THERMODYNAMIC PROPERTIES OF PURE FLUIDS
THERMODYNAMIC PROPERTIES OF PURE FLUIDS
Abstract and Keywords
This chapter gives an account of the statistical thermodynamics of pure fluids composed of non-spherical molecules. The early part of the chapter gives the derivation of equations for the various thermodynamic functions in terms of correlation functions and intermolecular forces, and includes discussion of quantum corrections and virial coefficients. The later parts of the chapter describe the application of molecular theory (perturbation theory, theory of hard non-spherical bodies, associating fluid theory) to the calculation of these properties, and compares these results with those from molecular simulations and experiments. The thermodynamics of nano-scale systems, in which some macroscopic laws and concepts may break down, are discussed at the end of the chapter. The mathematics of convex body geometry is given in an appendix.
Keywords: thermodynamic properties, convex body fluids, quantum corrections, associating fluids, nano-scale systems, virial coefficients, perturbation theory, scaled particle theory, geometry of convex bodies, molecular simulation
[Statistical thermodynamics] differs from classical thermodynamics in that the thermodynamic functions for the assemblies and phases with which we deal are not left unspecified, or to be derived solely from measurements, but are always constructed a priori by the application to particular molecular models of the fundamental theorems of statistical mechanics with which we start.
Ralph Fowler and E. A. Guggenheim, Introduction to Statistical
Thermodynamics (1949)
In this chapter we consider the thermodynamic properties of a pure isotropic, homogeneous fluid in the classical, rigid molecule approximation (quantum corrections are briefly discussed in § 6.9). In the canonical ensemble the fundamental relation is that for the configurational free energy (the configurational part, in the classical limit, of the total free energy), given by Eq. (3.11) of Vol. 1,
where Q _{c} is the configurational partition function, given by (3.82):
Here β = 1/kT, Ω = 4π or 8π^{2} for linear and nonlinear molecules, respectively, and Z _{c} is the configurational phase integral^{†}. Expressions for the other thermodynamic properties in terms of the partition function can be derived from (6.1) using the usual relations of classical thermodynamics, e.g.
From (6.1) and (6.3)–6.5 it is easy to write down the relations for the pressure p, configurational energy U _{c}, and configurational heat capacity C _{vc} in terms of the partition function Q _{c}. If, in addition to the classical and rigid-molecule approximations we assume the potentials to be pairwise additive (Eq. (1.13)), then we can also relate these properties to the pair correlation function g(r ω _{1} ω _{2}). These relations are derived in §§6.1–6.4 below. Alternative expressions for the pressure and chemical potential are provided by the compressibility relation (§ 6.3), and test particle expressions (also referred to as the potential distribution theorem, § 6.5), respectively. These expressions do not depend on either the rigid molecule or pairwise additivity approximations.
The energy, pressure, compressibility, free energy, and chemical potential equations ((6.9a), (6.15), (6.16), (6.24), (6.35), (6.37), and (6.49)) provide alternative routes to the thermodynamic properties. Four of these can be used to check the consistency of any theory of g(r ω _{1} ω _{2}) by comparing results obtained by the four methods; these should be identical if the theory is exact, so that the degree of inconsistency gives a rough measure of the inexactness of the theory. Consistency is not a sufficient condition for exactness, of course; thus if we use a perturbation expansion for g(r ω _{1} ω _{2}) identical results will be obtained by each of the four routes regardless of the order at which the series is truncated
In addition to the general expressions for p, U, χ, and A in terms of the partition function, intermolecular potential, and pair correlation function, it is often useful to express these properties in terms of the spherical harmonic expansion coefficients for the potential and the pair correlation function. Moreover, if the pair potential is of the site-site type some of these properties can be related to the simpler site-site correlation functions, rather than the full pair correlation function (in the case of χ such a relation exists irrespective of the type of potential). These expressions are therefore given in §§6.6 and 6.7. In §6.8 we note a rigorous inequality that exists for the free energy of a molecular fluid. Quantum corrections are briefly discussed in §6.9, and virial coefficients are considered in §6.10. Comparisons between theory and experiment are given in §6.11. The scaled particle approach for fluids of hard convex molecules, and its empirical extension to non-convex molecules, is described in §6.12, and theories for associating fluids are discussed in § 6.13.
6.1 The energy equation
The relation between the configurational energy and the partition function is obtained from (6.4) and (6.1) as
(p.629) On performing the differentiation, we get:
Using the pairwise additivity approximation and noting that the sum of the u(ij) ≡ u(r _{ij} ω _{i} ω _{j}) contains ½N(N – 1) terms, each of which gives the same result on averaging, we have U _{c} = ½N(N – 1) 〈u(12)〉, or
where (3.105) has been used in carrying out the last step. This integral can be further simplified by noting that the integrand is a function of r _{12} only, F(r _{12}) ≡ 〈u(12)g(12)〉_{ω 1 ω 2}. On changing integration variables from (r _{1}, r _{2}) to (r _{1}, r _{12}), where r _{12} = r _{2} – r _{1}, we have
Expressing dr _{12} in polar coordinates, ${\text{d}\text{r}}_{\text{12}}={r}_{12}^{2}\text{d}{r}_{12}\text{d}{\omega}_{\text{12}}$, and carrying out the integrations over the angles ω _{12} = θ _{12} ϕ _{12} gives
From (6.7) and (6.8b) we obtain the energy equation,
which provides a route to the thermodynamic properties if g(r ω _{1} ω _{2}) is known. For example, it can be used to derive an equation for the pressure by first using U _{c} = ∂(βA _{c})/∂β to derive an expression for A _{c}, followed by the use of p = –(∂A _{c}/∂V).
(p.630) For brevity (6.9a) is often written (since the integrand is independent of ω)
Similar relations hold for the pressure, free energy, and other equations given in subsequent sections.
6.2 The pressure equation
From (6.1) and (6.3) the pressure is given by
or
The evaluation of the volume derivative is complicated by the fact that V appears as the upper integration limit for the integrations over the r _{i}. Such integrals have already been considered in §3.4.1. The differentiation can be performed by a change of integration variables suggested by H. S. Green,^{1}
so that (6.11) gives
Assuming pairwise additivity of the potentials, we have
If we substitute (6.14) into (6.13), note that each of the N(N – 1) terms in the ij sum gives the same result on averaging, and transform back to the original r _{i} variables, we get p = ρkT – N(N – 1)/6V〈r _{12} u′(12)〉, or
(p.631) or
where u′ = ∂u/∂r, and in the last step we have used (6.8a). Equation (6.15) is the pressure equation. The pressure equation provides a second route to the thermodynamic properties. Like the energy equation, (6.9a), it assumes rigid molecules and pairwise additivity. The pressure equation can also be derived from the virial theorem (see Eq. (E.14) of Vol. 1).
We note that in regions of r for which ∂u/∂r 〉 0 (attractive region) the contribution of the intermolecular potential term on the right-hand side of (6.15) will result in a decrease in pressure, while the reverse is true where ∂u/∂r 〈 0 (repulsive region).
6.3 The compressibility equation
The compressibility equation provides a third route to the thermodynamic properties, and has been written down in Eq. (3.113). It is most easily derived^{2} by differentiating the grand canonical expression for the first order distribution function f(r _{1} ω _{1}) with respect to the chemical potential. Such derivatives have been considered in §§3.4.1 and 3.4.2, and (3.238) gives the general result of such a differentiation for the distribution function of order h. Putting h = 1 in (3.238) gives
which is the compressibility equation. Here χ = ρ ^{–1}(∂ρ/∂p)_{T} is the isothermal compressibility and g(r) is the centres pair correlation function. This expression does not rest on any assumptions concerning the form of the intermolecular potential energy (e.g. pairwise additivity or rigid molecules), and is therefore more general than the pressure equation. It holds for either isotropic or anisotropic fluids (e.g. liquid crystals), provided that they are homogeneous. Equation (6.16) relates χ to the centres correlation function, in contrast to the energy and pressure equations, where the full pair correlation function g(r ω _{1} ω _{2}) is needed.
As shown in Appendix 3E, it is also possible to express χ in terms of the direct correlation function, c(r ω _{1} ω _{2}), or the centres direct correlation function c(r) = 〈c(r ω _{1} ω _{2})_{〉ω 1 ω 2},
which holds for an isotropic fluid.
6.4 The free energy equation
In order to relate the free energy to the pair correlation function g(r ω _{1} ω _{2}) we use the coupling constant, or generalized charging process, approach.^{3} The intermole-cular potential energy is written
so that λ = 0 corresponds to the reference system and λ = 1 to the real system. The configurational free energy A _{cλ} for this system is
where
The difference A – A _{0} in free energies between the real and reference systems is given by
Combining (6.19) and (6.20) and differentiating gives
where 〈…〉_{λ} indicates an ensemble average for a system with potential u _{0} + λu _{p}. For pairwise additive u _{p}, (6.22) becomes (cf the derivation of (6.9a))
where g _{λ} is the pair correlation function corresponding to the system with pair potential u _{0} + λu _{p}. From (6.21), (6.22), and (6.23) we therefore have
(p.633) which is the free energy equation. It provides a fourth route from the pair correlation function to the thermodynamic properties. For example, using p = –(∂A/∂V) and (6.24) yields a fourth expression for the pressure, in addition to those obtained from the energy, pressure, and compressibility equations. Equation (6.24) can be used in computer simulation studies to calculate free energy changes.^{4} The Monte Carlo values of (A – A _{0}) given in Tables 4.1 and 4.2 were calculated in this way, as were the MSA and GMF values of (A – A _{0}) obtained in §§5.4.4 and 5.4.9.
6.5 The test particle expressions for thermodynamic properties
6.5.1 The Widom method^{5}
It is possible to relate the configurational thermodynamic properties to the interaction of a single molecule in the fluid with all of the other surrounding molecules.^{6} We call this single, specified molecule the ‘test particle’. These expressions are useful in both theoretical (see, for example, §6.12) and computer simulation (see later in this section) studies, and provide an alternative formalism to the expressions obtained in the previous four sections.
We first derive the expression for the chemical potential. Since we shall want to use these expressions for mixtures it is convenient to carry out the derivation for a mixture in which there are N _{A} molecules of componentA, N _{B} of component B, …, and N _{R} of component R ; the total number of molecules is N = ∑_{β} N _{β}. The configurational part of the chemical potential µ _{αc} of component α is given by
where N′ means all N _{β≠α}, and Q _{c} for mixtures is (cf. (3.82))
where x ^{N} = x _{1} x _{2} … x _{N} with x _{i} = r iω_{i}, and dx _{i} = dr i dω_{i}. Since the N _{β} are large we can approximate (6.25) by
where Q _{c}(N _{α}) is given by (6.26) and Q _{c}(N _{α} – 1) is the corresponding partition function for the system of N – 1 molecules (one α molecule removed),
(p.634) Here the product is over all components except α, u _{N–1} is the intermolecular potential energy for the (N – 1 ) molecule system, and we label the missing α molecule as molecule 1. The quantities Z _{c}(N _{α}) andZ _{c}(N _{α–1}) are the corresponding phase integrals given by (3.53). We now introduce u _{tα}(x _{1};x ^{N–1}), the intermolecular potential energy due to the interaction of the test particle of component α (molecule 1) with all other molecules in the system, by writing
The potentials u _{tα} and u _{N–1} can be expressed in terms of the coordinates of the molecules relative to those of the test particle, x _{21} x _{31} … x _{N1}, where x _{j1} = r _{j1} ω _{j1} denotes positional (r _{j1} = r _{j} – r _{1})^{†} and orientational ω _{j1} coordinates of molecule j relative to molecule 1; these coordinates are then independent of the coordinate x _{1}. We can therefore write Z _{c}(N _{α}) as
where 〈…〉_{N–1} means an ensemble average over the (N – 1) molecule system with intermolecular potential energy u _{N–1} (x _{2}… x _{N}). From (6.27) and (6.30) we have^{7} ^{,} ^{8}
where ρ _{α} = N _{α}/V is the number density of α molecules. The quantity 〈exp[–βu _{tα}]〉 can be interpreted as follows. The test particle is placed with some fixed coordinates x _{1} = r _{1} ω _{1} in the fluid of N – 1 molecules, and exp[–βu _{tα}] is then obtained as a function of the coordinates of all of the N – 1 molecules; a Boltzmann-weighted average over the coordinates x _{2} … x _{N} is then taken. Alternatively, we could imagine fixing the N – 1 molecules and moving the test particle freely among them; an unweighted average would then be taken over all possible positions and orientations of the test particle. In either case the test particle acts as a fictitious or ‘ghost’ particle, since it has no influence on the other N – 1 molecules.
(p.635) It is often more convenient to work with residual thermodynamic properties rather than configurational ones. The residual chemical potential μ _{αr} is given by ${\mu}_{\alpha \text{r}}={\mu}_{\alpha}-{\mu}_{\alpha}^{\text{id}}={\mu}_{\alpha c}-{\mu}_{\alpha \text{c}}^{\text{id}}$, where the superscript id signifies the ideal gas value at the same temperature, density, and composition as the real mixture. For an ideal gas u _{N} = 0 and ${Q}_{c}^{\text{id}}={V}^{N}/{\displaystyle {\prod}_{\beta}{N}_{\beta}!}$ from (6.26), so that, using Stirling’s approximation for ln N _{β}!,
we have
so that^{8}
From this result and (6.31) we have
An alternative but closely related expression for μ _{αr} can be obtained by noting that
where 〈…〉_{N} is an ensemble average over the N molecule system for which the intermolecular potential energy is u _{N}. From (6.35) and (6.36) we have^{9}
Note that in (6.37) the average is taken over a system in which the test particle does interact with the other N – 1 molecules, i.e. the test particle is not a ‘ghost’. We also note that both (6.35) and (6.37) are quite general. They do not rely on the assumption of pairwise additivity, nor on the rigid molecule approximation provided we suitably interpret the ensemble average, and are not restricted to uniform fluids^{7} They provide yet another route (the μ-route) to the thermodynamics; note that μ _{α} is not expressed in terms of g(12), however^{10}
Both (6.35) and (6.37) have been used in computer simulation studies to obtain the chemical potential^{11}, ^{12} In using (6.35) it is common to introduce a fixed lattice of test particles having various orientations, and to obtain the interaction energy of the test particles with the real molecules; the average value of exp[–αu _{tβ}] is then found. The major contribution to this average is from (p.636) configurations in which the interaction energy is attractive (u _{tα} negative), and the method works well for low and moderate densities. At high densities (much above ρσ^{3} = 0.5 for Lennard-Jones (LJ) molecules^{12}) the test particles overlap real molecules for the majority of configurations, leading to predominantly repulsive configurations; at such densities the usual Monte Carlo and molecular dynamics procedures do not effectively sample the regions of phase space most important in determining μ _{αr}, and the statistical errors become large. One way of overcoming this problem is to bias the sampling so as to effectively cover the desired regions of phase space^{12} Similar problems are encountered in using (6.37) in simulations. In that case the dominant contributions to the average of exp[βu _{tα}] are from configurations in which U _{tα} is repulsive (i.e. positive); since the test particle is a real molecule moving under the influence of forces from its neighbours, it tends to spend much of its time in attractive configurations, again leading to large statistical uncertainties in μ _{αr} as calculated from the simulation. The situation can be illustrated schematically by rewriting (6.35) and (6.37) in the forms (for a pure fluid):
where f(u _{t}) is the probability density for the test particle energy u _{t} in the N – 1 molecule system, and g(u _{t}) is the corresponding probability density in the N molecule system. As seen in Fig. 6.1, at high densities it is not enough to know f(u _{t}) or g(u _{t}) in the region of the maximum, since there are substantial contributions to the integrand for regions of u _{t} where f or g is small.
It should be noted that for fluids of hard bodies u _{tα} can only take the values 0 or ∞. The average of exp(–βu _{tα}) that appears in (6.35) is still well behaved under these conditions (see §6.12 for applications to hard bodies). However, the average of exp(βu _{tα}) in (6.37) is indeterminate, so that this expression is not useful for such fluids.
It is possible to derive equations for other properties in terms of u _{tα}(1) = u _{tα} (x _{1};x _{2} … x _{N}) by methods that closely follow that used above for the chemical potential^{6} For example, the configurational energy is u _{c} = 〈u _{N}〉_{N}. If the inter-molecular potential energy is pairwise additive we can write
and it is then easy to show that^{13}
or, using (6.27), (6.33), and (6.35),
By similar arguments, we can derive the following equation for the pressure, starting from (6.13)and (6.14):
where
Equation (6.42) has been used to determine pressures for hard core systems,^{14} including the tensorial components of the pressure for inhomogeneous systems,^{15} and for continuous potentials^{16}
The surface tension can also be expressed in terms of a test particle equation,^{17} and the pair correlation function g _{αβ}(r ω _{1} ω _{2}) can be expressed in terms of ${u}_{t\alpha \beta}^{\left(2\right)}\left({x}_{1}{x}_{2};{x}_{3}\cdots {x}_{N}\right)$, the intermolecular potential energy of interaction of two test particles of species α and β with all of the remaining molecules of the system.^{6},^{7}
6.5.2 The Kirkwood method
Starting from (6.27) it is possible to derive a further expression for the chemical potential, due originally to Kirkwood,^{3} ^{,} ^{18} that involves the pair correlation function g(12) ≡ g(r ω _{1} ω _{2}). In this approach molecule 1 of species α is coupled to the remaining N – 1 molecules by a coupling parameter λ. Equation (6.29) is now replaced by^{19}
where 0 ≤ λ ≤ 1 and x ^{N–1} represents the configuration of all the other (N – 1) molecules. Thus λ = 0 and λ = 1 correspond to the N – 1 and N molecule systems, respectively. We have Z _{c}(λ = 1) = Z _{c}(N _{α}) andZ _{c}(λ = 0) = VΩ_{α} Z _{c}(N _{α} – 1), so that from (6.27) and (6.34) we obtain
(p.639) where
We now introduce the approximation of pairwise additivity. We label the molecules of speciesA as 1,2,…, N _{A}, those of B as 1,2,…, N _{B}, etc., so that
Substituting this in (6.47), noting that the sum over j contains (N _{β} – δ _{αβ}) terms (each giving the same result on integration), and using the definition of g _{αβ}(12) given in (3.252), we get
which is the Kirkwood result. Here g _{αβ}(12; λ) is the pair correlation function for β molecules around a central α1 molecule when the α1 molecule is coupled to the extent λ. This equation is less general than (6.35) and (6.37) of the previous subsection, since it is based on the assumptions of pairwise additivity and rigid molecules. However, it is a useful additional route to the chemical potential, and has been widely used in the theory of solutions^{20}
6.6 Thermodynamic properties in terms of spherical harmonic expansion coefficients
The energy, pressure, compressibility, and free energy equations each involve integrals over functions of intermolecular distances and orientations. It is often convenient to expand these functions in spherical harmonics, and thus obtain an expression for the property in terms of the harmonic coefficients of g(r ω _{1} ω _{2}) and u(r ω _{1} ω _{2}). The property harmonic expansions are most readily derived using the generalized Parseval theorem; this is derived in Appendix B of Vol. 1. We consider here only the case of linear molecules, and employ space-fixed axes. The corresponding expressions for nonlinear molecules, and for the intermolecular r-frame, or for the space-fixed and k-frame expansions in k-space, can be obtained by using the appropriate form of the generalized Parseval theorem given in Appendix B.
(p.640) For linear molecules and space-fixed axes, the generalized Parseval theorem is
where A(r ω _{1} ω _{2}) and B(r ω _{1} ω _{2}) are real and A(ℓ _{1} ℓ _{2} ℓ;r) is the space-fixed harmonic coefficient defined by
where ω _{i} ≡ θ _{i} ϕ _{i} is the direction of the symmetry axis of molecule i, ω ≡ θϕ is the direction of r, and C is a Clebsch-Gordan coefficient (App. A).
We can now apply (6.50) to the energy equation (6.9b) by taking A = u and B = g, so that
where we have made use of the fact that u(ℓ _{1} ℓ _{2} ℓ;r) and g(ℓ _{1} ℓ _{2} ℓ;r) are real (see §§ 2.3 and 3.2.1) and depend only on r = |r|, and ∫ dr(…) here can be replaced by 4π ∫ r ^{2} dr(…). We note that the spherical harmonic expansion has reduced the original sevenfold integral to the sum of one-dimensional integrals in (6.52). Similarly, applying (6.50) to (6.15), (6.16), (6.17), and (6.24) gives
where u′(ℓ _{1} ℓ _{2} ℓ;r) = du(ℓ _{1} ℓ _{2} ℓ;r)/dr and h(000;r) = g(000;r) – (4π)^{3/2}. In deriving (6.54) and (6.55) we note that B = 1, so that B(ℓ _{1} ℓ _{2} ℓ;r) = (4π)^{3/2} δ _{ℓ 10} δ _{ℓ 20} δ _{ℓ 0}, and only the (000) harmonic of B is nonvanishing.
It is also possible to write down the corresponding expansions for U _{c}, p,χ, and A in terms of the intermolecular-frame harmonic cosoft12sofefficients, by using the appropriate form of the generalized Parseval theorem, (B.85), in place of (6.50).
(p.641) Thus, for the configurational energy we obtain^{21}
where A(ℓ _{1} ℓ _{2} m;r) is defined by
where ${{\displaystyle \omega}}_{i}^{\text{'}}$ refers to the r-frame and m ≡ –m. The space-fixed and r-frame coefficients of A are related by (3.147) and (3.148).
The intermolecular harmonics have been extensively used in computer simulation studies^{22} However, if the intermolecular potential consists of a small number of harmonic terms, the space-fixed harmonic expansion gives a simpler form for the properties. For example, if the anisotropic potential is u _{a} = u _{QQ}, the quadrupole-quadrupole interaction, then there is only one nonvanishing u(ℓ _{1} ℓ _{2} ℓ;r) harmonic, namely u(224; r), (see (2.177) and (2.24)):
Thus, the expression (6.52) for the anisotropic contribution to U _{c}, for example, reduces to a single term,
so that only one space-fixed harmonic of g, namely g(224; r), need be known. In the intermolecular-frame expansion, (6.57), for the same intermolecular potential one would need five harmonics u(ℓ _{1} ℓ _{2} m;r) and g(ℓ _{1} ℓ _{2} m;r), viz. ℓ _{1} ℓ _{2} m = 222,221,220,221,222. Also, since ru′ = –5u for the u _{QQ} interaction we see that ${p}^{a}=\frac{5}{3}\rho {U}_{c}^{a}/N$ for this case.
The convergence of the spherical harmonic expansion using the intermolecular frame has been studied^{23} for homonuclear diatomic molecules for an atom-atom LJ potential model together with molecular dynamics simulation results for the g(ℓ _{1} ℓ _{2} m;r). It is found that for moderate elongations, ℓ 〈 0.45 (with ℓ = ℓ/σ where ℓ is the bond length and σ is the atom diameter), the series converges quite rapidly, but that the convergence quickly gets poorer as ℓ increases above this value. Typical results are shown in Table 6.1. The convergence has been found^{22} to be rapid for central LJ + quadrupole-quadrupole (QQ) potentials. For potentials that are quite hard, and elongated, the convergence worsens considerably^{22}, ^{24} Of course, if the potential consists of only a few harmonic terms (e.g. LJ + QQ) speed of convergence is irrelevant for many properties, such as the energy, pressure, and G _{2}, which involve only particular low order harmonics. (p.642)
Table 6.1 Convergence of the spherical harmonic expansion for homonuclear diatomic molecules using an atom−atom Lennard-Jones model
U^{*}/N |
p ^{*} |
|||||||
---|---|---|---|---|---|---|---|---|
ℓ^{*} |
T^{*} |
ρ ^{*} |
Q^{*} |
Number of harmonics^{†} |
Series |
Exact |
Series |
Exact |
0.425 |
1.543 |
0.50 |
0 |
4 |
−13.06 |
−13.11 |
−2.41 |
−2.34 |
0.425 |
1.543 |
0.50 |
0 |
8 |
−13.18 |
−13.11 |
−2.84 |
−2.34 |
0.5471 |
1.29 |
0.56 |
0.50 |
−25 |
−16.74 |
−14.45 |
−1.11 |
−0.61 |
0.5471 |
1.29 |
0.52 |
1.00 |
25 |
−13.18 |
−14.13 |
−0.08 |
−1.38 |
0.5471 |
2.20 |
0.52 |
2.00 |
25 |
−13.07 |
−16.91 |
−0.63 |
−0.67 |
(*) Reduced quantities are ℓ=ℓ/σ; T = kT/ε; ρ = ρσ ^{3};Q = Q/(εσ ^{5})^{½}; U = U/ε; p = pσ/ε, where σ and ε are the atom-atom LJ parameters. Here ℓ is bond length.
^{†} Four terms include coefficients ℓ _{1} ℓ _{2} m = (000), (200), (020) and (220); eight terms include in addition to these (400), (040), (420), and (240); twenty-five terms include the additional harmonics (440), (600), (620), (260), (640), (460), (660), (800), (820), (280), (840), (480), (860), (680), (880), (10,0,0), and (0,10,0). From ref. 23.
6.7 Thermodynamic properties in terms of the site-site correlation functions
In this section we derive relations for the thermodynamic properties U _{c}, p, χ, and A _{c} in terms of site-site correlation functions. For the compressibility such relations can be derived for an arbitrary intermolecular potential, and are as general as the compressibility equation itself.^{†} The energy, pressure, and free energy equations can only be transformed to relations involving the g _{αβ} correlation functions if the intermolecular potential is a sum of site-site potentials, as in (2.5).
6.7.1 Compressibility
We first consider the compressibility, χ. We have seen in § 3.1.6 that h _{αβ}(k), the Fourier transform of h _{αβ}(r) = g _{αβ}(r) – 1, is given by
where h(k ω _{1} ω _{2}) is the transform of h(r ω _{1} ω _{2}) and r _{ciα} is the position of site α with respect to the molecular centre of molecule i. Taking the limit k → 0 in this relation gives (we use the notation$\tilde{f}\left(k\right)$ in place of f(k) when there is a possibility of ambiguity, as in (6.62))
and combining this result with (3.113) gives the compressibility equation
or
Hence χ can be obtained from any one of the site-site correlation functions g _{αβ} (equivalently, χ is unaffected by the choice of molecular centres used for g(r) in (6.16), as expected physically).
It is possible to derive a similar expression for χ in terms of the reference interaction site model (RISM) site-site direct correlation functions c _{αβ}, as follows. The (matrix) OZ relation between h _{αβ}(k) and c _{αβ}(k) is (5.238),
where ω(k) is the matrix with elements ω _{αα′}(k) = sin(kℓ _{αα′})/kℓ _{αα′}. In the limit k → 0, ω approaches the matrix with elements ω _{αα′} = 1 for all α, α′.
Hence from (6.65) we have:
where
and ${\tilde{c}}_{\alpha \beta}\left(0\right)\equiv {c}_{\alpha \beta}\left(k=0\right)$. Hence we have
(p.644) Thus the compressibility equation can be written
or
The number of ${\tilde{c}}_{\alpha \beta}$ in the sum over αβ sites in (6.69) and (6.70) corresponds to the number of ${\tilde{h}}_{\alpha \beta}$ in (6.65). In the event that only one site A is chosen in each molecule, e.g. at the molecular centre, (6.70) reduces to
as one expects from the atomic liquid case, (5.20). It should be stressed that c _{AA}(r) defined by the OZ equation (6.65) is not equal to c(r), where c(r) ≡ 〈c(r ω _{1} ω _{2})〉ω _{1} ω _{2} is the centres direct correlation function defined by (3.122), although their r-integrals are equal. In contrast, for this case h _{AA}(r) is equal to 〈h(r ω _{1} ω _{2})〉ω _{1} ω _{2} see also §§ 3.1.5 and 5.5.2).
For homonuclear diatomic AA molecules with two sites per molecule, (6.64) and (6.70) reduce to
which can also be derived directly from (5.251).
6.7.2 Internal energy ^{25} ^{–} ^{27}
We now turn to the properties U, A, and p for the particular case of a site-site pair potential, (2.5). The energy equation, (6.9a), becomes
where Ω = ∫ dω and r _{αβ} is the distance between site α in molecule 1 and site β in molecule 2. The integral in this equation can be written more simply in terms of the site-site correlation function g _{αβ}(r _{αβ}), which is proportional to the probability density that sites α and β (on different molecules) are at a distance r _{αβ} apart, irrespective of the molecular orientations (see § 3.1.6). To carry out this simplification we transform the integration variables in (6.72) from (r _{1} r _{2} ω _{1} ω _{2}) to (p.645)
Converting these specific distribution functions to (generic) correlation functions g (see (3.91) and (3.105)) we have, for the isotropic, homogeneous case,
where g(r ω _{1} ω _{2}) is the usual angular pair correlation function and g(r _{αβ} ω _{1} ω _{2}) is proportional to the probability density that a pair of molecules are at orientations ω _{1} and ω _{2}, with site-site separation r _{αβ} = r _{2β} – r _{1β}. The site-site correlation function is related to g(r _{αβ} ω _{1} ω _{2}) by (cf. (3.126))
Since (6.74) is true for each αβ term in the sum in (6.72) we can substitute (6.74) in (6.72); note that u _{αβ} (r _{αβ}) is independent of ω _{1} and ω _{1}, and use (6.75) to get
(p.646) Changing variables from r _{1α} r _{2β} to r _{1α} r _{αβ}, integrating over r _{1α}, and using polar coordinates for r _{αβ} (see (6.8a) and (6.8b)) gives
Thus a knowledge of the site-site correlation functions is sufficient to calculate the configurational energy, provided that the pair potential is of the site-site form. For the special case of homonuclear diatomic (AA) molecules, (6.77) becomes
For the case of atomic liquids, having one site per molecule, (6.77) reduces to the standard result
6.7.3 Pressure^{25} ^{–} ^{28}
For a site-site potential the pressure equation, (6.15), becomes
or, using (6.74)
The derivative ∂u _{αβ} (r _{αβ})/∂r _{12} is at fixed molecular orientations and fixed ω _{12}, and is given by
To evaluate (∂r _{αβ}/dr _{12}) we note that (see Fig. 3.8)
where r _{cαβ} = r _{c2β} –r _{c1α}. Thus we have
where ${\widehat{r}}_{12}={r}_{12}/{r}_{12}$ is the unit vector in the r _{12} direction. Differentiating (6.83), noting that r _{cαβ} and r _{cαβ} are fixed (since ω _{1} and ω _{2} are fixed), we have
(p.647) Thus, rearranging and using (6.82), we have
where ${\widehat{r}}_{\text{12}}={r}_{\text{12}}/{r}_{12}$ and ${\widehat{r}}_{\alpha \beta}={r}_{\alpha \beta}/{r}_{\alpha \beta}$ are the unit vectors along r _{12} and r _{αβ}, respectively, and cos ${\gamma}_{\alpha \beta}={\widehat{r}}_{12}\cdot {\widehat{r}}_{\alpha \beta}$ (i.e. γ _{αβ} is the angle between r _{12} and r _{αβ}). Substituting (6.81) and (6.84) into (6.80), and using (6.75), we get
where 〈…〉_{rαβ} denotes a weighted average over the orientations keeping the site-site distance r _{αβ} fixed, i.e.
Changing variables in (6.85) from r _{1α} r _{2β} to r _{1α} r _{αβ}, integrating over r _{1α}, and using polar coordinates for r _{αβ} gives
where ${u}_{\alpha \beta}\stackrel{}{\prime}\left({r}_{\alpha \beta}\right)=\text{d}{u}_{\alpha \beta}\left({r}_{\alpha \beta}\right)/\text{d}{r}_{\alpha \beta}$. This is the pressure equation for a site-site potential. Equation (6.87) is formally of the site-site type, but since 〈r _{12} cos γ _{αβ}〉_{r αβ} requires the complete g(r ω _{1} ω _{2}) for its evaluation, it is not completely analogous to the relations (6.64) and (6.77) and (6.98) for χ, U _{c}, and A _{c}. Thus p cannot be expressed directly in terms of g _{αβ} even for a site-site potential^{29} For an atomic liquid, with one site per atom, (6.87) reduces to the usual result, (5.23). For diatomic AA 4 molecules (6.87) becomes
We now consider the special case where the molecules are composed of hard spheres rigidly joined together (with sphere α of diameter σ _{αα}, etc.). The site-site potential is therefore
(p.648) where σ _{αβ} = ½ (σ _{αα} + σ _{ββ}). It follows that
and
where δ(r _{αβ} – σ _{αβ}) is the Dirac delta function. Equation (6.91) follows because exp[–βu _{αβ}(r _{αβ})] ≡ θ(r _{αβ}) is the unit step function. The relation dθ(x)/dx = δ (x) is verified by noting that the defining conditions (δ(x) = 0 at x ≠ 0, δ(x) = ∞ at x = 0, and ∫ dxδ(x) = 1) for the δ-function are satisfied (see Appendix B).
It follows from (6.91) that
or
Substituting this result into (6.85), and using the fact that y _{αβ} (r _{αβ}) = exp[βu _{αβ}(r _{αβ})]g _{αβ}(r _{αβ}) is continuous, gives
where ${g}_{\alpha \beta}\left({\sigma}_{\alpha \beta}^{+}\right)$ is the value of the site-site correlation function at contact.
For a hard sphere atomic fluid we have 〈r _{12} cos γ _{αβ}〉_{r αβ = σ} = σ, and (6.93) reduces to the usual result, (5.24). For a fluid of hard homonuclear diatomic AA molecules, each of the terms in the summation over α and β in (6.93) is the same, and the pressure equation becomes
Using (6.82) and (6.84) we find that
where ℓ = ℓ/σ, ℓ = 2r _{c1A} = 2r _{c2A} being the site-site distance, and ${\theta}_{1A}\stackrel{}{\prime}$ and ${\theta}_{2A}\stackrel{}{\prime}$ are the polar angles corresponding to r _{c1A} and r _{c2A} with Z chosen to lie along r _{AA′} (see Fig. 6.4). (p.649)
6.7.4 Free energy ^{28} ^{,} ^{30}
In § 6.4 we have shown that (see (6.21), (6.22))
where A is the Helmholtz free energy for the full system with potential energy u(r ^{N} ω ^{N}), A _{0} is the value for a reference system with potential energy, u _{0}(r ^{N} ω ^{N}), u _{p} ≡ u – u _{0} is the perturbing energy, and 〈…〉_{λ} is an average over configurations with Boltzmann weight corresponding to u _{λ} = u _{0} + λu _{p}. If u _{p} is pairwise additive, and the intermolecular pair potentials u _{p}(ij) are of site-site form, then from (6.77) we have
and substituting this result in (6.96) gives
where ${g}_{\alpha \beta}^{\lambda}\left(r\right)$ is the site-site pair correlation function for the system with potential energy u _{λ} = u _{0} + λu _{p}. If the reference system is the ideal gas, then ${u}_{\alpha \beta}^{p}$ is replaced by the full site-site potential u _{αβ} in the above expressions. For diatomic AA molecules (6.98) becomes
The above method of introducing the charging parameter λ is not suitable for fluids composed of fused hard sphere molecules. For such fluids it is more convenient to use a charging parameter that scales both^{28} the site-site diameters σ _{αβ} and the intramolecular centre-site distances r _{cα}. Thus molecules characterized by the value λ of this parameter have site-site diameters ${\sigma}_{\alpha \alpha}^{\lambda}$ and intramolecular distances ${\text{r}}_{c\alpha}^{\lambda}$ defined by
so that λ = 0 corresponds to the ideal gas of point noninteracting molecules. By a derivation analogous to that of (6.24) in § 6.4 we find that
where A _{0} is the Helmholtz free energy of the ideal gas of point particles at the same temperature, density, and number of molecules as the fluid of fused hard spheres. We note that (6.101) cannot be reduced to a form involving site-site correlation functions, since the term $\partial {u}_{\alpha \beta}^{\lambda}\left({r}_{\alpha \beta}^{\lambda}\right)/\partial \lambda $ involves the molecular orientations ω _{1} and ω _{2} (see below).
We must now evaluate $\partial {u}_{\alpha \beta}^{\lambda}/\partial \lambda $. The potential ${u}_{\alpha \beta}^{\lambda}$ depends on λ through both ${\sigma}_{\alpha \beta}^{\lambda}$ and ${r}_{\alpha \beta}^{\lambda}$, since (see (6.82))
where r _{cαβ} = r _{c2β} – r _{c1α}. Thus we have
From (6.92) we get
(p.651) Similarly $\partial {u}_{\alpha \beta}^{\lambda}/\partial {r}_{\alpha \beta}^{\lambda}$ is given by (6.92). To obtain $\partial {r}_{\alpha \beta}^{\lambda}/\partial \lambda $ we use (6.83),
where (6.82) and (6.100) have been used. Thus we have
where ${\widehat{r}}_{\alpha \beta}={r}_{\alpha \beta}/{r}_{\alpha \beta}$. Substituting these results into (6.103), noting that $\partial {\sigma}_{\alpha \beta}^{\lambda}/\partial \lambda ={\sigma}_{\alpha \beta}$ from (6.100), and substituting the resulting form for $\partial {u}_{\alpha \beta}^{\lambda}/\partial \lambda $ into (6.101) gives
We note that the first term on the right-hand side of this expression can be written in terms of ${g}_{\alpha \beta}^{\lambda}$, because the term $\left(\partial {u}_{\alpha \beta}^{\lambda}/\partial {\sigma}_{\alpha \beta}^{\lambda}\right)\left(\partial {\sigma}_{\alpha \beta}^{\lambda}/\partial \lambda \right)$ in (6.103) is independent of ω _{1}, ω _{2}; the same is not true of the second term on the right in (6.106) since r _{caβ} depends on ω _{1} and ω _{2}. In (6.106) both ${g}_{\alpha \beta}^{\lambda}$ and g ^{λ}(r ω _{1} ω _{2}) refer to the fluid with potential u _{λ}. By using the identity p = ∂A/∂V, together with the virial theorem, it is possible to derive^{28} the equation of state (6.93) from (6.106).
An alternative expression for the free energy can be obtained^{30} for fused hard sphere fluids by using a charging parameter λ′ that scales only the site diameters σ _{αβ}, so that (6.100) is replaced by
and the centre-site distances r _{cα} (and the angles between the r _{cα}) remain unchanged during the charging process and equal to the real values. In this case λ′ = 0 corresponds to an ideal gas composed of fused hard sphere molecules with infinitesimally small spheres. This gas, whose Helmholtz free energy is written A0′ ≡ A(λ′ = 0), differs from that used as the reference state in (6.106) in that the molecules possess rotational kinetic energy. By retracing the steps in deriving (6.106) it is clear that in this case the free energy will be given by
It should be noted that ${g}_{\alpha \beta}^{{\lambda}^{\prime}}\left({\sigma}_{\alpha \beta}^{{\lambda}^{\prime}+}\right)$ in (6.108) is the site-site correlation function for molecules with variable site diameters ${\sigma}_{\alpha \beta}^{{\lambda}^{\prime}}$, but with the fixed real values of the r _{cα}; in (6.106), by contrast, ${g}_{\alpha \beta}^{\lambda}\left({\sigma}_{\alpha \beta}^{\lambda +}\right)$ is the site-site correlation function when both the ${\sigma}_{\alpha \beta}^{\lambda}$ and ${\text{r}}_{c\alpha}^{\lambda}$ are variable. Equations (6.106) and (6.108) must give identical results if the exact correlation functions are used. If an approximate theory is used for the correlation functions they provide alternative routes to the thermodynamic properties which may give different results.
For a fluid of hard homonuclear diatomic AA molecules, (6.108) becomes
6.8 A rigorous inequality for the free energy^{31} ^{–} ^{41}
In this section we discuss an application of the classical Gibbs inequality^{31} for the configurational Helmholtz free energy. The quantum version of the inequality (due to Bogoliubov) can also be derived;^{32} ^{–} ^{35} the result then applies to the entire free energy, rather than just the configurational part.
If we write the total intermolecular potential energy as the sum of a reference and a perturbation part, u _{λ} = u _{0} + λu _{p}, then the free energy difference between the real and reference systems is given by (6.96). Moreover, we have
which is the analogue of (3.249) in the canonical ensemble. The right-hand side of (6.110) must be negative or zero. It follows that 〈u _{p}〉_{λ} is a decreasing function of λ, or a constant. Hence the maximum and minimum values of the integrand in (6.96) are 〈u _{p}〉_{λ = 0} and 〈u _{p}〉_{λ = 1}, respectively. Multiplying these by the integration interval in (6.96) (unity), we arrive at bounds for the integral itself,
where 〈u _{p}〉 = 〈u _{p}〉_{λ = 1}. The lower bound 〈u _{p}〉 in (6.111) is probably of little computational value, since it requires knowledge of the real fluid distribution functions. The upper bound
(p.653) is valid for angle-independent or dependent u _{p}, and is referred to as the Gibbs-Bogoliubov inequality.^{32} ^{,} ^{33} ^{,} ^{36} Atomic fluid applications are discussed by Mωnster^{36} When 〈u _{p}〉_{0} is nonzero it can be used^{39} ^{,} ^{40} as a basis for a variational determination of A – A _{0}.
If the fluid is molecular and the Pople reference system is chosen (§ 4.5), then 〈u _{p}〉_{0} = 0 and
i.e. the anisotropic perturbation rigorously lowers the free energy^{41} The inequality (6.113), which holds to all orders and for arbitrarily large Pople-type u _{p}, is manifestly satisfied for multipolar perturbations in second order (see (4.49)). For this case we also have, from (6.111), that 〈u _{p}〉 ≤ 0; this is expected, since the Boltzmann averaging places more weight on the negative parts of u _{p}.
The quantity 〈u _{p}〉_{0} equals the first-order perturbation term A _{1} (see (4.5)), so that A _{1} gives a rigorous upper bound for the free energy difference A – A _{0}.
It is clear from (6.110) that (for β ≠ 0) the equalities in (6.111) hold if and only if u _{p} is a constant, equal to its average value for all configurations. If 〈u _{p}〉_{0} = 0 the only possible value is u _{p} = 0. For β → 0 (infinite temperature) the situation is more subtle;^{41} we can have A – A _{0} = 0 or A – A _{0} = – ∞, depending on the nature of the potential.
General inequalities for the entropy or internal energy do not exist; it is possible that such inequalities can be derived for special potentials^{41}
6.9 Quantum corrections^{42}
When quantum corrections are small, as is expected for most liquids except He and H_{2}, (see § 1.2.2 and Table 1.1 of Vol. 1), their effect on the thermody-namic properties can be treated by expanding the partition function in powers of ħ^{2}. Details of this expansion are given in Appendix 3D of Vol. 1, and by Powles and Rickayzen^{42} The corresponding expansion for the Helmholtz free energy, A = –kT ln Q, gives the following results for molecules of the symmetries indicated:
Linear (I _{x} = I _{y} ≡ I):
Spherical top (I _{x} = I _{y} = I _{z} ≡ I):
(p.654) Symmetrie top (I _{x} = I _{y} ≡ I⊥, I _{z} ≡ I _{║}):
Asymmetric top (I _{x}, I _{y}, I _{z} not equal):
where A is the quantal and A _{Cℓ} the classical value of A, and 〈…〉 is a classical configurational average. Each of these expressions contains three terms of order ħ ^{2}; the first, involving 〈F ^{2}〉, arises from translational (diffraction) effects, the second involves 〈τ ^{2}〉 from rotational potential energy effects, and a final term (e.g. ħ ^{2} N/6I in (6.114)) arises from rotational kinetic energy effects (see § 1.2.2 for a qualitative discussion). Here 〈F ^{2}〉 and 〈τ ^{2}〉 are the classical mean squared force and mean squared torque (about the centre of mass) on a molecule, respectively, and can be obtained experimentally from the isotope separation factor (see § 1.2.2) or from spectral moments (infrared, Raman, or neutron), as discussed in Chapter 11. These quantities can also be calculated theoretically or by computer simulation (see Chapter 11). The quantities m and I _{α} are the molecular mass and principal αα-component of the moment of inertia, respectively, and in (6.117)
Also, τ _{α}, is the principal α component of τ,$\u3008{\tau}_{\parallel}^{2}\u3009\equiv \u3008{\tau}_{z}^{2}\u3009$, and $\u3008{\tau}_{\perp}^{2}\u3009=\u3008{\tau}_{x}^{2}\u3009+\u3008{\tau}_{y}^{2}\u3009$ is the total mean squared torque perpendicular to the symmetry axis z.
For linear molecules, (6.114) can be rearranged to the dimensionless form
or, alternatively,
where F ^{2} = F ^{2}/(ε/σ)^{2}, τ ^{2} = τ ^{2}/ε ^{2}, and I = I/mσ ^{2} are the reduced quantities, with ε and σ being the usual energy and distance parameters in the isotropie part of the potential, and m the molecular mass; the quantities Λ = h/2πσ(mε)^{½} and θ _{r} = ħ ^{2}/2Ik are the dimensionless de Broglie wavelength and the characteristic rotational temperature, respectively, introduced in § 1.2.2. The magnitude of these corrections for liquids with dipole-dipole and (p.655)
Table 6.2 Quantum corrections to configurational energy and pressure for ortho-baric liquids
Liquid |
T(K) |
$\frac{U-{U}_{c\ell}}{{U}_{c\ell}}$ |
$\frac{p-{p}_{c\ell}}{\text{\rho}kT}$ |
---|---|---|---|
N_{2} |
73 |
−5 |
0.8 |
F_{2} |
59 |
−1.7 |
0.26 |
Cl_{2} |
250 |
−0.7 |
0.07 |
HCl |
188 |
−2 |
0.16 |
${\text{CH}}_{\text{4}}^{\u2020}$ |
207 |
−4.4 |
0.09 |
† This state point is not orthobaric. The density is 24.9 mol ℓ ^{−1}. From ref. 42.
Rough estimates of the quantum corrections to the configurational energy and pressure have been made for several liquids by Powles and Rickayzen,^{42} and are shown in Table 6.2. These calculations are based on computer simulation values (p.656)
For hydrogen the quantum corrections are particularly large, so that the ħ ^{2} term alone in (6.114) is not sufficient at low temperatures, and a full quantal (p.657)
(p.658) where u _{LJ}(r) is the Lennard-Jones model of (2.2), u _{QQ}( 224) is the quadrupole-quadrupole potential given by (2.185), u _{dis} ( 202 + 022 + 224) is the anisotropic dispersion potential as given by the ℓ _{1} ℓ _{2} ℓ = 202, 022, and 224 terms (see (2.223) and also (2.228) and (2.232)), and u _{ov}(202 + 022) is the anisotropic overlap potential as given by the ℓ _{1} ℓ _{2} ℓ = 202 and 022 terms (see (2.247), (2.248)). Calculations are based on the Padé approximant of (4.47), with the anisotropic potential parameters Q = 0.65 × 10^{–26} esu, K = 0.125 (see (D.8)), and δ = 0.1 (values of Q and K are taken from sources listed in Appendix D of Vol. 1). Values of δ, ε, and σ are obtained by matching theory with experimental density and configurational energy data at higher temperatures, where quantum effects are small. It is seen from Figs. 6.6 and 6.7 that the quantum corrections increase at lower temperatures and higher pressures, as expected. Quantum corrections are appreciable at temperatures below 160 K, particularly at the higher pressures.
Quantum corrections to the critical constants have been calculated for LJ fluids^{47} The quantum corrections cause the critical temperature, pressure, and density to decrease.
6.10 Virial coefficients
The second virial coefficient is given by (3.268) and (3.269) as
where u′ = ∂u/∂r. These expressions are based on the assumptions that (a) the molecules are ‘rigid’ (see § 1.2.1), (b) quantum effects can be neglected (§ 1.2.2), and (c) the angle-averaged pair potential, 〈u(r ω _{1} ω _{2})〉_{ω 1 ω 2}, decreases faster than r ^{–3} for large r. Assumption (a) can lead to errors for molecules that have internal rotations or are otherwise flexible, while (b) may not hold for light gases at low temperature. Nevertheless, (a) and (b) lead to negligible errors for most small molecules that do not contain hydrogen. Condition (c) is necessary in order that the integrals appearing in (3.268) and (3.269) be independent of the shape and size of the system, and is required if the thermodynamic properties are to exist. Condition (c) will hold for all cases considered in this book; note that (c) is satisfied for a potential of the form u _{0}(r) + u _{μμ}, where u _{0} is shorter-ranged than r ^{–3} and u _{μμ} is the dipole-dipole potential (which varies as r ^{–3}), since 〈u _{μμ}〉_{ω 1 ω 2} = 0.
Qualitatively B _{2}(T) behaves as shown in Fig. 6.8 as a function of temperature T. At low temperature the attractive forces dominate molecular behaviour. On (p.659)
6.10.1 B2 and the pair potential
For the majority of cases, where the above conditions of rigidity and classical behaviour are fulfilled, (6.121a) shows that measurements of B _{2} provide a valuable source of information about the pair potential. Complications due to the influence of three-body potentials or the need for an approximate theory, always a concern in interpreting liquid state data, are not present here. Information about the pair potential can be extracted from B _{2} data either by inversion of the data, or by comparing calculated B _{2} values with the data for various model potentials. The first procedure is the most desirable, and is possible in principle for atomic gases.
(p.660) Thus, if the molecules are spherical, and the pair potential has a monotonically repulsive portion u _{+}(r _{+}) for r = r _{+} ≤ r _{m}, and a monotonically attractive portion u _{–}(r _{–}) for r ≡ r _{–} 〉 r _{m} (cf. Fig. 2.4), where r_{m} is the separation at the potential minimum, it is possible to rearrange (6.121b) to the form^{50} ^{,} ^{51}
where ϕ ≡ u + ε is the pair potential referred to the minimum of the well as zero, ε is the well depth, ℒ[ ] is the Laplace transform, and:
Thus, if accurate B _{2} data are available over a range of temperatures, it should be possible to invert the Laplace transform in (6.122) to obtain Δ (ϕ), i.e. the potential itself for the purely repulsive region (ϕ 〉 ε), and the well-width function $\left({r}_{+}^{3}+{r}_{-}^{3}\right)$ for ϕ ≤ ε. Such a procedure has been successfully used to obtain the repulsive part of the potential for helium;^{52} for this case data are available over a very wide range of temperatures and the attractive forces are weak. This inversion procedure has proved difficult for other atomic gases, since the inverse transform is highly sensitive to inaccuracies in the data, and because in many cases data are not available over a sufficient temperature range^{53} Less formal methods of inversion based on semiempirical techniques have been devised, and have proved successful for atomic gases^{51} ^{,} ^{53} ^{,} ^{54}
For molecular gases it is not possible to write (6.121a) or (3.272) as a Laplace transform that could be used to obtain information about u(12), so that a formal inversion procedure does not exist in this case^{55} The use of B _{2} data to investigate potentials has therefore been by the second of the methods mentioned earlier, i.e. comparisons of experiment with calculated B _{2} values using various models. Unfortunately, it is found that B 2 is only sensitive to the form of the intermolecular potential energy function when data are available over a wide temperature range. When data are available over only a limited range B _{2} is relatively insensitive to the form of the potential (in particular the anisotropic part), so that a variety of models often give an equally good fit to the data. Thus, for nitrogen and carbon dioxide it is found that the Kihara, LJ atom-atom (two sites), and Gaussian overlap potential models, with or without the addition of a quadrupole-quadrupole term, all give an equally good description of the available second virial coefficient data, although they do not work equally well for other properties (e.g. heat of sublimation, crystal lattice parameter)^{56} This situation is largely due to the insensitivity of the temperature-dependent pseudopotential u _{0}(r), defined by (3.273), to the nature of the anisotropic intermolecular forces; thus both multipolar and nonspherical shape forces tend to have a similar effect on u _{0}(r).^{57} Indeed, this insensitivity is (p.661)
Second virial coefficient data have often been used to estimate values for anisotropic potential parameters, by comparison of the data with calculations for a particular model; multipole moments and anisotropic overlap parameters have often been obtained in this way, by using a generalized Stockmayer potential model^{59} Multipole moments so obtained are often unreliable because of uncertainties in the potential model used (see Appendix D of Volume 1). A more stringent test of the potential model is to compare theory and experiment for both the pressure and the dielectric second virial coefficients^{60} (see § 10.6, and also the collision-induced absorption virial coefficients, Chapter 11), and where possible the mean squared torque for the gas^{61} (see Chapter 11).
Although second virial data alone are of limited value in testing potential models, a model that fits other data should, of course, also fit the virial data if it is the correct pair potential. A test of this sort on two potential functions whose parameters were obtained by comparing computer simulation results with experimental thermodynamic data for liquid hydrogen chloride is shown in Fig. 6.9. Either of these potential functions can be made to fit the B _{2} data if the site-site parameters ε _{αβ} and σ _{αβ} are suitably adjusted (here α, β are H or Cl); the solid line shows the result for the second potential, which includes multipole terms. When the parameters are adjusted to fit liquid data, however, the potentials no longer fit the virial data. The potential including multipole terms appears to be the better of the two. The failure of the liquid potentials to fit the gas data could be due to
(p.662) errors in the pair potential, the neglect of multibody potential terms in treating the liquid, or a combination of these two effects.
6.10.2 Calculations for simple model potentials
There have been many calculations of B _{2}, and in some cases higher virial coefficients, for a wide variety of simple model potentials of the types discussed in § 2.1; the earlier work has been reviewed elsewhere^{51} ^{,} ^{64} ^{,} ^{65} These have included calculations for the generalized Stockmayer model, site-site LJ models with and without point charges, the Kihara model, Gaussian Overlap model, etc. (These models are discussed in §2.1 of Vol. 1.) The addition of electrostatic forces causes B _{2} to decrease at any given temperature. This is because such forces are, on the average, attractive, and cause a decrease in the frequency of molecules hitting the wall, and hence a decrease in the pressure.
Fluids of hard nonspherical molecules are of particular interest since they are frequently used as reference fluids in perturbation theories for both isotropic and anisotropic (e.g. liquid crystal) fluids. For a fluid of hard spheres of diameter a it is easy to show from (6.121a) that
where v _{m} = πσ ^{3}/6 is the volume of a sphere. For a fluid of hard nonspherical molecules, each of volume v _{m}, it is convenient to write
where f _{B} is a factor whose difference from unity indicates the degree of departure from hard sphere behaviour.
The simplest molecules to treat are convex ones, i.e. ones in which any line segment whose endpoints are inside the body lies wholly within the body (see Appendix 6A); examples of such bodies are spherocylinders and ellipsoids (a nonconvex example is two fused hard spheres). For such molecules f _{B} is always greater than unity, so that convex nonspherical molecules always have second virial coefficients larger than those for spherical molecules of the same volume. The physical reason for f _{B} 〉 1 is that, as the molecule rotates, the effective volume swept out is greater than that of the molecule, thus tending to increase B _{2}. Isihara^{66} has derived the following general expression for f _{B} for the convex molecule case (see Appendix 6A.5 for derivation):
where γ = r _{m} s _{m}/v _{m} is the shape factor, and s _{m} and s _{m} are the mean radius of curvature and the surface area of the molecule, respectively (see Appendix 6A for an account of the geometry of hard convex bodies). From (6.125) and (6.126) we have, for hard convex molecules
(p.663) We note that γ, and hence B _{2}, is independent of temperature as in the case of hard spheres. For spheres of radius a we have r _{m} = a, s _{m} = 4πa ^{2}, v _{m} = 4πa ^{3}/3, γ = 3, and f _{B} = 1. Formulae for r _{m}, s _{m}, v _{m}, and f _{B} for other shapes have been worked out;^{65} ^{,} ^{66} some of these are given in Table 6A.1.
For general nonconvex molecules there is no simple expression analogous to (6.127). For homonuclear (AA) hard sphere diatomic molecules (dumbbells), Isihara has derived an exact analytic expression for B _{2}.^{67} This result has been generalized to heteronuclear hard diatomics (AB), unlike pair diatomics (AB/CD), and symmetrical linear triatomics (ACA) by Wertheim.^{68} Numerical calculations of B _{2} have been made for other fused hard sphere molecules, and som