- #1

EE18

- 112

- 13

(1) The mechanical equation of state is of the form ##PV = NRT##.

(2)For a single-component ideal gas the temperature is a function only

of the molar energy (and inversely). ##u = u(T,v) = u(T)## in particular.

(3) The Helmholtz potential ##F(T, V,N_1, N_2, \dots , N_r)## of a multicomponent ideal gas is additive over the components ("Gibbs's Theorem"):

$$F(T, V, N_1, ... , N_r)= F_1(T, V, N_1)+ F_2 (T, V, N_2)

+ \dots +F_r(T,V,N_r).$$

Of course, all of these can be proved as a theorem of statistical mechanics given a no-interaction assumption.

At any rate, my claim is about Callen's claim that a single component ##j## of general ideal gas has the following expression for its entropy:

$$S_j = N_js_{j0} + N_j\int_{T_0}^T \frac{c_{vj}}{T'} \, dT' + N_j R \ln \left(\frac{v}{v_0}\frac{N_0}{N_j}\right).$$

I supply a derivation below but want to confirm:

Let ##S_{j0} = N_js_{j0}## be the entropy in some reference state, and consider taking this substance to some other ##T,V## (at fixed ##N_j##). Then we have

$$dS_j = N_j\frac{c_{vj}}{T}dT + \left(\frac{\partial S}{\partial V}\right)_{T,N}dV = N_j\frac{c_{vj}}{T}dT + \left(\frac{\partial P}{\partial T}\right)_{V,N}dV $$

$$ = N_j\frac{c_{vj}}{T}dT + \left(\frac{\partial (N_jRT/V)}{\partial T}\right)_{V,N}dV = N_j\frac{c_{vj}}{T}dT + (N_jR/V)dV$$

so that

$$\int dS_j = S_j - S_{j0} = S_J - N_js_{j0} = \int_{T_0}^T N_j\frac{c_{vj}}{T'}dT' + \int_{V_0}^V(N_jR/V')dV' \implies S_j = N_js_{j0} + N_j\int_{T_0}^T \frac{c_{vj}}{T'} \, dT' + N_j R \ln \left(\frac{V}{V_0}\right).$$

But this is surely different than what Callen gets. Does Callen perhaps write ##v \equiv V/N_j## and ##v_0 \equiv V_0/N_0## (where ##N_0## is some randomly chosen number of moles)? Then his result is just

$$S_j = N_js_{j0} + N_j\int_{T_0}^T \frac{c_{vj}}{T'} \, dT' + N_j R \ln \left(\frac{v}{v_0}\frac{N_j}{N_0}\right).$$

But things are similar enough that I wonder if I've made an error?