Ideal Gas Entropy: Solving Kardar's Equation IV.33

[Jolb]

I'm working through Kardar's Statistical Physics of Particles, and I'm in chapter 4 on the part about the ideal gas. Here's a link to that chapter from the book:

http://web.mit.edu/8.333/www/lectures/lec13.pdf

I think he has an error in equation IV.33 but I'd like you guys to make sure of it.

http://img694.imageshack.us/img694/467/kardar.jpg

I think in the top equation of IV.33, the 2∏mE should be 4∏mE, so the final equation should have an 8 instead of a 4. Here's why I think that:
$$ln\left (V^N\frac{2\pi^\frac{3N}{2}}{\left (\frac{3N}{2}-1 \right )!} \left ( 2mE \right )^\frac{3N-1}{2}\Delta _R\right )$$

$$=Nln(V)+\frac{3N}{2}ln(2\pi)-\left (\frac{3N}{2}-1 \right )ln\left ( \frac{3N}{2}-1 \right )+\left ( \frac{3N}{2}-1 \right )+\frac{3N-1}{2}ln(2mE)+ln\Delta _R$$

eliminating terms of order 1 or lnN,

$$=Nln(V)+\frac{3N}{2}ln(2\pi)-\left (\frac{3N}{2} \right )ln\left ( \frac{3N}{2} \right )+\left ( \frac{3N}{2} \right )+\frac{3N}{2}ln(2mE)$$
$$=N\left (ln(V)+\frac{3}{2}ln(2\pi)-\left (\frac{3}{2} \right )ln\left ( \frac{3N}{2} \right )+\left ( \frac{3}{2} \right )ln(e)+\frac{3}{2}ln(2mE) \right )$$
$$=N\left (ln(V)+ln(2\pi)^\frac{3}{2}-ln\left ( \frac{3N}{2} \right )^\frac{3}{2}+ln(e)^\frac{3}{2}+ln(2mE)^\frac{3}{2} \right )$$
$$=Nln\left (V\left [\frac{(2\pi)(e)(2mE)}{\frac{3N}{2}} \right ]^\frac{3}{2} \right )$$

Did I make a mistake eliminating the terms of lower order? Please help!

 

[Rap]

It depends on whether you interpret $$2\pi^{3 N/2}$$ to mean $$2(\pi^{3 N/2})$$ or $$(2\pi)^{3 N/2}$$.

It doesn't much matter, the entire expression is in error. The correct expression is

$$S=N k_B\log\left[V \left(\frac{4 \pi m E}{3 h^4}\right)^{3/2}\left(\frac{e}{N}\right)^{5/2}\right]$$

also known as the Sackur-Tetrode equation for a monatomic ideal gas.

 

[Jolb]

Great, thanks! Silly mistake. The 2 next to the pi vanishes when you neglect terms of order 1.

Kardar's approach is an interesting one--he uses classical physics to derive that incorrect expression and then shows how indistinguishability can be introduced to resolve the error. Putting 1/N! next to omega leads to something closer to the Tetrode-Sackur equation, but without any h. I'm guessing this follows exactly Gibbs' formulation of the Gibbs paradox. I think it's worthwhile to approach statmech from the classical point of view, so I do want to worry about this derivation.

 

[Rap]

Great, thanks! Silly mistake. The 2 next to the pi vanishes when you neglect terms of order 1.

Kardar's approach is an interesting one--he uses classical physics to derive that incorrect expression and then shows how indistinguishability can be introduced to resolve the error. Putting 1/N! next to omega leads to something closer to the Tetrode-Sackur equation, but without any h. I'm guessing this follows exactly Gibbs' formulation of the Gibbs paradox. I think it's worthwhile to approach statmech from the classical point of view, so I do want to worry about this derivation.

Ok, yes that will help to fix things. Note that in order for the ST equation to be physically meaningful, the argument of the logarithm must be dimensionless. Without the h, its not. Let me know how, in this derivation, the h is finally introduced.

 

[Jolb]

Well this "derivation" is meant to illustrate where and how the classical approach breaks down. The incorrect equation above is not extensive, so Kardar motivates the introduction of 1/N! in the expression for Ω by noting that it ensures extensivity, and he argues its physical appropriateness by discussing indistinguishability.

Here's what he says about h. "Yet another difficulty with the expression IV.47, [which is the 1/N! corrected version of entropy] resolved in quantum statistical mechanics, is the arbitrary constant that appears in changing the units of measurement for q and p. The volume of phase space involves products pq, of coordinates and conjugate moment, and hence has dimensions of (action)^N. Quantum mechanics provides the appropriate measure of action in Planck's constant h. Anticipating these quantum results, we shall henceforth set the measure of phase space for identical particles to..." So he sticks in a 1/h^3N. Not quite as satisfying as the demonstration of why N! was needed.

 

[Rap]

Well this "derivation" is meant to illustrate where and how the classical approach breaks down. The incorrect equation above is not extensive, so Kardar motivates the introduction of 1/N! in the expression for Ω by noting that it ensures extensivity, and he argues its physical appropriateness by discussing indistinguishability.

Here's what he says about h. "Yet another difficulty with the expression IV.47, [which is the 1/N! corrected version of entropy] resolved in quantum statistical mechanics, is the arbitrary constant that appears in changing the units of measurement for q and p. The volume of phase space involves products pq, of coordinates and conjugate moment, and hence has dimensions of (action)^N. Quantum mechanics provides the appropriate measure of action in Planck's constant h. Anticipating these quantum results, we shall henceforth set the measure of phase space for identical particles to..." So he sticks in a 1/h^3N. Not quite as satisfying as the demonstration of why N! was needed.

It sure isn't. I've been trying to understand Arieh Ben-Naim's derivation of the STE, in "A Farewell to Entropy". He derives it using information theory, and its really informative (LOL - no pun), up to the point where he introduces the h term, then it becomes just as vague.