The argument for partial molar data for force field parameterization of mixtures

The Gibbs free energy of a mixture is \sum N_i \mu_i(x,T,P), where x is the vector of composition.

Therefore, if the \mu_i(x,T,P) are correct, then the free energy of the mixture will be correct as a function of composition. S and V are derivatives of the free energy (and S_i and V_i derivatives of the chemical potential, and the molar volume is of course equivalent to the 1/density), and other properties are 2nd derivatives (with some tweaks and transformations) of the free energy -- specifically, heat capacity is related to the d^2/dT^2, isothermal compressibility related to d^2/dP^2, and thermal expansion coefficient related to d^2/dTdP. Getting getting the x,T,P dependent derivatives sufficiently correct will get these properties correct as well. Note that numerically, however, one might want to improve the fit by directly fitting the 2nd derivatives experimental properties, though first things first!

Define a reduced \mu_i = rmu_i, and reduced enthalpy \beta H = h, with reduced partial molar enthalpy h_i. Let v_i be the partial molar volume. Then:

d(rmu_i)/d(\beta P) = -v_i (from dG/dP = -V, then take derivatives wrt N_i)

Then \beta (d(rmu_i)/d(\beta)) = h_i (from d(G/T)/d(1/T) = H -> d(\beta G)/d(\beta) = H, then take derivatives wrt N_i).

Knowing the partial molar enthalpy and partial molar volume (density) at all compositions is equivalent to knowing the derivatives of rmui everywhere, which is itself the derivative wrt composition. If one knows the derivative everywhere, then up to a constant, one knows the function everywhere, since we can integrate:

f(x,y) - f(x_0,y_0) = int_(x',y'=x_0,y_0)^(x,y_0) df/dx dx + int_(x',y'=x,y_0)^(x,y) df/dy dy

Clearly, if one knows rmu_i at all T, then one knows mu_i at all T as well, up to a reference point.

We still need to be more specific about the reference. One possibility is to use the pure substances as a reference. Getting the pure property references right means getting mu_pure = G/N = (H-TS)/N. However, since we are dealing with classical models, we can't find an absolute free energy - we would have to find a free energy difference. This could be G_Vap = H_vap - TS_vap, but that would require polarization corrections, etc. That would be less elegant; we would prefer to stick to liquid phase properties.

However, one could just to get the excess partial molar enthaply and excess partial molar volume right, i.e. h_i(x,T,P) - h_i(T,P) and v_i(x,T,P) - v_i(T,P), but that may leave some degeneracy in the parameters. The pure substance values the don't enter directly into the excess properties.