Derivation of ln Z for Bose-Einstein case

[VnHorn]

1. Problem From Fitzpatrick we need to derive[itex]ln(Z)=αN-\sum ln(1-e^{-\alpha-\betaε_{r}})[/itex] (Equation 8.45)

Homework Equations

This is claimed to be derived from Equations 8.20, 8.30, and 8.43

Eq 8.20
[itex]\overline{n}_{s}=-\frac{1}{\beta}\frac{\partial ln(Z)}{\partial\epsilon_{s}}[/itex]

Eq 8.30
[itex]\alpha\cong\frac{\partial ln(Z)}{\partial N}[/itex]

Eq 8.43
[itex]\overline{n}_{s}=\frac{1}{e^{\alpha+\betaε_{r}}-1}[/itex]

The Attempt at a Solution

My initial attempt involved setting the RHS of 8.20 to the RHS of 8.43 and then integrating to solve for ln(Z)
but this ultimately gave me

[itex]ln(Z)=-\alpha-ln(1-e^{-\alpha-\betaε_{r}})[/itex]

I'm not so concerned with the lack of N and the missing summation since that should come when I apply it to more particles, but as hard as I try everything I do ends up with that -[itex]\alpha[/itex]

Nevermind, solved it. Needed to consider the contribution from all the different portions of the partial and make sure it met all requirements. Thanks

 

[BigJDubs]

for the help. Here's what I ended up withln(Z)=αN-\sum \left(\alpha+ln(1-e^{-\alpha-\betaε_{r}})\right)