Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)
6. Describing Liquids: Structure and Energetics
If we assume that the correlation function g(r) is known and the pairwise additivity (of spherical potentials) holds for the potential energy, that is, Φ(r) = ½Σijϕij(r), we can write down for the internal energy almost by inspection the energy equation
where the first term is the kinetic contribution. More formally, the canonical average, as given by Eq. (6.9), results in
and using Φ = ½Σijϕij leads to N(N − 1)/2 identical terms so that
to which the kinetic term has to be added. In the thermodynamic limit (N → ∞, V → ∞ but N/V → ρ), the factor (N − 1)/V becomes ρ, so that the result is Eq. (6.32).
Similarly, though in a somewhat more complex fashion, an expression for the pressure P can be obtained. We start with the previously derived expression (see Chapter 5)
with the thermal wavelength Λ ≡ (h2/2πmkT)1/2. Before differentiating, we consider that the pressure is independent of the shape of the container, so that we can take a cube and employ the coordinate transformation ri = V1/3ri′. The expression for QN then becomes
while for the intermolecular distance rij = [(xi − xj)2 + (yi − yj)2 + (zi − zj)2]1/2 we obtain the expression . Therefore ∂QN/∂V becomes
Transforming back, noting that we have N(N − 1)/2 identical terms and taking the thermodynamic limit, one obtains the pressure (or virial) equation (see Section 3.8)
In Eqs (6.32) and (6.39), the first terms are due to the momenta (i.e., the kinetics), while the second term is due to the coordinates (i.e., the configuration). Thus, both equations in principle relate a thermodynamic quantity to the structure, as represented by g(r).
For future reference we note that the compressibility κT can be calculated using the grand canonical ensemble (see Justification 6.1). The final result
is known as the compressibility equation, and this expression is derived without assuming any pairwise additivity for the potential energy of the system, and is thus generally valid.
Justification 6.1: The compressibility equation*
The derivation of the compressibility equation requires the grand canonical ensemble. We recall first the definition of the isothermal compressibility
From PV = kT lnΞ we have, using ,
We further need the absolute activity given by λ = exp(βμ), and note that ∂lnΞ/∂ρ = (∂lnΞ/∂λ)/(∂ρ/∂λ). For the numerator ∂lnΞ/∂λ we obtain, defining IN ≡ ∫ … ∫exp(−βΦ)dr1 … drN,
where the distribution as defined for the grand canonical ensemble
is used. The denominator ∂ρ/∂λ is given by
where for the last step the definition g(r1,r2) = ρ(2)(r1,r2)/ρ(1)(r1)ρ(2)(r2) = ρ(2)(r1,r2)/ρ2 is used. Combining the expressions for ∂lnΞ/∂λ and ∂ρ/∂λ yields
The isothermal compressibility is then given by
Note that the structure factor S(s) for s = 0 reduces to
and therefore S(0) = ρkTκT, a fact that can be used for reconstructing g(r).
One might wonder whether the compressibility equation and the pressure equation yield the same result for the pressure. On integrating Eq. (6.40), one obtains
to be compared with PP as obtained from Eq. (6.39). For the ideal gas g(r) = 1, leading for both expressions to the ideal gas EoS P = ρkT. This is no longer true for a nonideal fluid for which ϕ(r) ≠ 0, and thus g(r) is unknown. Using approximations for g(r) usually leads to different values for PP and Pκ, indicating a loss of thermodynamic consistency (see Section 6.3).
In order to obtain a complete description of the thermodynamic state, some extra information is required. This is probably most clear from the Gibbs equation
which shows that information is required about the energy U, the pressure P, and the chemical potential μ to obtain a complete description of the thermodynamic state of the system.
If the correlation function g(r) were to be known over a wide temperature range, we could use the energy Eq. (6.32) and integrate the Gibbs–Helmholtz equation and so obtain the required information, since the chemical potential μ can be calculated from the Helmholtz energy F. Similarly, if g(r) were to be known over a wide density range, we could use the pressure Eq. (6.39) and integrate ∂(F/N)/∂(1/ρ) = ρ2P (as obtained from dF = −PdV − SdT + μdN). Alas, neither the temperature nor density dependence of g(r) is usually available.
As an alternative we introduce the coupling parameter approach, also known as thermodynamic integration. A coupling parameter ξ is defined, varying from 0 to 1, which controls the interaction between the reference molecule 1 and another, say j, by replacing ϕ1j by ξϕ1j. The potential energy then becomes
and a molecule is added to the system when ξ varies from 0 to 1. The partition function ZN(ρ,T) reads now ZN(ρ,T,ξ) while the correlation function g(r;ρ,T) becomes g(r;ρ,T,ξ). Because the number of particles N is very large, we can write
From F = −kTlnZ, we obtain −βF = ln QN − ln N! − 3N ln Λ, and hence we get
Further, we have QN(ξ = 1) = QN and QN(ξ = 0) = VQN−1. Therefore,
Calculating ∂QN(ξ)/∂ξ using Eq. (6.43) results in
Further, employing ρ(N)(ξ) = N!n(N)(ξ) with n(N)(ξ) = QN(ξ)−1exp[−βΦ(ξ)], and recognizing that we have N − 1 identical integrals, we obtain
where in the last step ρ(2)(ξ) = ρ2g(ξ) is employed. Substitution in Eq. (6.45) yields the required result
From , P and μ, we can calculate all of the thermodynamic properties if g(r;ξ) is known. The theoretical calculation of g(r;ξ) is the topic of Chapter 7.
Check the steps necessary to obtain Eqs (6.39), (6.40), and (6.50).