Simpler derivation of Sackur-Tetrode equation

Question

Is there a reason the following derivation for the Sackur-Tetrode equation is not common? I am teaching a lower undergraduate level class and would like to derive it with simpler terms of only using relative volume.

The equation is: $$ kn \ln \frac V {n\lambda^3}, $$ where $\lambda^3$ is the thermal wavelength cubed, or the quantum volume for one particle.

Since each particle has a volume of $\lambda^3$, the total of number of positions in the volume for a particle is $N = \frac V {\lambda^3}$, and $n$ is the total number of particles.

1. Using the binomial distribution, the definition of $S$ from Boltzmann's equation is: $$S = k\ln \Omega = k\ln \biggl[\frac {N!}{n!(N-n)!}\biggr]$$
2. Substituting for $N = \frac V {\lambda^3}$, $$S = k \ln\left[\frac {\bigl(\frac V {\lambda^3}\bigr)!}{n!\bigl(\frac V {\lambda^3}-n\bigr)!}\right]$$
3. Using Stirling's approximation: $$S = k \biggl[ \frac V {\lambda^3} \ln \biggl(\frac V {\lambda^3}\biggr) - \biggl(\frac V {\lambda^3} - n \biggr) \ln \biggl(\frac V {\lambda^3}-n\biggr) - n \ln (n)\biggr]$$
4. Using the approximation $\ln \bigl(\frac V {\lambda^3}-n\bigr) = \ln\bigl(\frac V {\lambda^3}\bigr) $ for $\frac V {\lambda^3} \gg n$ $$S = k \biggl[ n \ln \biggl(\frac V {\lambda^3}\biggr) - n \ln (n)\biggr]$$
5. Manipulate algebra. $$S = kn \ln \frac V {n\lambda^3}$$

I'd like to use this in my class because it is simpler and cleaner and develops a sort of chemical intuition based on binomial distribution. However, I want to know if this is correct. I have seen this equation without the $5/2$ term before.

Edit: The physical assumption that I think might be invalid is the use of N as $\frac V {\lambda^3}$. The binomial distribution is valid when you have n particles that fit into N DISCREET positions. That would work fine for a box of volume V with N slots separated by partitions. But in this case, the volume V does not have any partitions and is continuous. In this case, a volume $\lambda^3$ can fit into positions that are not discreetly defined. This creates a technically infinite number of N's. So, would this still be a valid $\Omega$ to be used in the Boltzmann's equation?

Answer 1

Issues with that derivation:

• You're missing the extra term $\frac 52 k N,$ which may matter if you have to do any work with chemical potentials.
• Your students will not necessarily know why to parcel the space into volumes of size $\lambda^3$. Starting from the definition of entropy and deriving that the thermal volume $\lambda^3$ is important seems preferable.
• Your students may benefit from knowing that half-factorials exist via $n! = \int_0^\infty dx~x^n~e^{-x},$ that they start from $(-1/2)! = \sqrt{\pi},$ and that the volume of an $n$-ball of radius $r$ is $\pi^{n/2} ~ r^n / (n/2)!.$ It is only about fifteen minutes or less on a chalkboard of mathematical trickery, but it can help to facilitate some later calculations, plus an "oh, that's how that works" explaining why Gaussians have these mysterious factors of $\sqrt{\pi}$ in them.
• The fact that quantum mechanics makes particles even-in-principle indistinguishable is huge, and it resolves the Gibbs paradox, leading to the Sackur-Tetrode equation. Robbing a student of this fact, and the amount of brain-warp that it creates, is not a bad thing necessarily, but it's also not necessarily a good thing. This is an opportunity to plant a seed of cognitive dissonance that only really will get resolved when the student learns quantum field theory. It is maybe even the central problem with particle-centric understandings of the world.
• If you're teaching statistical mechanics, your students just barely are beginning to understand what temperature really is. I'm not sure I personally would make it a "load-bearing beam" of their understanding of statistical mechanics. For learning thermodynamics prior to a statistical mechanics course, sure, just treat temperature as a phenomenological given, we can measure it with a thermometer, who cares what it is? But now that you have an ability to talk about how $\beta = (k_B T)^{-1}$ deals much better with "negative temperatures" than $T$ does, everything that the student knows from earlier thermodynamics courses is painfully incomplete and you may not want to rest too much material on a good understanding of temperature and the thermal wavelength.

With all of this said: your approach is certainly simple and clean, and might make a good heuristic introduction to the topic if you do not want to devote a whole lecture to the Sackur-Tetrode equation.