Calculating free energy from partition function

So I have $N$ particles and I've determined the partition function for one of them to be $Z_1=\lambda e^{\beta\varepsilon_1} + 2\lambda e^{\beta\varepsilon_2}$. I know the free energy is $A=-kT\log(Z)$, and I want to find the chemical potential from this. I know the chemical potential is the derivative of the free energy with respect to $N$ evaluated at constant temperature and volume. What confuses me is, if I input this $Z_1$, then my free energy does not vary with $N$, which seems wrong. It could be I have to use $Z=Z_1^N$, but the textbook seems to suggest otherwise. Is there something I'm missing?

• Looks like the free energy you wrote down is really the free energy per particle. – Samuel Weir May 15 '18 at 19:42
• Ah, I was just trying to get the wrong free energy. I see. – Radagast May 15 '18 at 19:50
• The expression for chemical potential I then calculate contains the chemical potential. Do I rearrange algebraically? Seems odd. – Radagast May 15 '18 at 19:55