Introduction

ANTHONY R. H. GOODWIN, JAN V. SENGERS AND CORNELIS J. PETERS

In the series Experimental Thermodynamics the first and only volume concerned with equations of state for fluids and fluid mixtures was Volume V.5 This volume, which was published during 2000 in two parts, provided descriptions of equations of state, over an entire range of approaches and range of variables for their wide use in science, engineering and industry. Reference 5 included methods required to develop equations of state, including their theoretical bases and practical uses along with their strengths and limitations. Furthermore, the volume contained not only equations of state for simple fluids and fluid mixtures, but also for and more important classes of complex fluids and mixtures. In particular, the volume included associating fluids, ionic fluids, poly-disperse systems, polymers, and micelle-forming and other self-organizing systems. However, some subject matter of importance to the practitioner was omitted including equations of state specifically for chemically reacting fluids and methods applicable to non-equilibrium thermodynamics.

Since the year 2000, there have been over 15,000 publications in the academic scientific and engineering archival literature that in some form or other are concerned with equations of state. In addition, equations of state for chemically reacting and non-equilibrium fluids have received additional theoretical and practical attention and now deserve more detailed explanations than hitherto provided.

Both developments in the field and the desire to provide the content omitted in the previous volume motivated this text entitled Applied Thermodynamics of Fluids. The change in title also reflects a greater emphasis that has been placed on the application of theory, without recourse to derivations of the constitutive equations, while retaining the fundamental aspects. The volume includes thermodynamics at the nano and meso scale in Chapter 7, chemically reacting systems in Chapter 13 and the application of non-equilibrium thermodynamics in Chapter 14. This volume is intended to address the needs of practitioners within academia, government and industry. However, chapters from reference 5 regarding self-assembled systems and analytical solvable integral equations have been omitted in this work and as have in both volumes the use of computer simulations for the calculation of thermodynamic properties. The latter would deserve an in-depth coverage of its own and because of size limitations along with a recent special issue of Fluid Phase Equilibria and other publications reporting the Industrial Fluid Property Simulation Challenges, it was decided not to include the topic in the present volume.

Some chapters from Volume V5 have been revised and updated and are included here because of their fundamental importance to the topic: these appear in this text as Chapter 2, regarding Fundamental Considerations that is essential to determine the validity of any method adopted, Chapter 3, entitled Virial Equation of State, Chapter 4 concerned with Cubic and Generalized van der Waals Equations, Chapter 5, Mixing and Combining Rules, Chapter 6 on Corresponding States. Significantly greater prominence has been is given to Statistical Associating Fluid Theory (SAFT) in Chapter 8, while Chapter 9 provides and update on Poly-disperse fluids, Chapter 10 is concerned with the more general topic of Critical Behaviour, Chapter 11 reports on Ionic Fluids and Chapter 12 on the Multi-parameter Equations of State.

Applied Thermodynamics is published under the auspices of the Physical and BioPhysical Division (I) of the International Union of Pure and Applied Chemistry as a project proposed by the International Association of Chemical Thermodynamics (IACT) in its capacity as an organization affiliated with IUPAC. Consequently, throughout the text we have adopted the quantities, units and symbols of physical chemistry defined by IUPAC in the text commonly known as the Green Book. We have also adopted the ISO guidelines for the expression of uncertainty and vocabulary in metrology. Values of the fundamental constants and atomic masses of the elements have been obtained from references 21 and 22, respectively.

Fundamental Considerations

ANGEL MARTÍN MARTÍNEZ AND COR J. PETERS

2.1 Introduction

This chapter provides a thermodynamic toolbox and contains most of the important basic relations that are used in other chapters. The scope is restricted almost exclusively to the second law of thermodynamics and its consequence, but the treatment is still intended to be exemplary rather than definitive. New results are not presented as befits a discussion of fundamentals which are necessarily invariant with time.

2.2 Basic Thermodynamics

The state of a system may be described in terms of a small number of variables. For a phase in the absence of any external field, the second law of thermodynamics may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

This equation shows that the change dU in the energy U may be described in terms of simultaneous changes dS in the entropy S, dV in the volume V, and dni in the amount of substance ni of the i components. It is often convenient to eliminate the size of the phase by writing eq 2.1 in terms of intensive variables. For example, division by the total amount n

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

where the subscript m denotes a molar quantity and the mole fraction of component i is defined by

xi = ni/N, (2.4)

Equation 2.1, or eq 2.3 for intensive variables, is the fundamental expression of the second law of thermodynamics. However, entropy, in particular, is not a very convenient experimental variable and, consequently, alternative forms have been derived from the fundamental eq 2.1. Introduction of the following characteristic functions:

H = U + pV, (2.5)

A = U - TS, (2.6)

and

G = U + pV - TS = H - TS = A + pV. (2.7)

For enthalpy, Helmholtz and Gibbs functions and use of Legendre transformations with the fundamental eq 2.1 gives the following alternative forms of the second law

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

Another modification of eq 2.1, frequently used in statistical mechanics, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

or, as the Gibbs-Duhem equation, in the study of phase equilibria

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12)

From eq 2.1 and eqs 2.8 to 2.10 we find the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.13)

where the subscript [??]j means that the amount of substance nj of all the components are constant except for component i. The quantity µi is the chemical potential of species i. In terms of intensive variables these equations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.18)

For a system of constant composition, eq 2.1 and eqs 2.8 to 2.10 reduce to

dU = TdS - pdV, (2.19)

dH = TdS + Vdp, (2.20)

da = -SdT - pdV, (2.21)

and

dG = -SdT + Vdp, (2.22)

A large number of thermodynamic relations may be derived from the above equations by conventional manipulations. Table 2.1 summarizes the most frequently used equations.

2.2.1 Homogeneous Functions

A homogeneous function F of the first order in any number of the variables x, y, z, ... is defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.23)

where λ is an arbitrary number. If each independent variable is made λ times larger, the function F increases λ times. For large enough systems, all extensive thermodynamic functions are homogeneous and of the first order in amount of substance ni at fixed temperature and pressure. For homogeneous functions of the first order, Euler's theorem on homogeneous functions applies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.24)

Equation 2.24 relates the value of the function to the values of its derivatives. For λ = 1, eq 2.24 reduces to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.25)

By applying Euler's theorem to the various characteristic functions U = U (S, V, n1, n2, ···), H = H (S, p, n1, n2, ···), A = A (V, T, n1, n2, ···), G = G (p, T, n1, n2, ···), respectively, the following expressions result:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.28)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.29)

For further details on Euler's theorem see references 1 to 3.

2.2.2 Thermodynamic Properties from Differentiation of Fundamental Equations

The quantities U = U (S, V, n1, n2, ···, xi), H = H (S, p, n1, n2, ···, xi), A = A(V, T, n1, n2, ···, xi), and G = G (p, T, n1, n2, ·· , xi) are examples of thermodynamic potentials from which all properties of a system can be obtained without the need for integration. For example, eq 2.10 gives directly the heat capacity at constant pressure

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.30)

and the isothermal compressibility

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.31)

Avoiding integration is often advantageous in theoretical applications of thermodynamics because there are no constants of integration. On the other hand, very often the derivatives needed involve variables that are difficult to measure experimentally. Even with the Gibbs function surface, which is closely linked to the convenient experimental variables of temperature, pressure and composition, differences in the Gibbs function can be studied only at equilibrium. At constant composition the characteristic functions U = U (S, V, n1, n2, ···, xi), H = H (S, p, n1, n2, ···, xi), A = A (V, T, n1, n2, ···, xi), and G = G (p, T, n1, n2, ···, xi) reduce to U = U (S, V), H = H (S, p), A = A (V, T) and G = G (p, T). By differentiation of these functions and use of eqs 2.5 and 2.19 to 2.21, the value of any thermodynamic property can be expressed in terms of the derivatives of each characteristic function. Table 2.2 summarizes the most relevant results. Further details can be found elsewhere.

2.3 Deviation Functions

Almost all definitions of molar properties for mixtures lack an unambiguous definition in the sense that they can be related directly to measurable properties. Therefore, it is common practice to compare an actual mixture property with its corresponding value obtained from an arbitrary model, for instance, an equation of state. This approach leads to the introduction of deviation functions. For a general mixture molar property Mm, the deviation function is defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.32)

An important aspect in this definition is the choice of the independent variables. Many analytical equations of state are expressions explicit in pressure: that is, temperature, molar volume (or density) and composition x = x1, x2, ···, xi are the natural independent variables. Therefore, eq 2.32 can be rewritten into:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.33)

where n denotes the amounts n1, n2, ·· , ni. The value of M obtained from the model is evaluated at the same values of the independent variables as used for the mixture property. Alternatively, temperature, pressure and composition may be a suitable choice as independent variables for deviation functions, for example:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.34)

In this case the value of M obtained from the model is evaluated at the same values of T, p and n as used for the actual mixture property. Both approaches are interrelated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.35)

In this equation, pr is the reference pressure at which the molar volume of the mixture obtained from the model is equal to the molar volume of the actual mixture at the same temperature and composition as the mixture. A necessary feature of the model is that such a pr exists.

2.3.1 Residual Functions

As pointed out in the previous section, the calculation of deviation functions requires a choice of an appropriate model. If the model system is chosen to be an ideal gas mixture, which is an obvious choice for fluid mixtures, then the deviation functions are called residual functions. With temperature, volume and composition as independent variables eq 2.33 becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.36)

or with T and p as independent variables

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.37)

which is a particular form of eq 2.34. The two sets of residual functions are related by eq 2.35 in the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.38)

where the reference pressure pr=RT/Vm is sufficiently low for the thermodynamic property M of the real fluid to have the ideal-gas value.

Some thermodynamic properties (U, H, CV and Cp) of an ideal gas are independent of pressure, while others like S, A and G are not. Consequently, from eq 2.37, it can be easily seen that the following equations hold:

UR(T, V, n) = UR(T, p, n), (2.39)

HR(T, V, n) = HR(T, p, n), (2.40)

CRV(T, V, n) = CRV(T, p, n), (2.41)

CRp(T, V, n) = CRp(T, p, n), (2.42)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.43)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.44)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.45)

where Z=pV/nRT and is known as the compression or compressibility factor.