Does the Logarithm Correction Resolve the Gibbs Paradox?

[nucleartear]

What I have not seen in books about the Gibbs paradox is that it doesn't exist if we make the Gibbs correction at the logarithm of the Z function, not at the Z function itself, in that way:

$$\ln Z_{i} - \ln N_{i} !$$

where $N_{i}$ is the number of identical particles of class $i$, where there are a total of $p$ classes.

With this definition of the correction, we can generalize the three cases of gas mixture (diferent gases, identical gases with different density and identical gases with identical density) in one expresion!:

$$F = - k_{B} T \sum_{i=1}^{p} \Left( \ln Z_{i} - \ln N_{i} ! \Right)$$

You could say that it's the same but... I think it's not the same!

If you find some error, please, let me know!

Thanx!

 

[baffledMatt]

$$F = - k_{B} T \sum_{i=1}^{p} \Left( \ln Z_{i} - \ln N_{i} ! \Right)$$

That's fine except that generally you can't simply sum over the logarithm of each term of the partition function.

Matt