Entropy function additivity

Question

I am reading Herbert Callen's book Thermodynamics, which proposes a postulatory treatment to the subject.

The postulate number 3 states the properties of the "entropy" function, one of which is the additive property: the entropy of a system composed of multiple subsystems separated via internal constraints is sum of the entropy of the individual subsystems.

Consider two systems with internal energies $U_1, U_2$, volumes $V_1$, $V_2$ and number of molecules $N_1$, $N_2$. Doesn't the above statement mean

$S(U_1+U_2,V_1+V_2,N_1+N_2) = S(U_1,V_1,N_1)+S(U_2,V_2,N_2)$

? In which case the entropy function is linear. But the entropy function for an ideal gas is nonlinear in the preceding sense.

For two identical systems, it would mean the entropy function must be a homogenous function of order one, which makes sense. But this doesn't imply linearity.

So, I would like to know the lapse in my understanding. Thanks!

Edit: The above equation doesn't qualify for the entropy function to be linear. I think the additivity of entropy can mathematically be represented by this equation.

Answer 1

I believe it rather means: $$ S_{tot}(U_1,U_2,V_1,V_2,N_1,N_2) = S_1(U_1,V_1,N_1) + S_2(U_2,V_2,N_2) $$ Think, for instance, of two isolated containers. If the statement, as you wrote it down, would be true, the mixing of two ideal gases would be reversible.

Answer 2

A late answer, extending previous answer by denklo.

If the two subsystems are made by the same material, the additivity of the volumes, energies and number of particles allows to write $$ S_{tot}(U_1+U_2,V_1+V_2,N_1+N_2;U_1,V_1,N_1) = S(U_1,V_1,N_1)+S(U_2,V_2,N_2), $$ where the dependence on $6$ arguments of $S_{tot}$ makes clear that i) the entropy of the compound system is not the same function, in general, as the entropy of each subsystem; ii) that there is a dependence of the total entropy on the constraints of fixed energy, volume and number of particles in each subsystem, which can be encoded in the value of those quantities in only one of the two subsystems, since their sum is fixed.

Written in this form, the effect of removing all the constraints, implies to maximize $S_{tot}$ with respect to the constraint variables ($U_1,V_1,N_1$). The resulting extremum conditions express the condition of equal temperature, pressure and chemical potentials in the two subsystems.

Therefore, it is trivial to see that only in the case of thermodynamic equilibrium $$ S_{tot}(U_1+U_2,V_1+V_2,N_1+N_2;U^*_1,V^*_1,N^*_1) =S(U^*_1,V^*_1,N^*_1)+S(U^*_2,V^*_2,N^*_2)=S(U^*_1+U^*_2,V^*_1+V^*_2,N^*_1+N^*_2). $$ Indeed, it is enough to recall that homogeneity of degree one of the entropy of the two subsystems at the equilibrium implies $$ \begin{eqnarray} S(U^*_1,V^*_1,N^*_1)&=&\frac{U^*_1}{T}+\frac{V^*_1P}{T}-\frac{N^*_1\mu}{T}\\ S(U^*_2,V^*_2,N^*_2)&=&\frac{U^*_2}{T}+\frac{V^*_2P}{T}-\frac{N^*_2\mu}{T} \end{eqnarray} $$ from which the conclusion follows.