Simpler derivation of Sackur-Tetrode equation 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.


*

*Using the binomial distribution, the definition of $S$ from Boltzmann's equation is:
$$S = k\ln \Omega = k\ln \biggl[\frac {N!}{n!(N-n)!}\biggr]$$

*Substituting for $N = \frac V {\lambda^3}$,
$$S = k \ln\left[\frac {\bigl(\frac V {\lambda^3}\bigr)!}{n!\bigl(\frac V {\lambda^3}-n\bigr)!}\right]$$

*Using Stirling's approximation:
$$S = k \biggl[ \frac V {\lambda^3} \ln \biggl(\frac V {\lambda^3}\biggr) -  \biggl(\frac V {\lambda^3}  - n \biggr) \ln \biggl(\frac V {\lambda^3}-n\biggr)  - n \ln (n)\biggr]$$

*Using the approximation $\ln \bigl(\frac V {\lambda^3}-n\bigr) =  \ln\bigl(\frac V {\lambda^3}\bigr) $ for $\frac V  {\lambda^3} \gg n$
$$S = k \biggl[ n \ln \biggl(\frac V {\lambda^3}\biggr)  - n \ln (n)\biggr]$$

*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?
 A: 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.
