- #1

EE18

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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):(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 can't see how to go further in terms of identifying one with the otherWe 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).

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##?