Gibbs Paradox: Exploring the Logarithm Correction

[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

 

[GSXR750]

Thank you for bringing up the Gibbs Paradox and the logarithm correction. It is an interesting topic that is often overlooked in books. Your suggestion to make the Gibbs correction at the logarithm of the Z function instead of the Z function itself is a valid approach and can indeed generalize the three cases of gas mixtures into one expression. This approach also highlights the importance of considering the number of identical particles in each class, rather than just the total number of particles in the system.

I agree that this is not the same as the traditional approach and it could potentially avoid the paradox altogether. However, it is important to note that this approach may not be applicable in all cases and may lead to different results. It would be interesting to see further studies and discussions on the implications of this correction method.

Overall, I appreciate your contribution to this topic and your willingness to share your thoughts and ideas. It is through open discussions and different perspectives that we can continue to deepen our understanding of complex concepts like the Gibbs Paradox. Thank you again for your insight!