Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

11.4. Ideal Behavior in Statistical Terms

Having reviewed some basics of ideal solutions in the previous sections, in the present section we reiterate some results for ideal behavior in statistical terms. The starting point is to use a crystal lattice, with coordination number z, on which at each lattice cell a molecule is located. The size of the solvent and solute molecules is taken to be approximately the same, so that each cell is occupied by either a solvent or solute molecule. We also take only nearest-neighbor interactions into account. In this model, the internal energy U for a lattice of N similar molecules can be written as U = ½zNw₁₁, where w₁₁ is the “bond energy” between a pair of molecules of type 1. For a binary system of components 1 and 2, we then have to consider the 1–1, 2–2 and 1–2 pair interactions with an interaction³⁾ w₁₁, w₂₂, and w₁₂, respectively. If we then create two 1–2 bonds from a 1–1 and a 2–2 bond, the interaction change per bond is given by

(11.44)

The parameter w so defined is often referred to as the interchange energy. Now, for such a mixed crystal to be ideal we assert that w should be zero. In this case any interchange of molecule 1 and 2 leaves the internal energy unchanged. Moreover, the distribution of molecules 1 and 2 will be random over the lattice cells. If we have a mixed crystal with N = N₁ + N₂ with N₁ ≡ Nx₁ molecules of type 1 (matrix) and N₂ ≡ Nx₂ molecules of type 2 (solute), the energy U = ½z(N₁w₁₁ + N₂w₂₂). Furthermore, the number of distinguishable ways W of arranging these molecules on the lattice is W = N!/N₁!N₂! = N!/(Nx₁)!(Nx₂)!, and therefore the entropy S = k lnW = −Nk (x₁lnx₁ + x₂lnx₂). The configurational Helmholtz energy F = U − TS and hence the configurational partition function Q = exp(−βF) with, as before β = 1/kT, reads

(11.45)

The question is now: can we use this lattice model for liquid solutions? Obviously we can, but what are the limitations? The structure of liquids as described by the pair-correlation function g(r) is far from static and continuously changing, although the average structure is constant and mainly determined by T and P. The average structure can be considered to be closer to an almost random packing of molecules with a certain coordination for the first and possibly second coordination shell than to lattice-like structure. A lattice-like structure is rare, although in some liquids much more order is present than in others, water (H₂O) being a good example. A lattice model is thus not a good reference structure for liquids. Nevertheless, for solutions lattice models were and still are frequently used, and we do so here. If we regard the liquid as quasi-crystalline (an image used previously in Chapter 8), the size of the molecules should be not too different⁴⁾. Moreover, the internal degrees of freedom in the solution should be separable from the translational degrees of freedom and behave in the same way as in the pure liquid. Overall, the concept of a lattice should not be taken too seriously: it is used to represent the average coordination in the first coordination shell, and we are dealing with nearest-neighbor interactions mainly.

If we accept this image, bearing in mind that so far we have w = 0, the configurational Helmholtz energy F, using Stirling's theorem, reads

(11.46)

Since in this model the Helmholtz energy of a pure liquid, labeled α, is⁵⁾

(11.47)

the Helmholtz energy of mixing Δ_mixF = F − x₁F₁ − x₂F₂ is given by

(11.48)

For the entropy of mixing Δ_mixS we thus regain, as expected,

(11.49)

The chemical potential μ_α = ∂F/∂N_α becomes

(11.50)

and since for a pure component we have , we find

(11.51)

Finally, we recall that the (absolute) activity λ is given by μ = kT lnλ, and obtain

(11.52)

so that, if we may regard the vapor phase as a perfect gas and use the partial pressure P_α = x_αP for the activity (see Chapter 2), the final result becomes

(11.53)

which we recognize as Raoult's law.

Problem 11.8

Using the lattice model with N = N₁ + N₂ sites, show that the entropy of mixing can be estimated, using the Boltzmann expression, as

with W the number of possible configurations and that this estimate leads to