Helmholtz entropy of ideal gas mixture is additive?

[EE18]

In his classic textbook, Callen remarks that

(1) The Helmholtz potential of a mixture of simple ideal gases is the sum of the Helmholtz potentials of each individual gas:$$ F(T,V,N_1, ... ,N_m)=F(T,V,N_1)+ ··· +F(T,V,N_m). $$
(2) An analogous additivity does not hold for any other potential expressed in terms of its natural variables.

I have labelled the claims (1) and (2). I am not sure about either. For the first, I have tried to proceed as follows (all equations are from Callen's second edition and all 0 subscripts are with respect to some reference state of an ideal gas):

We begin by noting that for a single component of the mixture (in the volume ##V## of the overall gas per the given formula in this problem) we have from (3.34) that
$$F_i(T,V,N_i) = U(T,V,N_i) - TS(T,V,N_i) $$
$$= c_iN_iRT -T\left(N_is_{i0} + c_iN_iR \ln (T/T_0) + N_iR \ln(V/V_0) - N_iR(c_i+1) \ln (N_i/N_0) \right)$$
Now for the mixture we have
$$F = U - TS $$
$$\stackrel{(1)}{\equiv} (\sum N_j c_j)RT - T\left( \sum N_js_{j0} + (\sum N_j c_j)R \ln (T/T_0) + (\sum N_j) R \ln(V/V_0) -R \sum N_j \ln(N_i/N)\right).$$
where (1) is from (3.39) and (3.40) (i.e. from the definition of ##F## that for a mixture of ideal gases (i.e. in the same vessel), we form ##U-TS## for ##U## and ##S## of the entire system).

I can't see how to go further in terms of identifying one with the other

But even supposing I can show that, what does claim (2) mean? Is Callen saying that there is no other thermodynamic potential (partial Legendre transform of the energy ##U##) which is such that this additivity holds in terms of natural variables? Obviously ##U = \sum N_jc_j RT## which is additive, but I guess this isn't a counter example since ##T## isn't a natural variable of ##U##?

 

[Chestermiller]

I think you mean Helmholtz free energy, not Helmholtz entropy.

I think it might help to consider the following questions:

What is Gibbs theorem for the partial molar properties of the components in an ideal gas mixture?

What is the equation for the partial pressure of a gas component in an ideal gas mixture in terms of Nj, R, T, and V.

What is the equation for the partial molar entropy of a gas component in an ideal gas mixture as a function of temperature and partial pressure?

What is the equation for the partial molar Helmholtz free energy of a gas component in an ideal gas mixture in terms of temperature and partial pressure?

 

[EE18]

I think you mean Helmholtz free energy, not Helmholtz entropy.

Oy, yes I did mean Helmholtz (free) energy -- I've been thinking too much about the entropy because of another question I had clearly! If you do get the chance, that question is here. I've been very grateful for your insights on thermodynamics as I work through Callen, both now and in the past!

For these:

What is Gibbs theorem for the partial molar properties of the components in an ideal gas mixture?

What is the equation for the partial pressure of a gas component in an ideal gas mixture in terms of Nj, R, T, and V.

"The contribution to a property of a mixture of ideal gases is the sum of the properties that each component gas would have if it alone were to occupy the volume V at temperature T"
and
##P_i = N_iRT/V.##

I don't think either of these is relevant for this particular question though?

For these:

What is the equation for the partial molar entropy of a gas component in an ideal gas mixture as a function of temperature and partial pressure?

What is the equation for the partial molar Helmholtz free energy of a gas component in an ideal gas mixture in terms of temperature and partial pressure?

I believe I included the answer to both in my OP. Indeed, it seems like the problem I'm having is massaging a sum (on ##j##) over the first equation in my OP to be of the second equation's form, hence why I'm not sure if I'm making a mistake at the outset.

 

[Chestermiller]

Maybe something like this: $$F_j((T,V,N_j)=N_ju_j(T_0,v_0)+N_jC_{v,j}(T-T_0)-$$$$T\left[N_js_j(T_0,v_0)+N_jC_{v,j}\ln{(T/T_0)}+N_jR\ln{\left(\frac{V}{N_jv_0}\right)}\right]$$where ##T_0## and ##v_0## are the temperature and molar volume in a reference state, and ##s_j## and ##u_j## are the molar entropy and molar internal energy of species j in the reference state. Then substitute $$\ln\left(\frac{V}{N_jv_0}\right)=\ln\left(\frac{V}{x_jNv_0}\right)=\ln\left(\frac{V}{Nv_0}\right)-\ln{x_j}$$