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 $\frac{5}{2}NK_B$?



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