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## Main Question or Discussion Point

Hi all,

I was brushing up on statistical ensembles, and I found something apparently weird in

microcanonical treatment of the ideal "classical" gas. I'm mainly following K. Huang's

So there's a first approach to the problem in which the MC entropy is evaluated

via an integral in the phase space (chapters 6.5, 6.6).

In particular, in chapter 6.6 the Gibbs paradox and the correct Boltzmann

counting are discussed, and the Sackur-Tetrode equation (6.62) is derived

[see eq (C) below].

A different approach is illustrated in chapter 8.5. The system is discretized and

the number of states corresponding to a set of occupation numbers is

evaluated as

[tex]W\{n_j\} = \prod \frac{g_j^{n_j}}{n_j!}[/tex]

where [tex]g_j[/tex] is the degeneracy of the jth level. The entropy is then

calculated by recalling that it should coincide with [tex]\log W\{\bar n_j\}[/tex],

where

[tex]\bar n_j = z g_j e^{-\beta \epsilon_j}[/tex]

is the set of occupation numbers maximizing W.

After some straightforward manipulations one obtains

[tex]S = \frac{1}{T} (E-\mu N) = \frac{3}{2} N k - N k \log\left[\frac{N}{V} \lambda^{-3}\right] \qquad (A)[/tex]

This is basically eq (8.54), and here's the point that looks weird to me.

From the one hand, this equation seems to provide the correct equation of state

[tex]PV = NkT[/tex]. However, there's a thermodynamic equation stating

[tex]E = TS - PV + \mu N \qquad (B)[/tex]

(Gibbs-Duhem equation?). Now if I plug (B) into the first equality in (A) I obtain

[tex]TS = TS - PV[/tex] i.e [tex]PV = 0[/tex], which disagrees with the equation

of state. In other words, it should be

[tex]S = \frac{1}{T} (E +PV -\mu N) [/tex]

Furthermore eq. (A), is also referred to as Sackur-Tetrode equation, but differs from

eq. (6.63) which, after some algebra, reads

[tex]S = \frac{5}{2} N k - N k \log\left[\frac{N}{V} \lambda^{-3}\right] \qquad (C)[/tex]

Now both (A) and (C) are free from the Gibbs paradox, and hence I understand that they

are both Sackur-Tetrode equations.

However (B) has an extra [tex]NkT=PV[/tex] term which seems to exclude a clash with the

thermodynamic equation (B).

What does this mean? Should we conclude that the "maximization approach" in section 8.5

does not really work for the classical gas (which, anyway, does not even exist...).

Or perhaps eq. (B) has a more limited scope, and cannot be used in this approach?

For some reason both of these answer sound too dismissive to me...

Also, should eq. (B) be unavailable, I'd have some trouble in solving problem 8.2 (see also https://www.physicsforums.com/showthread.php?t=272585" )

Thanks a lot for any insight and comment.

Franz

PS I was not sure where to post this. There's no statistical physics forum, and after all this is the

"classical gas", so it could fit in classical physics. Apologies in case another forum was more appropriate.

I was brushing up on statistical ensembles, and I found something apparently weird in

microcanonical treatment of the ideal "classical" gas. I'm mainly following K. Huang's

*Statistical Mechanics*.So there's a first approach to the problem in which the MC entropy is evaluated

via an integral in the phase space (chapters 6.5, 6.6).

In particular, in chapter 6.6 the Gibbs paradox and the correct Boltzmann

counting are discussed, and the Sackur-Tetrode equation (6.62) is derived

[see eq (C) below].

A different approach is illustrated in chapter 8.5. The system is discretized and

the number of states corresponding to a set of occupation numbers is

evaluated as

[tex]W\{n_j\} = \prod \frac{g_j^{n_j}}{n_j!}[/tex]

where [tex]g_j[/tex] is the degeneracy of the jth level. The entropy is then

calculated by recalling that it should coincide with [tex]\log W\{\bar n_j\}[/tex],

where

[tex]\bar n_j = z g_j e^{-\beta \epsilon_j}[/tex]

is the set of occupation numbers maximizing W.

After some straightforward manipulations one obtains

[tex]S = \frac{1}{T} (E-\mu N) = \frac{3}{2} N k - N k \log\left[\frac{N}{V} \lambda^{-3}\right] \qquad (A)[/tex]

This is basically eq (8.54), and here's the point that looks weird to me.

From the one hand, this equation seems to provide the correct equation of state

[tex]PV = NkT[/tex]. However, there's a thermodynamic equation stating

[tex]E = TS - PV + \mu N \qquad (B)[/tex]

(Gibbs-Duhem equation?). Now if I plug (B) into the first equality in (A) I obtain

[tex]TS = TS - PV[/tex] i.e [tex]PV = 0[/tex], which disagrees with the equation

of state. In other words, it should be

[tex]S = \frac{1}{T} (E +PV -\mu N) [/tex]

Furthermore eq. (A), is also referred to as Sackur-Tetrode equation, but differs from

eq. (6.63) which, after some algebra, reads

[tex]S = \frac{5}{2} N k - N k \log\left[\frac{N}{V} \lambda^{-3}\right] \qquad (C)[/tex]

Now both (A) and (C) are free from the Gibbs paradox, and hence I understand that they

are both Sackur-Tetrode equations.

However (B) has an extra [tex]NkT=PV[/tex] term which seems to exclude a clash with the

thermodynamic equation (B).

What does this mean? Should we conclude that the "maximization approach" in section 8.5

does not really work for the classical gas (which, anyway, does not even exist...).

Or perhaps eq. (B) has a more limited scope, and cannot be used in this approach?

For some reason both of these answer sound too dismissive to me...

Also, should eq. (B) be unavailable, I'd have some trouble in solving problem 8.2 (see also https://www.physicsforums.com/showthread.php?t=272585" )

Thanks a lot for any insight and comment.

Franz

PS I was not sure where to post this. There's no statistical physics forum, and after all this is the

"classical gas", so it could fit in classical physics. Apologies in case another forum was more appropriate.

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