1
$\begingroup$

I want to obtain the Sackur-Tetrode expression for the entropy of an ideal gas based on:

$$S = K_B ln \Omega$$

We have an isolated system with a fixed volume V and fixed number of particles N and there is no energy exchange with the external world (what statistical mechanics calls micro-canonical ensemble).

We know that $\Omega$ is:

$$\Omega(N,V,E)=\frac 1 {N! h^{3N}} \int_{\mathcal H<E} d^N\mathbf p \ d^N \mathbf q \tag{1}\label{1} = \frac{V^N}{N!h^{3N}} \frac{(2mE \pi)^{3N/2}}{(3N/2)!}$$

When I plugged equation 1 into $S = K_B ln \Omega$ I got:

$$S = K_B ln \Omega = Nk_B ln [\frac{V}{N}(\frac{4mE \pi}{3 h^2 N})^{3/2}]$$

Which is incorrect as I should have gotten:

$$S = K_B ln \Omega = Nk_B ln [\frac{V}{N}(\frac{4mE \pi}{3 h^2 N})^{3/2}] + \frac{5}{2}NK_B$$

Why I am not getting the term $\frac{5}{2}NK_B$?

$\endgroup$
2

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.