, thus precluding a closed mathematical description of the entire fluid range (P = \(P_{{\upsigma }}\) denotes the vapor pressure, with the subscript σ indicating saturation conditions) [63, 64]. In contradistinction, formulations based on Helmholtz energybased fundamental equations, using canonical variables T and V (or T and \(\rho_{n}\)), are suitable for describing the entire fluid region. They are valid for liquid and vapor states, for equilibria between them, and the description of supercritical states is included. Their validity range terminates at the melting curve, thus allowing the calculation of liquidphase properties, but not of properties of the coexisting solid phase. Modern fundamental equations are usually based on the Helmholtz energy [63, 64, 106,107,108]. However, in practical applications the dimensionless property \({{F\left( {T,V} \right)} \mathord{\left/ {\vphantom {{F\left( {T,V} \right)} {RT}}} \right. \kern\nulldelimiterspace} {RT}}\) (see Eq. 73) is usually replaced by \({{F\left( {T,\rho_{n} } \right)} \mathord{\left/ {\vphantom {{F\left( {T,\rho_{n} } \right)} {RT}}} \right. \kern\nulldelimiterspace} {RT}}\), \(\rho_{n} = {1 \mathord{\left/ {\vphantom {1 V}} \right. \kern\nulldelimiterspace} V}\), which quantity is split into a residual part and a perfectgas (idealgas) part, where both are empirically expressed by dimensionless functions of the inverse reduced temperature \(\tau \equiv {{T_{{\text{c}}} } \mathord{\left/ {\vphantom {{T_{{\text{c}}} } T}} \right. \kern\nulldelimiterspace} T}\) and the reduced amount density, i.e., the inverse reduced molar volume, \(\delta \equiv {{\rho_{n} } \mathord{\left/ {\vphantom {{\rho_{n} } {\rho_{{n,{\text{c}}}} = {{V_{{\text{c}}} } \mathord{\left/ {\vphantom {{V_{{\text{c}}} } V}} \right. \kern\nulldelimiterspace} V}}}} \right. \kern\nulldelimiterspace} {\rho_{{n,{\text{c}}}} = {{V_{{\text{c}}} } \mathord{\left/ {\vphantom {{V_{{\text{c}}} } V}} \right. \kern\nulldelimiterspace} V}}}:\)
The commonly used functional form of the residual term is that found in modified BWR equations. For details, see Refs. [63, 64, 106,107,108]. The most important application of the approach represented by Eq. 187 is the IAPWS (International Association for the Properties of Water and Steam) formulation 1995 for the thermodynamic properties of water prepared by Wagner and Pruß [108]. By using highprecision experimental data and applying modern strategies for optimizing the functional form of the EOS and for the simultaneous nonlinear data fitting, they obtained the socalled IAPWS95 formulation covering the temperature range from the melting line to 1273 K and pressures up to 1000 MPa. In this range, IAPWS95 represents all data to within their experimental uncertainty. In the critical region, data are represented very well, and IAPWS95 can be reasonably extrapolated up to extremely high pressures and temperatures. The equation for the dimensionless perfectgas part \(\alpha^{{{\text{pg}}}} \left( {\tau ,\delta } \right)\) contains 8 terms, while the equation for the dimensionless residual part \(\alpha^{{\text{r}}} \left( {\tau ,\delta } \right)\) contains 56 terms.
Note that the dimensionless property \({{F\left( {T,V} \right)} \mathord{\left/ {\vphantom {{F\left( {T,V} \right)} {RT}}} \right. \kern\nulldelimiterspace} {RT}}\) is related to the molar Massieu function divided by R (see Eq. 55):
Quite recently, generalized models for calculating thermodynamic mixture properties using Helmholtz energybased fundamental equations have been developed [109,110,111,112,113]. For instance, the GERG2008 equation developed by Kunz and Wagner [113] is based on data of 21 natural gas components. Over the entire composition range, it covers the liquid phase, the gas phase, the supercritical region, and VLE within the range 90 K to 450 K, and up to 35 MPa. However, when applied to significantly asymmetric binary mixtures, say, methane + pentane, it predicts critical curves with physically unreasonable temperature maxima [114]; thus, further work is indicated.
2.4 Property Changes on Mixing and Excess Properties
Experimental determination, correlation and prediction of mixture properties are topics of central importance in chemical thermodynamics, and data on binary liquid mixtures are of particular interest. Primarily, they are useful for testing and guiding theories that attempt to predict thermodynamic mixture properties from the properties of the constituent pure components, and the experimental results provide information on parameters characterizing interactions between unlike species. In turn, these data constitute the very foundation for the development of predictive methods for properties of liquid multicomponent mixtures that, on the application side, are indispensable for the calculation of phase equilibria. At present, no generally satisfactory theory exists that provides a reliable basis for prediction/correlation of thermodynamic data for binary liquid mixture, and thus, a fortiori, for ternary and multinary mixtures. Fortunately, large numbers of (critically) evaluated experimental data of excess properties (for a definition see below) are available in systematic data collections, such as the Solubility Data Series (IUPACNIST) [115, 116], NIST ThermoData Engine [117,118,119,120] or LandoltBörnstein [121,122,123,124,125,126,127,128,129], or in data banks, such as the Dortmund Data Bank [70]. Based on this wealth of experimental information, wellhoned semiempirical models, such as UNIFAC [130,131,132,133,134,135,136], DISQUAC [137,138,139,140], and the recent MOQUAC model [141] have been developed for correlating, extrapolating and predicting, in particular, molar excess Gibbs energies \(G^{{\text{E}}}\) and molar excess enthalpies \(H^{{\text{E}}}\), over reasonably large temperature ranges. However, estimated infinitedilution properties, aqueous solubilities of hydrocarbons, and excess heat capacities \(C_{P}^{{\text{E}}}\) of liquid mixtures are frequently not satisfactory [142]. Similar comments apply to COSMORS and related models [143,144,145,146,147].
When describing reality, instead of considering total mixture properties \(M^{{\text{t}}} = nM\left( {T,P,\left\{ {x_{i} } \right\}} \right)\) it is helpful to discuss the corresponding molar mixture properties in relation to the properties of the pure liquid constituents at the same T, P and \(\left\{ {x_{i} } \right\}\), i.e., to focus on difference measures. Discussion is thus based on a new class of thermodynamic functions known as property changes of mixing, customarily designated by the symbol \(\Delta\) and, on a molar basis, defined by
The corresponding new class of partial molar property changes of mixing (see Eq. 52c) is defined by
With the corresponding summability relation at constant T and P, see Eq. 52d, we have
or
and the exact differential of the extensive property \(n\Delta M\) is
From Eq. 192 a differential change in \(n\Delta M\) is given by
Hence, comparison with Eq. 193, and after division by n,
is obtained: this is another form of the general Gibbs–Duhem equation. In this section, the focus will be on \(M = G, \, S, \, V, \, H\), and because of direct measurability, \(\Delta H\)(via calorimetry) and \(\Delta V\)(via dilatometry) are the molar property changes of mixing of special interest.
Applying residualfunction approaches or equationsofstate methods to liquid mixtures, frequently consisting of chemically quite complex components, is often unsatisfactory. Thus, for many applications discussed in this symposium, one way to proceed is to select in place of the perfect gas (ideal gas) a reference basis that is more appropriate for condensed phases: the most commonly selected basis is the ideal solution (indicated by a superscript id). In fact, this approach represents the classic method in solution thermodynamics [148,149,150]. Compared to the perfectgas basis, we recognize that changes in composition affect liquid mixture properties much stronger than changes in pressure.
Consider a liquid equilibrium phase with composition \(\left\{ {x_{i} } \right\} \equiv \left\{ {x_{i}^{{\text{L}}} } \right\}\) at uniform temperature and pressure. Using the idealized composition dependence of the component fugacity as represented by the Lewis–Randall (LR) rule [17, 66, 151],
one obtains for the partial molar Gibbs energy.
Discussion of realsolution behavior may now be based on deviations from Lewis–Randall idealsolution behavior, i.e., on the differences between property values of real solutions and property values calculated for the Lewis–Randall idealsolution model at the same T, P and \(\left\{ {x_{i} } \right\}\) based on Eq. 197. That is, the partial molar Gibbs energy \(G_{i}^{{{\text{id}}}}\) of component i serves as a generating function for other partial molar properties of an LRideal solution that henceforward will be indicated by a superscript id (alternative idealsolution models are possible, and are indeed used, though the LRideal solution is the conventional reference for mixture behavior of liquid phases). For instance, the temperature derivative and the pressure derivative yield the partial molar entropy and the partial molar volume, respectively,
while the Gibbs–Helmholtz equation yields the LRideal partial molar enthalpy
The LRideal molar properties corresponding to the partial molar properties of Eqs. 197–200 are obtained with the summability relation:
The molar property changes of mixing for LRideal solutions, \(\Delta M^{{{\text{id}}}}\), may be obtained as special cases from the general defining equation, Eq. 189:
That is, by substituting the corresponding expression for \(M^{{{\text{id}}}}\), Eqs. 201–204, into Eq. 205 (or the corresponding expressions for \(M_{i}^{{{\text{id}}}}\), Eqs. 197–200), we obtain
The general property \(\Delta M_{i}^{{{\text{id}}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) of Eq. 205 denotes a partial molar property change of mixing for LRideal solutions, such as those appearing in Eqs. 206–209:
Quantities measuring deviations of real solution properties \(M\left( {T,P,\left\{ {x_{i} } \right\}} \right)\) from LRideal solution properties \(M^{{{\text{id}}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) at the same T, P and \(\left\{ {x_{i} } \right\}\) (see Eqs. 201–204), constitute another highly useful new class of functions called molar excess properties. They are designated by a superscript E and are defined by
The corresponding partial molar excess properties for component i in solution are defined by
and with the summability relation we have
Substituting the idealsolution expressions Eqs. 201–204 into Eq. 211, and taking into account Eq. 189, we obtain the following relations:
Firstlaw excess properties are identical to the property changes of mixing.
As a generating function, the molar excess Gibbs energy \(G^{{\text{E}}}\) is of central interest. As a matter of convenience, Eq. 197 may be generalized in such a manner that an expression for the partial molar Gibbs energy \(G_{i}\) is obtained that is valid for any real mixture per definition. Thus, we may write
where \(\gamma_{i}^{{{\text{LR}}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) is known as the Lewis–Randall (LR) activity coefficient of component i in solution. With the defining prescription Eq. 212, the partial molar excess Gibbs energy is thus given by
In view of Eq. 213, the molar excess Gibbs energy reads
Since \(S_{i} =  \left( {{{\partial G_{i} } \mathord{\left/ {\vphantom {{\partial G_{i} } {\partial T}}} \right. \kern\nulldelimiterspace} {\partial T}}} \right)_{{P,\left\{ {x_{i} } \right\}}}\), we have for the molar excess entropy
With \(V_{i} = \left( {{{\partial G_{i} } \mathord{\left/ {\vphantom {{\partial G_{i} } {\partial P}}} \right. \kern\nulldelimiterspace} {\partial P}}} \right)_{{_{{T,\left\{ {x_{i} } \right\}}} }}\), the molar excess volume is given by
and finally, with the Gibbs–Helmholtz equation or via \(H^{{\text{E}}} = G^{{\text{E}}} + TS^{{\text{E}}}\) we obtain for the molar excess enthalpy
The idealsolution equations for \(M^{{{\text{id}}}}\) (M = G, S, V, H) which follow from the Lewis–Randall rule are given by Eqs. 201–204. I reiterate that Eqs. 220–223 only apply when the Lewis–Randall model for the ideal solution is used.
Central to the development of useful relations is the fundamental property relation
and, of course,
and so forth. The definition of an excess property is not restricted to any phase, though they are predominantly used for liquid mixtures.
Excess properties and property changes of mixing are closely related, and one may readily calculate \(M^{{\text{E}}}\) from \(\Delta M\) and vice versa. By combining the definitions Eqs. 211 and 189, in conjunction with \(\Delta M^{{{\text{id}}}}\) defined by Eq. 205, the important relation
is obtained, with a similar one holding for the corresponding partial molar quantities:
Rearranging Eq. 226, the difference \(\Delta M  M^{{\text{E}}} = \Delta M^{{{\text{id}}}}\) is zero except for the secondlaw properties M = G, F (= G—PV) and S, see Eqs. 206–209, and similarly for the partial molar properties in Eq. 227, the difference \(\Delta M_{i}  M_{i}^{{\text{E}}} = \Delta M_{i}^{{{\text{id}}}}\) is zero except for the secondlaw properties \(M_{i} = G_{i}\), \(F_{i} \, \left( { = G_{i}  PV_{i} } \right)\) and \(S_{i}\). Further, from Eq. 226 we see immediately that since a molar excess property represents also the difference between the real change of a property of mixing and the LRidealsolution change of a property of mixing, we may identify it, alternatively to the defining equation Eq. 211, as a molar excess property change of mixing
Analogously, from Eq. 228 we may identify, alternatively to defining equation Eq. 212, a partial molar excess property as a partial molar excess property change of mixing
Evidently, the terms molar excess property and molar excess 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 \(M^{{\text{E}}}\) and \(M_{i}^{{\text{E}}}\) are preferred, while for mixing processes the notations \(\left( {\Delta M} \right)^{{\text{E}}} = \Delta M^{{\text{E}}}\) and \(\left( {\Delta M_{i} } \right)^{{\text{E}}} = \Delta M_{i}^{{\text{E}}}\) may be regarded as being more appropriate.
For the four quantities selected for a more detailed discussion, Eqs. 226 and 227 yield the following: for the secondlaw properties
is obtained, while for the firstlaw properties we have
in accord with Eqs. 214–217. Depending on the point of view, \(V^{{\text{E}}} = \Delta V\) is known as either the molar excess volume or the molar volume change of mixing, and \(H^{{\text{E}}} = \Delta H\) is called either the molar excess enthalpy or the molar enthalpy change of mixing.
In analogy to Eq. 193, the exact differential of the extensive property \(nM^{{\text{E}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) is given by
while Eq. 213 yields for a differential change in \(nM^{{\text{E}}}\)
Comparison with Eq. 234 and division by n results in
which is still another form of the general Gibbs–Duhem equation.
For convenience, instead of \(G_{i}^{{\text{E}}}\) the nondimensional group \({{G_{i}^{{\text{E}}} } \mathord{\left/ {\vphantom {{G_{i}^{{\text{E}}} } {RT}}} \right. \kern\nulldelimiterspace} {RT}}\) is frequently used, which quantity is directly related to the LRbased dimensionless state function \(\ln \gamma_{i}^{{{\text{LR}}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) by
Using the summability relation, we have
The corresponding fundamental excessproperty relation for a singlephase system, in which the amounts \(n_{i}\) may vary either through interchange of matter with its surroundings (open phase) or because of chemical reactions within the system or both, reads
in complete analogy to Eq. 70. By inspection we find
The corresponding Gibbs–Duhem equation reads
which at constant T and P reduces to
Rewriting it for a binary mixture
we recognize that if in a binary mixture \(\gamma_{1}^{{{\text{LR}}}}\) increases (or decreases) with increasing \(x_{1}\), then \(\gamma_{2}^{{{\text{LR}}}}\) must simultaneously decrease (or increase) with increasing \(x_{1}\). In addition, when \(x_{2} \to 0\), and thus \(x_{1} \to 1\), the slope of the curve \(\ln \gamma_{1}^{{{\text{LR}}}}\) vs. \(x_{1}\) is zero, and vice versa, when \(x_{1} \to 0\), and thus \(x_{2} \to 1\), the slope of the curve \(\ln \gamma_{2}^{{{\text{LR}}}}\) vs. \(x_{1}\) is zero: each \(\ln \gamma_{i}^{{{\text{LR}}}}\) curve (i = 1, 2) terminates at \(x_{i}^{{{\text{LR}}}} = 1\) with zero value and zero slope.
The fundamental excessproperty relation Eq. 239 with canonical variables T, P and \(\left\{ {n_{i} } \right\}\) supplies complete information on excess properties. It is of central importance in solution chemistry because \(H^{{\text{E}}}\), and its temperature dependence \(C_{P}^{{\text{E}}}\), the molar excess heat capacity at constant pressure (molar excess isobaric heat capacity)
the molar excess volume \(V^{{\text{E}}}\) and \(\ln \gamma_{i}^{{{\text{LR}}}}\) are experimentally accessible quantities: excess enthalpies and excess isobaric heat capacities may be obtained more or less directly via calorimetry, and excess volumes may be obtained more or less directly via dilatometry, or pycnometry, or vibratingtube densimetry, or buoyancy methods (exploiting the Archimedes principle), respectively [37, 46, 66, 68, 152,153,154,155,156,157,158,159,160]. The excess Gibbs energy, in principle the key excess property, is a conceptual property that cannot be measured directly (nor can the excess entropy), though it can be quantitatively deduced from vapor–liquid equilibrium (VLE) measurements via determination of LR activity coefficients [17, 20, 46, 161,162,163,164,165,166,167,168,169,170,171,172]. For mixtures at low to moderate pressure, that is, well below the critical pressure, the conventional highly effective and refined “phi/gamma (\({\phi \mathord{\left/ {\vphantom {\phi \gamma }} \right. \kern\nulldelimiterspace} \gamma }\))” approach to VLE is commonly used: note that isothermal measurements are advantageous [17]. For applications at pressures up to a few bars, to an excellent approximation the equilibrium relation for a binary system reads
and
Here, \(\delta_{ij}\) is defined by
where the identically subscripted second virial coefficients \(B_{ii}\) and \(B_{jj}\) refer to the pure vapor of component i or j, respectively, and \(B_{ij}\) is known as the interaction virial coefficient, or crosscoefficient. Note that we have \(\delta_{ij} = \delta_{ji}\). In Eq. 247a, \(\phi_{i}^{{\text{V}}} \left( {T,P,y_{i} } \right)\) denotes the vaporphase fugacity coefficient of component i, and \(\phi_{i}^{{{\text{V,}} * }} \left( {T,P_{{{\upsigma ,}i}} } \right)\) is the fugacity coefficient of pure saturated vapor at T and \(P_{{{\upsigma ,}i}}\). These phaseequilibrium relations are easily extended to multicomponent mixtures.
The classical data reduction approach uses activity coefficients directly determined by Eq. 246. Insertion of the so obtained \(\gamma_{i}^{{{\text{LR}}}}\) s into Eq. 220 or Eq. 238 yields values of \({{G^{{\text{E}}} } \mathord{\left/ {\vphantom {{G^{{\text{E}}} } {RT}}} \right. \kern\nulldelimiterspace} {RT}} = \sum\limits_{i}^{c} {x_{i} \ln \gamma_{i}^{{{\text{LR}}}} }\) that are then fit to an appropriate analytical correlating equation representing the composition dependence of \({{G^{{\text{E}}} } \mathord{\left/ {\vphantom {{G^{{\text{E}}} } {RT}}} \right. \kern\nulldelimiterspace} {RT}}\)(see next section). For more recent methods see Van Ness and Abbott [17]. Combination with calorimetrically measured excess enthalpies yields the molar excess entropy
For one mole of a constantcomposition mixture,
and for the corresponding partial molar excess properties, see Eq. 237,
is obtained. Hence the partial molar analogues of Eqs. 240 and 241, respectively, are
and
Thus, the partial molar excess entropy is given by
Finally, I list the useful relations
and, important for calorimetric work at high pressure
and
Modern calorimeters allow reliable measurements of \(H^{{\text{E}}}\) and \(C_{P}\) (and thus of \(C_{P}^{{\text{E}}}\)) at elevated T and P, and the results have to be consistent with experimentally determined volumetric properties, as indicated by Eqs. 256 and 257, respectively. However, outside the critical region the influence of pressure on excess properties is usually rather small.
Focusing now on the nondimensional excess property \({{G^{{\text{E}}} } \mathord{\left/ {\vphantom {{G^{{\text{E}}} } {\left( {x_{1} x_{2} RT} \right)}}} \right. \kern\nulldelimiterspace} {\left( {x_{1} x_{2} RT} \right)}}\) for a binary mixture, we find this quantity of considerable practical utility, especially when a graphical (visual) evaluation of experimentally determined \(G^{{\text{E}}}\) s is intended. Note that
where \(\gamma_{1}^{{{\text{LR,}}\infty }}\) and \(\gamma_{2}^{{{\text{LR,}}\infty }}\) are the LR activity coefficients at infinite dilution. These quantities play an important role in solution chemistry and have found many applications in the characterization of liquid solution behavior. In general, for binary mixtures extrapolation of \({{M^{{\text{E}}} } \mathord{\left/ {\vphantom {{M^{{\text{E}}} } {x_{1} x_{2} }}} \right. \kern\nulldelimiterspace} {x_{1} x_{2} }}\) to \(x_{1} = 0\) and \(x_{2} = 0\), respectively, is the most convenient and reliable graphical method for determining the infinitedilution partial molar excess properties \(M_{1}^{{{\text{E,}}\infty }}\) and \(M_{2}^{{{\text{E,}}\infty }}.\)
2.5 Correlation of Experimental Data
In general, experimental data have to be treated on their way from experiment in the laboratory to the place of application, be it in support of theory development, or model checking, or plant operation. Clearly, the mathematical treatment should aim to eliminate inconsistencies without distorting the results, it should correlate the data to within experimental error, provide at least realistic first derivatives upon differentiation, and it should combine adequate flexibility with reasonable simplicity. Unfortunately, no theoretical model of any general validity exists that satisfactorily describes the composition dependence of excess properties of liquid mixtures, and relations commonly used are semiempirical at best [1, 16,17,18]. Focusing now on binary liquid mixtures, one procedure is to express \({{M^{{\text{E}}} } \mathord{\left/ {\vphantom {{M^{{\text{E}}} } {x_{1} x_{2} }}} \right. \kern\nulldelimiterspace} {x_{1} x_{2} }}\) at constant temperature and pressure as a power series in \(x_{1}\) (since \(x_{2} = 1  x_{1}\), \(x_{1}\) can be selected as the single independent variable). However, an equivalent power series, with certain advantages, is the popular empirical expansion due to Redlich and Kister [173,174,175],
where the parameters \(A_{0}\) and \(A_{m}\), \(m = 1,2,3,\ldots\) are, in general, functions of T and P; truncations of Eq. 259 with \(a \le 3\) are frequently used in the literature. The partial molar excess property values at infinite dilution, \(M_{i}^{{{\text{E,}}\infty }} = \mathop {\lim }\limits_{{x_{i} \to 0}} M_{i}^{{\text{E}}}\), are given by
For highly skewed data, using more than four terms may cause spurious oscillations of the \(M_{i}^{{\text{E}}}\) s, and may yield unreliable \(M_{i}^{{{\text{E,}}\infty }}\) s. Superior flexibility for fitting strongly unsymmetrical curves is provided by Padé approximants [1, 17, 176,177,178] of order [a/b], where the denominator must never become zero:
As alternatives, expressions based on orthogonal polynomials have been suggested [179,180,181], e.g., expansions based on Legendre polynomials in \(z_{12} \equiv x_{1}  x_{2}\):
with \(L_{0} \left( {z_{12} } \right) = 1\), \(L_{1} \left( {z_{12} } \right) = z_{12}\), \(L_{2} \left( {z_{12} } \right) = {{\left( {3z_{12}^{2}  1} \right)} \mathord{\left/ {\vphantom {{\left( {3z_{12}^{2}  1} \right)} 2}} \right. \kern\nulldelimiterspace} 2}\), \(L_{3} \left( {z_{12} } \right) = {{\left( {5z_{12}^{3}  3z_{12} } \right)} \mathord{\left/ {\vphantom {{\left( {5z_{12}^{3}  3z_{12} } \right)} 2}} \right. \kern\nulldelimiterspace} 2}\), and so forth. The summation limit \(n_{p}\) is selected as required to fit the available experimental data. If \(H^{{\text{E}}}\) data are available at several temperatures, the temperature dependence of the coefficients \(a_{p}\) has to be incorporated via, say,
or an exponential temperature dependence [182].
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 lowerorder terms. As pointed out by Pelton and Bale [180, 181], using Legendre expansions in terms of \(L_{p} \left( {z_{12} } \right)\) instead in terms of \(L_{p} \left( {x_{1} } \right)\) 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 [181], Howald and Eliezer [183], and Tomiska [184].
When the number of components increases to three and beyond, experimental work to determine excess properties increases sharply, thus explaining the scarcity of experimental data for multicomponent mixtures. The situation is aggravated by less reliable empirical/semiempirical correlating functions describing the composition dependence of multinary excess properties, though, for instance, the Redlich–Kister expansion can be generalized without difficulty. However, predictions of multicomponent solution properties from results pertaining to the constituent binaries alone, without ternary (or higher) terms, are always approximate, the most successful correlation of this type being Kohler’s equation [185]: it relates the excess molar Gibbs energy \(G^{{\text{E,123}}}\) of a ternary liquid mixture with mole fractions \(\left\{ {x_{1} ,x_{2} ,x_{3} } \right\}\), \(\sum\limits_{i = 1}^{3} {x_{i} = 1}\), to the excess molar Gibbs energies \(G^{{{\text{E,}}ij}}\) of the three binary subsystems with composition \(\left\{ {x_{i}^{\prime } ,x_{j}^{\prime } = 1  x_{i}^{\prime } } \right\},\) where the mole fractions characterized by a superscript prime are defined by
Based on the reasonable approximation that pairwise molecular interactions \(i \Leftrightarrow j\) remain constant along lines representing mixtures having a constant composition ratio \({{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } {x_{j} }}} \right. \kern\nulldelimiterspace} {x_{j} }}\), the binary quantities \(G^{{{\text{E,}}ij}}\) are assumed to depend only on \(\left\{ {x_{i}^{\prime } ,x_{j}^{\prime } = 1  x_{i}^{\prime } } \right\}\), and.
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 \(G^{{{\text{E,}}ij}}\) data. Similar comments apply, of course, to \(H^{{\text{E,123}}}\), \(C_{P}^{{\text{E,123}}}\) and \(V^{{\text{E,123}}}\). Kohler’s equation can be generalized to correlate/predict the composition dependence of molar excess properties of multicomponent systems with four or more components. Assuming again that pairwise molecular interactions \(i \Leftrightarrow j\) remain constant at conditions imposing a constant composition ratio \({{x_{i} } \mathord{\left/ {\vphantom {{x_{i} } {x_{j} }}} \right. \kern\nulldelimiterspace} {x_{j} }} = {{x_{i}^{\prime } } \mathord{\left/ {\vphantom {{x_{i}^{\prime } } {x_{j}^{\prime } }}} \right. \kern\nulldelimiterspace} {x_{j}^{\prime } }}{,}\) and using Eq. 262 for the binary subsystems with \(z_{ij}^{\prime } \equiv x_{i}^{\prime }  x_{j}^{\prime }\), that is,
such a generalized equation for the molar excess enthalpy \(H^{{{\text{E,123}} \ldots c}}\) of a ccomponent system reads [37]
For the composition dependence of the excess molar enthalpies \(H^{{{\text{E,}}ij}} \left( {x_{i}^{\prime } ,x_{j}^{\prime } } \right)\) of the binary subsystems, any function, say, Redlich–Kister, Padé or Legendre polynomial (see Eq. 262), may be used. In fact, each binary system could be represented by a different correlating equation.
Traditionally, the thermodynamic description of real liquid solutions is based on the excessproperty formalism presented above. As already pointed out, large numbers of (critically) evaluated experimental data on \(G^{{\text{E}}}\) and \(\ln \gamma_{i}^{{{\text{LR}}}}\), \(H^{{\text{E}}}\), \(C_{P}^{{\text{E}}}\) and \(V^{{\text{E}}}\) of binary liquid mixtures are now available in systematic data collections [70, 115,116,117,118,119,120,121,122,123,124,125,126,127,128,129], and are at the disposal of liquidstate physical chemists and chemical engineers.
For the global thermodynamic description of liquid nonelectrolyte mixtures, \(C_{P}^{{\text{E}}}\) s are pivotal properties, and taking advantage of the exact relations of Eq. 255, considerable economy in experimental effort can be attained. In order to obtain precise values of \(G^{{\text{E}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) (as well as of \(H^{{\text{E}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) and \(S^{{\text{E}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\)) over wide ranges of temperature at constant pressure P (though well below the vapor–liquid critical region), it suffices to determine \(G^{{\text{E}}} \left( {T = T_{{{\text{ref}}}} ,P,\left\{ {x_{i} } \right\}} \right)\) and \(H^{{\text{E}}} \left( {T = T_{{{\text{ref}}}} ,P,\left\{ {x_{i} } \right\}} \right)\) at one suitably selected reference temperature (frequently,\(T_{{{\text{ref}}}} = 298.15{\text{ K}}\)), and to measure the temperature dependence of the molar excess heat capacity \(C_{P}^{{\text{E}}} \left( {T,P,\left\{ {x_{i} } \right\}} \right)\) over a large enough temperature range of interest at constant pressure and at well distributed compositions. These data then permit the consistent determination of the remaining excess properties \(G^{{\text{E}}}\), \(H^{{\text{E}}}\) and \(S^{{\text{E}}}\) over this temperature range by integration of the relevant differential equations. Compared to the converse approach of determining \(G^{{\text{E}}}\) over a reasonably large temperature range and deriving \(H^{{\text{E}}}\), \(S^{{\text{E}}}\) and \(C_{P}^{{\text{E}}}\) by differentiation with respect to T, it is an obviously advantageous route that will yield more reliable results. This will be especially the case when \(G^{{\text{E}}}\) values are needed at low temperatures where the vapor pressures of the mixtures are small and the conventional determination of \(G^{{\text{E}}}\) is difficult. Well below the vapor–liquid critical region, \(C_{P}^{{\text{E}}}\) of a constantcomposition mixture frequently shows a simple temperature dependence, that is, at constant pressure it can be approximated satisfactorily by [157],
where \(\tau \equiv {{T_{{{\text{ref}}}} } \mathord{\left/ {\vphantom {{T_{{{\text{ref}}}} } T}} \right. \kern\nulldelimiterspace} T}\). Using the differential equations presented in Eq. 255, integration over temperature at constant pressure and constant composition yields
and
The dimensionless coefficients \(a_{i}\) depend, of course, on pressure and composition; they are related to the corresponding molar excess quantities at \(\left\{ {T = T_{{{\text{ref}}}} ,P,\left\{ {x_{i} } \right\}} \right\}\) as follows: \({{C_{P}^{{\text{E}}} \left( {T_{{{\text{ref}}}} } \right)} \mathord{\left/ {\vphantom {{C_{P}^{{\text{E}}} \left( {T_{{{\text{ref}}}} } \right)} R}} \right. \kern\nulldelimiterspace} R} = a_{3} + a_{4} + a_{5}\), \({{H^{{\text{E}}} \left( {T_{{{\text{ref}}}} } \right)} \mathord{\left/ {\vphantom {{H^{{\text{E}}} \left( {T_{{{\text{ref}}}} } \right)} {RT_{{{\text{ref}}}} = a_{2} }}} \right. \kern\nulldelimiterspace} {RT_{{{\text{ref}}}} = a_{2} }}\), \({{S^{{\text{E}}} \left( {T_{{{\text{ref}}}} } \right)} \mathord{\left/ {\vphantom {{S^{{\text{E}}} \left( {T_{{{\text{ref}}}} } \right)} {R = a_{1} }}} \right. \kern\nulldelimiterspace} {R = a_{1} }}\), and \({{G^{{\text{E}}} \left( {T_{{{\text{ref}}}} } \right)} \mathord{\left/ {\vphantom {{G^{{\text{E}}} \left( {T_{{{\text{ref}}}} } \right)} {RT_{{{\text{ref}}}} }}} \right. \kern\nulldelimiterspace} {RT_{{{\text{ref}}}} }} =  a_{1} + a_{2}\). Analogous expressions may be derived if the temperature dependence of \(C_{P}^{{\text{E}}}\), at constant pressure and composition, is described by a polynomial in T instead of \(T^{  1}\). Global studies of this kind are, however, quite rare, with some of the most careful investigations being those of Ziegler and colleagues [186, 187].
Classical thermodynamics does not constrain the signs of particular excess properties, that is, \(G^{E} ,\) \(H^{{\text{E}}}\) and \(S^{{\text{E}}}\) can each be positive or negative, and quite a few systems are known where the sign changes with composition (sigmoidal composition dependence: one maximum, one minimum). Very rarely socalled Mshaped excess enthalpies have been reported: for the binary liquid mixture {benzonitrile + benzene}, at 298.15 K, 308.15 K, and 318.15 K and ambient pressure, \(H^{{\text{E}}}\) changes its sign with composition two times, that is, it exhibits two maxima and one minimum [188]. Thermodynamics restricts, however, the admissible combinations of signs for these three excess properties: Eq. 249 clearly shows that a positive \(G^{{\text{E}}}\) with a negative \(H^{{\text{E}}}\) and a positive \(S^{{\text{E}}}\) is not possible, nor a negative \(G^{E}\) with a positive \(H^{{\text{E}}}\) and a negative \(S^{{\text{E}}}\). In passing I note that our research group in France was the first to report (in 1982) Wshaped excess heat capacities \(C_{P}^{{\text{E}}} {\text{ vs. } }{ } x_{1}\) for {1,4dioxane + nC_{n}H_{2n+2}} at 298.15 K, n = 7, 10, 14 [189]: \(C_{P}^{{\text{E}}}\) exhibits two minima and one maximum, see Fig. 1.
2.6 Excess Molar Gibbs Energy and Lewis–Randall Activity Coefficients (in Particular at Infinite Dilution)
Activity coefficients \(\gamma_{i}^{{{\text{LR,}}\infty }}\) at infinite dilution characterize the thermodynamic behavior of a single solute molecule completely surrounded by solvent molecules, hence it usually indicates maximum nonideality and—in the absence of \(i \Leftrightarrow i\) interactions—it provides important information on solute–solvent interactions \(i \Leftrightarrow j\). Thus, they are of great value in chemical engineering and are key parameters in the discussion of dilute solutions encountered in environmental studies [190,191,192]. In fact, given the infinitedilution activity coefficients of each component in the other in a binary mixture, values of parameters in popular twoparameter activity coefficient models can be easily obtained; in turn, these can be used for phase equilibrium predictions over the entire composition range [193]. However, prediction quality depends strongly on the model used, and whether the mixture is only slightly or strongly nonideal. With improved experimental techniques [170, 171, 194], precise measurements at low concentrations can be made with less effort compared to conventional VLE measurements, and with greater accuracy since extrapolation of activity coefficients obtained at higher mole fractions to infinite dilution is quite demanding.
For the prediction of the composition dependence of LRbased \(G^{{\text{E}}}\), many empirical equations have been proposed, and for binary mixtures, some of the simpler ones are special cases of one of the following power series expansions in the mole fractions:
see Eq. 259 (Redlich–Kister expansion), or
With restriction to two parameters and on rearrangement, i.e., \(B^{\prime} + C^{\prime} = A_{21}^{\prime }\) and \(B^{\prime}  C^{\prime} = A_{12}^{\prime }\), Eq. 272 yields the equivalent (twoparameter) 3suffix Margules^{Footnote 3}equation
with the following expressions for the LR activity coefficients:
With restriction to two parameters and on rearrangement, i.e., \({1 \mathord{\left/ {\vphantom {1 {\left( {B  C} \right) = A_{12} }}} \right. \kern\nulldelimiterspace} {\left( {B  C} \right) = A_{12} }}\), and \({1 \mathord{\left/ {\vphantom {1 {\left( {B + C} \right) = A_{21} }}} \right. \kern\nulldelimiterspace} {\left( {B + C} \right) = A_{21} }}\), Eq. 273 yields the equivalent twoparameter van Laar equation [196, 197]
with the following expressions for the LR activity coefficients:
If for a binary system precise and plentiful data are available, the use of a 3parameter Redlich–Kister equation (see Eqs. 259 or 272) is frequently indicated, i.e.,
with the following expressions for the LR activity coefficients:
Combining the results for \(\ln \gamma_{1}^{{{\text{LR,}}\infty }}\) and \(\ln \gamma_{2}^{{{\text{LR,}}\infty }}\) with the value of \({{G^{{\text{E}}} } \mathord{\left/ {\vphantom {{G^{{\text{E}}} } {RT}}} \right. \kern\nulldelimiterspace} {RT}}\) at \(x_{1} = x_{2} = 0.5\) yields.
Note that the Redlich–Kister equation, Eq. 280, may also be transformed into an alternative, equivalent polynomial known as the 4suffix Margules equation [1, 17]
In 1964 Wilson suggested a novel equation for \(G^{{\text{E}}}\) by introducing the local mole fraction of component i in a mixture {i + j} [198], a concept which has been developed impressively since then [137, 199,200,201,202,203,204,205,206,207,208,209]. For a binary mixture, the molar excess Gibbs energy is given by
and the activity coefficients are
Thus, at infinite dilution we obtain
An iterative procedure is required to evaluate the adjustable parameters \(\Lambda_{12}\) and \(\Lambda_{21}\). In Wilson’s derivation, they are related to the purecomponent molar volumes and to characteristic interaction energy differences, i.e.,
Numerical values of the parameters \(\lambda_{ij}  \lambda_{ii}\) can only be found through reduction of experimental VLE data. The Wilson equation is a very flexible equation with a builtin temperature dependence; it is able to represent mixtures exhibiting strong deviations from ideality and is easily generalized to describe multicomponent behavior using only binary parameters. However, Wilson’s equation is unable to predict limited miscibility, and should therefore be used only for liquid systems where the components are completely miscible.
Enormous research efforts have been invested in developing the local composition concept, for instance in developing the NRTL equation, the UNIQUAC and UNIFAC formalism, and the DISQUAC model [137, 199,200,201,202,203,204,205,206,207,208,209]. This topic definitely deserves a review of its own.
2.7 Henry’s Law: An Alternative IdealSolution Model for Liquid Systems
In Sect. 2.4, I presented an idealsolution model derived from the Lewis–Randall rule, that serves as the basis of the excess property formalism. This approach for the discussion of liquid solution properties is entirely appropriate as long as the constituent components are stable as pure liquids at T and P of the solution. However, solutions of gases in liquids pose a problem, since the dissolved gas does not exist in the pure state as a liquid at T and P of the solution, thereby precluding any measurements of its properties in this state. In addition, the solubility of gases in liquids is usually quite small, hence experimental data are usually collected only over a rather small part of the composition range [70, 115, 116, 210,211,212,213,214,215].
Consider a binary system of solute 2 dissolved in solvent 1. A general criterion for phase equilibrium at T and P is the equality of the fugacity \(f_{i}^{{\uppi }}\) of each one of the components in the coexisting phases. Thus, for the specific case of VLE, where π = V or L,
The link with experiment may be formally established by expressing the component fugacities in the vapor phase in terms of the respective fugacity coefficients \(\phi_{i}^{{\text{V}}}\),
while the liquidphase fugacities of the components are expressed with the help of appropriately normalized liquidphase activity coefficients. Two entirely equivalent approaches are in use: the symmetric convention is based on the Lewis–Randall rule, Eq. 196, and leads to (i.e., for both solvent and solute)
where
denotes the fugacity of pure component i in either a real or a hypothetical liquid state at T and P of the liquid solution, and \(\phi_{i}^{{{\text{L,}} * }} \left( {T.P} \right)\) is the corresponding puresubstance fugacity coefficient (see Eqs. 131–133). As I have already indicated, \(\gamma_{i}^{{{\text{LR}}}}\) is predominantly used for the thermodynamic description of liquid mixtures when all components are considered stable as pure liquids at T and P of interest, and
applies to all components of the solution: the activity coefficients are said to be normalized symmetrically. At all other compositions \(x_{i}^{{\text{L}}} \ne 1\), deviations of \(\gamma_{i}^{{{\text{LR}}}}\) from unity measure deviations of real solution behavior from ideal LR behavior (ratio measure). This \(\left( {\phi ,\gamma } \right)\)approach is also known, somewhat imprecisely, as being based on Raoult’s law.
Alternatively, idealsolution behavior may be based on Henry’s law (HL), that is, the validity of the linear relation
over the entire composition range \(0 \le x_{i}^{{\text{L}}} \le 1\) is assumed, where \(h_{i,j} \left( {T,P} \right)\) denotes the Henry fugacity (also known as Henry’s law constant) as defined below by Eq. 300a. Traditionally, for compact notation, no superscript L is attached to the Henry fugacity. The corresponding activity coefficient \(\gamma_{i}^{{{\text{HL}}}}\) measures how much the component fugacity of i in the real solution differs from the component fugacity of i corresponding to the HLidealization Eq. 296 (ratio measure) [17, 66, 169]. Thus,
The value of the Henry fugacity depends on T and P and also on the identity of the solvent (the other component), hence the double subscript \(i,j\) has been added; and \(h_{i,j} \left( {T,P} \right)\) has to be determined for each binary system. HLbased activity coefficients are predominantly used for the thermodynamic description of dilute liquid solutions and, in particular, if the experimental temperature is above the critical temperature of one component of the solution (gas solubility). From Eqs. 297 and 300a (presented below), it follows that
That is, at nonzero compositions, \(x_{i}^{{\text{L}}} \ne 0\), deviations of \(\gamma_{i}^{{{\text{HL}}}}\) from unity quantify real solution behavior visàvis ideal HL behavior.
The activity coefficients are said to be normalized unsymmetrically, when for the solvent (i = 1) the LR convention Eq. 295 applies, and for the (frequently supercritical) solute (i = 2) the HL convention Eq. 298 is used. The limiting behavior of the activity coefficients characterizing such a binary liquid solution is thus given by
For obvious reasons, this approach is also called a \(\left( {\phi ,\gamma } \right)\)method. For details, see Refs. 1, 17, 66, 169 and 216,217,218,219.
Figure 2 presents schematically the composition dependence of the component fugacity \(f_{2} \left( {T,P,x_{2} } \right)\), at constant temperature and pressure, in a binary liquid solution (solid curve) exhibiting a positive deviation from the Lewis–Randall rule, Eq. 196. \(f_{2}^{ * } \left( {T,P} \right)\) is the fugacity of pure liquid component 2, and \(h_{2,1} \left( {T,P} \right)\) is the Henry fugacity (also known as Henry’s law constant) of solute 2 dissolved in solvent 1. For the sake of simplicity, the superscripts L have been omitted. The dashed lines represent the two most popular idealsolution models discussed in this review: the one labeled LR corresponds to Eq. 196, and the other, labeled HL, corresponds to Eq. 296. As shown below via Eq. 300a, the HL line becomes a tangent to \(f_{2} \left( {T,P,x_{2} } \right)\) in the limit \(x_{2} \to 0\), while according to Eq. 300b, the LR line becomes a tangent to \(f_{2} \left( {T,P,x_{2} } \right)\) in the limit \(x_{2} \to 1\).
Since the limiting value of the ratio \({{f_{i}^{{\uppi }} } \mathord{\left/ {\vphantom {{f_{i}^{{\uppi }} } {x_{i}^{{\uppi }} }}} \right. \kern\nulldelimiterspace} {x_{i}^{{\uppi }} }}\) for \(x_{i}^{{\uppi }} \to 0\) at constant T and P is known from experiment to be finite, \(f_{i}^{{\uppi }}\) must also become zero at \(x_{i}^{{\uppi }} = 0\), and by de l’Hôpital’s rule
is obtained. Equation 300a summarizes Henry’s law; it defines the Henry fugacity \(h_{i,j}^{{\uppi }} \left( {T,P} \right)\) of component i dissolved in component j for any phase π (L or V) and identifies the limiting slope of the curve \(f_{i}^{{\uppi }}\) vs. \(x_{i}^{{\uppi }}\) at constant T and P as \(h_{i,j}^{{\uppi }} \left( {T,P} \right)\). Henry’s law is a limiting law, and for real solutions it is approximately valid for small values of \(x_{i}^{{\uppi }}\), with the experimental precision determining the observed apparent validity range [17, 66, 169, 216, 219].
As shown in Refs. 17, 66, 169 and 219, at the other end of the composition range, that is, for \(x_{i}^{{\uppi }} \to 1,\) at constant temperature and pressure we obtain
where \(f_{i}^{{{\pi ,} * }} \left( {T,P} \right)\) denotes the fugacity of pure component i at T and P of the solution and in the same physical state. Equation 300b summarizes the Lewis–Randall rule; it is valid in any phase \(\pi\) (L or V) and shows that in the limit \(x_{i}^{{\uppi }} \to 1\) both \(f_{i}^{\pi }\) and the limiting slope of the curve \(f_{i}^{{\uppi }}\) vs. \(x_{i}^{{\uppi }}\) at constant T and P become equal to the fugacity of pure i in phase \(\pi\). The Lewis–Randall rule is a limiting law, and for real solutions it is approximately valid for values of \(x_{i}^{{\uppi }}\) near unity, with the experimental precision determining the observed apparent validity range [17, 66, 169, 216, 219].
Focusing now on gas solubilities, the Henry fugacity of supercritical solute 2 dissolved in liquid solvent 1 is defined by Eq. 300a, with π = L. Because of vapor–liquid phase equilibrium, see Eq. 291, in conjunction with Eqs. 292, 297 and 298, and adopting the widely used notation \(x_{i}^{{\text{V}}} \equiv y_{i}\) and \(x_{i}^{{\text{L}}} \equiv x_{i} ,\) we arrive at the classical experimental prescription for the determination of \(h_{2,1} \left( {T,P_{{{{\upsigma ,1}}}} } \right)\) from isothermal VLE experiments (measurement of P, x_{2} and y_{2}) at decreasing total pressures \(P \to P_{{{{\upsigma ,1}}}} \left( T \right)\) and concomitantly decreasing \(x_{2} \to 0\) and \(y_{2} P \to 0\):
That is to say, the Henry fugacity referring to solute 2 dissolved in the liquid phase (consisting essentially of solvent 1) is obtained as the intercept of a plot, at constant T, of \({{\phi_{2}^{{\text{V}}} \left( {T,P,y_{2} } \right)y_{2} P} \mathord{\left/ {\vphantom {{\phi_{2}^{{\text{V}}} \left( {T,P,y_{2} } \right)y_{2} P} {x_{2} }}} \right. \kern\nulldelimiterspace} {x_{2} }}\) against \(x_{2}\) or \(y_{2} P\). Entirely equivalent expressions relating the Henry fugacity to limiting slopes (see Eq. 300a) may be derived. When applying Eq. 301 to experimental data, the vaporphase fugacity coefficient \(\phi_{2}^{{\text{V}}} \left( {T,P,y_{2} } \right)\) must be calculated with a suitable vaporphase EOS. Since the majority of gassolubility measurements is performed in the low to moderate pressure domain, the virial equation of state is the equation of choice: it has a sound theoretical basis, and it is superior to cubic equations of state. The computational convenience associated with a volumeexplicit version leads to the excellent, widely used approximation
where \(B\left( {T,\left\{ {y_{i} } \right\}} \right)\) is the second mixture virial coefficient [49, 220,221,222,223] (note that B refers to the virial equation in amount density and is usually the quantity listed in data compilations). In turn, for the calculation of the fugacity coefficient of the gas in the binary vapor mixture Eq. 153a is used, yielding (see Eq. 153b)
where \(\delta_{12}\) is defined by Eq. 248.
The temperature dependence of the Henry fugacity is given by [17, 66, 169, 216, 219]
and its pressure dependence is given by [17, 66, 169, 216, 219]
Here, \(H_{2}^{{{\text{L,}}\infty }}\) is the partial molar enthalpy of solute 2 at infinite dilution in liquid solvent 1, \(H_{2}^{{{\text{pg,}} * }}\) denotes the molar enthalpy of the pure solute in the perfectgas state, and \(\Delta H_{2}^{\infty } \left( {T,P} \right)\) is the partial molar enthalpy change on solution (this notation is generally preferred over \(H_{2}^{{{\text{R,L,}}\infty }}\), the partial molar residual enthalpy at infinite dilution [66, 169]). \(V_{2}^{{{\text{L,}}\infty }}\) is the partial molar volume of the solute at infinite dilution in the liquid solvent. Equation 304 provides the basis for obtaining partial molar enthalpy changes on solution via van’t Hoff analysis of highprecision solubility data of gases in liquids (see below).
For the reduction, correlation, and further use of highprecision gas solubility data, as suggested by Eq. 301, it is advantageous to select at each temperature the vapor pressure \(P_{{{{\upsigma,1}}}} \left( T \right)\) of the solvent as reference pressure. With this convention, the Henry fugacity of solute 2 dissolved in solvent 1 at any other pressure is obtained via integration of Eq. 305:
The exponential in Eq. 306 is known as Poynting correction factor; its evaluation requires reliable data on the partial molar volume of the solute at infinite dilution [224,225,226,227,228,229,230,231,232,233,234,235,236].
Since the approaches to the thermodynamic description of VLE introduced above are equivalent, various key quantities associated with them are, of course, connected with each other by exact relations [66, 169, 216,217,218,219, 237,238,239]. For instance, focusing on binary liquid mixtures, comparison of Eq. 293 with Eq. 297 yields (i = 1 or 2, j = 1 or 2, \(i \ne j\))
independent of composition. Thus, for the LR activity coefficient at infinite dilution one obtains
and hence
For compact notation, the specifications \(\left( {T,P,x_{i} } \right)\), etc. have been omitted.
From the definitions Eqs. 129 and 300a follows the important, generally valid relation [66, 169], here specified to apply to the liquid solution phase (that is, π = L):
\(\phi_{i}^{{{\text{L,}}\infty }} \left( {T,P} \right)\) denotes the fugacity coefficient of component i at infinite dilution in solvent j, \(G_{i}^{{{\text{R,L,}}\infty }} \left( {T,P} \right)\) denotes the partial molar isobaric residual Gibbs energy of component i at infinite dilution, and \(\mu_{i}^{{{\text{R,L,}}\infty }} \left( {T,P} \right)\) is its isobaric residual chemical potential at infinite dilution. Equation 310 provides an exact link between the thermodynamic description of any solution based on Henry’s law and a description based on fugacity coefficients and thus on an equation of state: for solute 2 we have.
In addition, using Eqs. 129, 132, 293, 297 and 310 as needed, the following rigorous relations between quantities used in a description based on activity coefficients and one based on fugacity coefficients (and hence on an EOS), are obtained (all quantities refer to the same temperature and pressure):
and in the limit \(x_{i} \to 0\), for the LR activity coefficient at infinite dilution we have
For the sake of a more compact notation, the specifications \(\left( {T,P,x_{i} } \right)\) etc., have been omitted. Equations 310–314 allow straightforward transition of system description from one using fugacity coefficients to one using activity coefficients, and vice versa.
The last topic I shall discuss briefly in this review concerns the determination of partial molar enthalpy changes on solution \(\Delta H_{2}^{\infty }\) and partial molar heat capacity changes on solution \(\Delta C_{P,2}^{\infty }\) of sparingly soluble gases in liquids. The latter quantity is defined by
where \(C_{P,2}^{{{\text{L,}}\infty }}\) is the partial molar heat capacity at constant pressure of the solute at infinite dilution in the liquid solvent, and \(C_{P,2}^{{{\text{pg,}} * }}\) is the molar heat capacity at constant pressure of the pure solute in the perfectgas state. The notation \(\Delta C_{P,2}^{\infty }\) is generally preferred over \(C_{P,2}^{{{\text{R,L,}}\infty }}\), the partial molar residual isobaric heat capacity at infinite dilution [66, 169].
In fact, until quite recently highprecision measurements of Henry fugacities over temperature ranges large enough to permit van’t Hoff analysis of the solubility data, constituted the only reliable source of information on these quantities [211, 214, 240, 241]. Once experimental Henry fugacities \(h_{2,1} \left( {T,P_{{{{\upsigma ,1}}}} \left( T \right)} \right)\) for a binary solution have been determined over a reasonably large temperature range (but not too close to the critical temperature of solvent 1), the most popular correlating equations are either the ClarkeGlew (CG) equation [242,243,244]
or the BensonKrause (BK) equation [245,246,247], conventionally expressed as
Based on the ability to fit highprecision Henry fugacity data over fairly large temperature ranges, and of simplicity, the BK power series in 1/T appears to be superior. In passing I note that the 3term version of Eq. 316 is the wellknown Valentiner equation [248].
At this juncture I would like to emphasize once more that the frequently found sweeping statement “the solubility of a gas in a liquid decreases with increasing temperature” is misleading/incorrect when the entire liquid range of the solvent between its triple point temperature and its critical temperature is considered. For many binary solutions of gases in liquids, the following behavior is well documented [1, 66, 157, 169, 216, 249,250,251]: at low temperatures, the Henry fugacity \(h_{2,1} \left( {T,P_{{{{\upsigma ,1}}}} } \right)\) typically increases with increasing temperature, passes through a maximum, and then decreases towards its limiting value at the solvent’s critical point (with critical temperature \(T_{{\text{c,1}}}\) and critical pressure \(P_{{\text{c,1}}}\)), which, according to Eq. 311, is given by [66, 157, 169, 216,217,218,219]
Figure 3 provides two examples of such a solubility behavior: Henry fugacities, i.e. \(\ln \left[ {{{h_{2,1} \left( {T,P_{{{{\upsigma ,1}}}} } \right)} \mathord{\left/ {\vphantom {{h_{2,1} \left( {T,P_{{{{ ,1}}}} } \right)} {{\text{GPa}}}}} \right. \kern\nulldelimiterspace} {{\text{GPa}}}}} \right]\), of methane dissolved in liquid water, and of krypton dissolved in liquid water, are plotted against temperature [249, 252]. Note that the system {x_{1}H_{2}O + x_{2}CH_{4}} is important in the discussion of hydrophobic effects [253,254,255,256,257,258,259,260,261,262].
While the Henry fugacity remains finite at \(T_{{\text{c,1}}}\), for volatile solutes the limiting slope approaches \( \infty\) when the critical point of the solvent is approached along the coexistence curve [263]:
Since the experimental Henry fugacities at different temperatures refer to different vapor pressures \(P_{{{{\upsigma ,1}}}} \left( T \right)\), we obtain with Eqs. 304 and 305 [66, 157, 169, 216, 219, 237,238,239]
and by analogous arguments
The second term on the righthand side of Eq. 320 and the third, fourth and fifth term on the righthand side of Eq. 321, respectively, containing the partial molar volume of the solute at infinite dilution \(V_{2}^{{{\text{L,}}\infty }}\) and its derivatives with respect to T and P, together with \({{{\text{d}}P_{{{{\upsigma ,1}}}} } \mathord{\left/ {\vphantom {{{\text{d}}P_{{{{\upsigma ,1}}}} } {{\text{d}}T}}} \right. \kern\nulldelimiterspace} {{\text{d}}T}}\), \(\left( {{{{\text{d}}P_{{{{\upsigma ,1}}}} } \mathord{\left/ {\vphantom {{{\text{d}}P_{{{{\upsigma ,1}}}} } {{\text{d}}T}}} \right. \kern\nulldelimiterspace} {{\text{d}}T}}} \right)^{2}\) and \({{{\text{d}}^{2} P_{{{{\upsigma ,1}}}} } \mathord{\left/ {\vphantom {{{\text{d}}^{2} P_{{{{\upsigma ,1}}}} } {{\text{d}}T^{2} }}} \right. \kern\nulldelimiterspace} {{\text{d}}T^{2} }}\), are referred to in the literature as Wilhelm terms [247, 263, 264]. For aqueous solutions, say, of the rare gases below 373 K, their contributions are small [247], usually smaller than the experimental error associated with current highprecision measurements, though they become more important in the critical region. Worldwide, there were essentially only two groups that could provide the necessary highprecision experimental data on \(h_{2,1} \left( {T,P_{{{{\upsigma ,1}}}} } \right)\) with an imprecision of, say, ± 0.05%, that allow reliable van ’t Hofftype analysis: the group of B. B. Benson and D. Krause, Jr. (BK), Department of Physics, Amherst College, Amherst, MA, USA, and the group of R. Battino, E. Wilhelm and T. R. Rettich (BWR), Department of Chemistry, Wright State University, Dayton, OH, USA.
Direct calorimetric determinations of the highdilution molar enthalpy change on solution \(\Delta_{{{\text{sol}}}} H\) of a gas in a liquid have been carried out by only a very limited number of researchers [265,266,267,268,269,270,271,272,273,274,275,276,277], simply because of the experimental difficulties associated with accurate measurements of very small heat effects in very dilute solutions (typically, aqueous mole fraction solubilities at ambient temperature and 0.1 MPa gas pressure are about 10^{−5} to 10^{−4}). Recently, various calorimeter designs used in this field were discussed in detail by Wilhelm and Battino [241]. The most prominent methodological step forward in measuring enthalpies of solution of gases in liquids with a precision high enough to allow reliable determination of the corresponding molar heat capacity changes on solution
is closely connected with the development of microcalorimeters in the Thermochemistry Laboratory of the University of Lund, Sweden (I. Wadsö’s group), and in the Chemistry Department of the University of Colorado in Boulder, CO, USA (S. J. Gill’s group), respectively. A fortiori, because of the formidable experimental difficulties encountered in the case of direct measurements of the molar heat capacity changes on solution of nonreacting gases dissolved in water at very low concentration, there exist only five sets of such data, all originating from the laboratory of R. H. Wood at the University of Delaware in Newark, DE, USA. Wood and collaborators determined the apparent molar isobaric heat capacities at very small solute mole fractions of the following four gaseous solutes in water: argon [278, 279], xenon [279], methane [280] and ethene [279]; in addition, they measured the apparent molar isobaric heat capacities at small mole fractions of aqueous solutions of CO_{2}, H_{2}S and NH_{3} [280]. The sophisticatedly constructed flow calorimeter developed by Wood and his collaborators was used over very large temperature ranges from about 305 K to well into the supercritical region of water up to T ≈ 720 K. Measurements of the apparent molar heat capacities of acid gases in liquid water near ambient conditions using a commercial Picker calorimeter [281] were reported by Barbero et al.: aqueous solutions of hydrogen sulfide at 10 °C, 25 °C and 40 °C [282]; aqueous solutions of carbon dioxide and sulfur dioxide at 25 °C [283]; note that T/K = t/°C + 273.15.
Although the measurements of Wood et al. were all performed at somewhat elevated pressures between, roughly, 17 MPa and 32 MPa, the mole fractions of the dissolved gases are still small enough to make the apparent molar heat capacities to a good approximation, that is, within experimental error, equal to the partial molar isobaric heat capacities at infinite dilution (this statement is valid at temperatures below ca. 500 K). At constant T and P we have
hence
Since
we obtain
Except for the few sets of direct heat capacity data on aqueous solutions of nonreacting gases as reported by Wood et al., all other calorimetrybased determinations of molar heat capacity changes on solution were obtained from the temperature dependence of the enthalpies of solution via Eq. 322.
Evidently, comparing van’t Hoff derived enthalpy changes (one differentiation level, see Eq. 320) and heat capacity changes (two differentiation levels, see Eq. 321) with directly obtained highprecision calorimetric results constitutes a particularly severe test of solubility data. Recently, Wilhelm [169], Wilhelm and Battino [240, 241], and Battino [284] presented essentially comprehensive compilations of van ’t Hoffbased partial molar enthalpy changes on solution \(\Delta H_{2}^{\infty }\) and partial molar heat capacity changes on solution \(\Delta C_{P,2}^{\infty }\) for gases dissolved in liquid water at \(T = 298.15{\text{ K}}\) and \(P_{{{{\upsigma ,1}}}} \left( {{\text{H}}_{2} {\text{O; 298}}{\text{.15 K}}} \right)\) = 3.1691 kPa, and compared them with calorimetrically determined results for \(\Delta_{{{\text{sol}}}} H\) and \(\Delta_{{{\text{sol}}}} C_{P}\) at high dilution and ambient pressure. Note that highprecision solubility data for nitrogen and carbon monoxide dissolved in water [285, 286] are available, but somewhat surprisingly no calorimetric results have been reported so far. For the sake of brevity, in Table 5 I have limited such a comparison to a few representative systems at 298.15 K, though essentially the same overall picture emerges for other systems at other temperatures too: agreement between these two approaches is highly satisfactory, i.e., it is usually within the combined experimental error: what a credit to experimental ingenuity and to the skills of solution thermodynamicists!
3 Concluding Remarks, Future Directions and Acknowledgments
By common consent, the liquid state of matter houses by far the largest group of unsolved/crudely solved problems in modern physical chemistry, especially when biophysical chemistry is included: indeed, molecular thermodynamics of aqueous solutions of biorelevance is an exciting, dynamically evolving field. With increasingly sophisticated computer simulations, new insights at microscopic, mesoscopic, and macroscopic levels are obtained, providing stimulating connections with a growing number of neighboring fields, in particular with biomedical research. Another major driving force is the unabated progress in instrumentation: increasing the applicability range, that is, larger temperature and pressure ranges, ever smaller concentrations, improving precision and accuracy, and increasing the speed of measurements together with facilitating application and data transfer now allows tackling problems which, say, twenty years ago would have been considered to be quite impossible tasks. In this review, I have focused on the rigorous and concise presentation of the thermodynamic formalism relevant for the study of liquid solutions of nonelectrolytes, in particular of dilute liquid solutions, and I have tried to cover, in a coherent fashion, the most important aspects of its practical implementation, indicating areas where more and/or more detailed experimental work would be desirable. Special emphasis was placed on binary aqueous solutions of supercritical solutes (gases), a field I have worked in with Rubin Battino for more than 50 years. Two closely related areas have been dealt with prominently:

discussion of solution behavior in terms of the Henry fugacity \(h_{2,1} \left( {T,P_{{{{\upsigma ,1}}}} } \right)\) (also known as Henry’s law constant) and related quantities, such as fugacity coefficients and activity coefficients, and exposing some frequently encountered misconceptions;

discussion of the equivalency of results for caloric quantities (e.g., \(\Delta H_{2}^{\infty }\) and \(\Delta C_{P,2}^{\infty }\)) derived from solubility measurements via van’t Hoff analysis with those measured directly with calorimeters.
These two topics present the state of the art for the (potential) experimentalist. Results obtained for aqueous systems by the BK and BWR groups (VLE), by Wadsö et al. and by Gill et.al. (calorimetry), have greatly contributed to our understanding and appreciation of hydrophobic effects. They are thought to play an important role in biology, though in many complex biochemical processes hydrophilic effects may be more important [290, 291]. A rich research field lies ahead, as evidenced by the increasing number of investigations with a strong biophysical and/or biomedical flavor. The research of Wood et al., based on the instruments he developed, made possible pioneering measurements of the density and the heat capacity of dilute aqueous solutions of nonelectrolytes as well as electrolytes from ambient temperatures to the critical temperature of water, \(T_{{\text{c}}} = 647.1{\text{ K}}\), and beyond, well into the single fluid region, yielding novel, spectacular results [292].
What I hope to have also communicated is my conviction that advances in science inevitably lead to a broadening and merging of neighboring areas of research, and that crossdisciplinary fundamental research is always a potent stimulus in science as well as for technological innovation.
Studies on solubility in general, and on solubility in liquid water in particular, have come a long way. The field has grown too big to be covered in one modest review article, but for the foundations and the few subsections covered I hope to have succeeded in providing a feeling for their scope, current position in physical chemistry, and future potential. Related to these aspects I find the statement by Gilbert Newton Lewis (1875–1946) on the practical philosophy of science most appropriate and encouraging. In fact, it is one of my favorite quotes, and although I’ve cited it before, the insight contained makes it appropriate to repeat it here:
The scientist is a practical man and his are practical aims. He does not seek the ultimate but the proximate. He does not speak of the last analysis but rather of the next approximation....On the whole, he is satisfied with his work, for while science may never be wholly right it certainly is never wholly wrong; and it seems to be improving from decade to decade.
My view of chemical thermodynamics has evolved over the years, and I have benefitted from collaborations as well as discussions with many friends and colleagues. Quite early in my career I was asked to participate at various IUPAC activities, and I did indeed devote an appreciable part of my “scientific life” to further the Union’s goals, both nationally and internationally (see Appendix). Thus, I was particularly pleased to participate, as an Invited Lecturer, at the 47th IUPAC World Chemistry Congress: Frontiers in Chemistry, celebrating 100 Years with IUPAC, in Paris, France, 7–12 July 2019. This Invited Lecture was part of a Special Symposium dedicated to Professor Eduard Hála (1919–1989), who was one of the great pioneers of modern vapor–liquid experimental science and theory. Incidentally, Professor Hála and I were both founding members of the editorial board of Fluid Phase Equilibria, one of the internationally leading journals in this field. This Special Symposium was admirably organized by M. Bendova, ICPF/CAS, Prague, Czech Republic, and J. Jacquemin, Université de Tours, Tours, France.
Chemical thermodynamics is a vast field of science with many applications to realworld problems (hence many advances originated from work of chemical engineers). For me it is a great pleasure to acknowledge here (a) the initiation to this topic, many decades ago, by Professor Friedrich Kohler (†), my PhD advisor (on adiabatic calorimetry and applied statistical mechanics) at the University of Wien, Austria, via his courses and seminars [293]; (b) the more than 50 years of fruitful scientific collaboration with Rubin Battino, now Emeritus Professor at Wright State University (WSU), Dayton, Ohio, USA, who introduced me to the fascinating field of the solubility of gases in liquids; (c) the work with JeanPierre E. Grolier, now Emeritus Professor at Université Blaise Pascal, ClermontFerrand, France, with whom I enjoyed about 45 years of exciting scientific collaboration with more than 25 years of continuously funded research in solution thermodynamics (that is, research on \(H^{{\text{E}}} ,C_{P}^{{\text{E}}} {\text{ and }}V^{{\text{E}}}\)), and many memorable firstclass dinners; (d) the cooperation with Henry V. Kehiaian (†) [294, 295], former Directeur de Recherches at ITODYS (CNRS), Paris, France, and an old friend since 1967 (since his first visit from Warsaw, Poland, to the”west”, that is, to the Institute of Physical Chemistry, University of Wien, Vienna, Austria), on group contribution theory and many enjoyable dinners (often together with JeanPierre Grolier); (e) the research (since 1981) with Augustinus Asenbaum, now Retired Professor of Experimental Physics at the University of Salzburg, Salzburg, Austria, who opened the doors to the fascinating extrathermodynamic field of Brillouin scattering at ambient and elevated pressures; and (f) the satisfying and frequently stimulating work with about 80 colleagues, postdoctoral fellows and students from 17 countries. Without them, many projects would have been difficult to carry out. Fortunately, over the years the University of Wien (Vienna), Austria, was generous enough to grant repeatedly extended Leaves of Absence for scientific research abroad, and in the present context, the most important were two extended stays in the United States of America at WSU, Dayton, Ohio (first as a Senior Fulbright Research Associate, then as Visiting Associate Professor: about four years altogether), which I greatly enjoyed. In fact, most of the experimental work on gas solubility was supported by the Public Health Service of the United States of America via grants from the National Institute of General Medical Sciences, and a smaller part was supported by the Petroleum Research Fund, administered by the American Chemical Society. Research on fluid phase chemical thermodynamics funded within the frame of the AustrianFrench Program on Scientific and Technical Cooperation also led to many extended stays in France, altogether for more than five years: first at the CRMT/CNRS in Marseille (TianCalvet Institute), and then at the Université Blaise Pascal in ClermontFerrand and at ITODYS in Paris, all of which I also greatly enjoyed. When visiting JeanPierre Grolier in ClermontFerrand, I frequently had also the pleasure to meet Robert H. Wood on leave from the University of Delaware, Newark, DE, USA, to whom a Special Issue of The Journal of Solution Chemistry (Volume 44, Number 5), a Festschrift to celebrate his eightieth birthday, was dedicated in May 2015. These joint stays provided a welcome opportunity to discuss molecular thermodynamics problems in general, and hot topics in solution thermodynamics in particular. In appreciation of our scientific research activities at the Université Blaise Pascal, Bob Wood and I were recipients of honorary doctorates. Besides science, for me those years in Marseille, ClermontFerrand and Paris provided unforgettable experiences associated with France: indeed, these were the years of St. Nectair and Bleu d’Auvergne, of SaintEmilion and Pomerol!
Notes
The thermodynamic systems treated in this review are generally referred to as simple systems. By definition, such systems are macroscopically homogeneous, isotropic, uncharged, nonreactive, and large enough to neglect surface effects. In addition, simple systems are not acted upon, for instance, by electrostatic or magnetic fields; in addition, the fluid samples are small enough so that the influence of the earth’s gravitational field is not detected in a variation of properties with the height of the containing vessel. Pressure is the only mechanical force considered, causing contraction or dilation of the fluid. Near the critical point, many thermodynamic properties (and transport properties as well) show anomalies linked to the divergence of the fluid’s isothermal compressibility [29,30,31].
$$\beta_{T} =  V^{  1} \left( {{{\partial V} \mathord{\left/ {\vphantom {{\partial V} {\partial P}}} \right. \kern\nulldelimiterspace} {\partial P}}} \right)_{T} = \rho^{  1} \left( {{{\partial \rho } \mathord{\left/ {\vphantom {{\partial \rho } {\partial P}}} \right. \kern\nulldelimiterspace} {\partial P}}} \right)_{T}$$where \(\rho\) denotes the mass density. Thus, associated with the large compressibility in a nearcritical fluid (say, within 1 K from the critical point), the presence of the gravitational field will cause the local value of the density to vary with height, i.e., macroscopic density gradients (gravitational sedimentation) develop due to compression under the fluid’s own weight. Over the height of a few centimeters, they can cause density variations as large as 10% [32,33,34]. Note that the divergence of the compressibility gives rise to divergencies of the isobaric expansion coefficient and the isobaric heat capacity, etc., and measurements close to the critical point become significantly distorted. For experiments on fluid systems under terrestrial laboratory conditions and sufficiently removed from the critical region the influence of the earth’s gravitational field is generally ignored. However, for interpreting precision measurements in the critical region and to test theoretical predictions of critical phenomena, experiments taking advantage of the microgravity environment of space (space laboratories) are indispensable and have indeed been performed [35].
With the Avogadro constant \(N_{{\text{A}}}\) and the Boltzmann constant \(k_{{\text{B}}}\) now being exactly defined [44, 45], that is, \(N_{{\text{A}}} = 6.022 \, 140 \, 76 \times 10^{23} {\text{ mol}}^{  1}\) and \(k_{{\text{B}}} = 1.380 \, 649 \times 10^{  23} {\text{ J}}\;{\text{K}}^{  1}\), the (molar) gas constant is \(R = N_{{\text{A}}} k_{{\text{B}}} = 8.314 \, 462 \, 618{\text{ J}}\;{\text{K}}^{  1} \;{\text{mol}}^{  1}\).
Max Margules was an Austrian physicist. Born in Brody, Galicia, a former crownland of the AustroHungarian Empire, on April 23, 1856, he started his studies in mathematics, physics and chemistry at the University of Wien (Vienna), Austria, in 1872. Among his teachers were Ludwig Boltzmann and Josef Loschmidt. After a few years as Assistant at the Central Institute of Meteorology in Wien, he went to Berlin for additional studies in mathematical physics (1879/1880) and was habilitated at the University of Wien after his return. However, in 1882 Margules resigned from this academic position and rejoined the Central Institute of Meteorology. During the first years at this institution, he continued to pursue physical and physical–chemical research parallel to his work on theoretical meteorology. In 1906 he voluntarily retired at the age of fifty on a very modest pension. This was also the year of his last meteorological publication, dedicated to the theory of storms:
Margules, M.: Zur Sturmtheorie. Meteorolog. Z. 23, 481–497 (1906).
His small pension and the inflation after the end of World War I led to a life in poverty, which was compounded by the general state of malnutrition of the Austrian population and his refusal to accept help from colleagues and/or the Austrian Meteorological Society (he was awarded the Hann Medal in 1919 but declined the associated honorarium). Max Margules died of starvation in Perchtoldsdorf near Wien (Vienna), Austria, on October 4, 1920. He contributed significantly and lastingly to meteorology and thermodynamics [195]. Obituaries were prepared by F. M. Exner (Meteorolog. Z. 37, 322–324 (1920)) and E. Gold (Nature 106, 286–287 (1920)).
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