proving that the free energy is extensive

Question

If I have two system of an Ideal gas A and B Each of these system has a partition function:

$Z_{A,B} = \left ( \frac{V_{A,B}}{\lambda_T} \right )^{N_{A,B}}$

Where:

$\lambda_T = \left ( \frac{m}{2\pi\beta \hbar } \right )^{\frac{1}{2}} $

The free energy is:

$F_{A,B} = -kT \ln \left ( Z_{A,B} \right ) = -kT N_{A,B}\ln \left (\frac{V_{A,B}}{\lambda_T}\right)$

For the free energy to be extensive the following must be true:

$F_{A} + F_B = F_{A+B} \Rightarrow Z_A \cdot Z_B = Z_{A+B}$

However:

$Z_A \cdot Z_B = \left (\frac{V_A}{\lambda_T} \right )^{N_A} \left ( \frac{V_B}{\lambda_T} \right)^{N_B}$

and:

$Z_{A+B} = \left ( \frac{V_{A+B}}{\lambda_T} \right )^{N_A + N_B}$

So, for $Z_A + Z_B = Z_{A+B}$ to be true $V_A^{N_A} V_{B}^{N_B} = \left(V_A + V_B \right)^{N_A + N_B}$ must be true as well but this isn't true for any system.

Since we cannot create energy by mixing two containers of Argon in the same pressure and temperature, something in my understanding is wrong. Where is my fault?

Answer 1

What you observe is Gibbs' paradox. The resolution comes about by postulating that the particles are indistinguishable (and thereby introducing a factor $1/N!$). Then the entropy and the free energy becomes extensive (in the thermodynamic limit $N,V\to\infty$, $N/V=\text{const}$).