Musings

We start with Clausius’s definition¹ of the entropy

the First Law of Thermodynamics

(where $W= p d V$) and the Ideal Gas Law

where $U$ is the internal energy of the gas and $\alpha = 3/2$ for a monatomic ideal gas.

Let’s consider an isothermal process. Since $T=\text{const}$, $d U =0$ and $$Q = N k T \ln(V_f/V_i)$$ Comparing this with (1), we conclude that

where $f(T)$ is some volume-independent function of the temperature.

Repeating the same analysis for an adiabatic process, $Q=0$, and hence $$d U = \alpha N k d T = -W = -\frac{N k T}{V} d V$$ or

Since $d S=0$, we can solve for the previously unknown function $f(T)$

where the constant is independent of both $V$ and $T$.

This is (almost) the answer we are after. But it behoves us to pause and note that it has a very suggestive interpretation. We don’t know where any particular gas molecule is located. But we do know that it must be somewhere within the volume $V$. Similarly, we don’t know what the velocity of any particular gas molecule is. But baby kinetic theory² tells us that $$v_{\text{RMS}} \propto T^{1/2}$$ So $T^{3/2}$ is (proportional to) the volume in “velocity space” in which we expect to find the molecule of gas and $V T^{3/2}$ is the volume in “phase space” for a single molecule. It represents, in other words, our lack of knowledge of the state of that gas molecule. For $N$ molecules, the volume in phase space is ${(V T^{3/2})}^N$, which is what appears as the argument of the logarithm in (6).

So the Boltzmann entropy is $k$ times the natural logarithm of the volume in phase space of the system.

That’s about as far as I got in my lecture, but one can go a little further. (6) is wrong because it isn’t extensive. If we take two container of the same gas, at the same temperature and pressure, we should find that the total entropy $S= S_1 + S_2$. Instead, with (6), we find

$${(S- S_1 - S_2)}^{\text{naïve}} = N k \ln(N) - N_1 k \ln(N_1) - N_2 k \ln(N_2)$$

where $N=N_1+N_2$.

But this discrepancy is easy to fix. The quantity that behaves extensively is

where $c_{1,2}$ are constants. Using Stirling’s formula, for large $N$, we can then write this as

That is, we should treat the gas molecules as identical particles, and take $k$ times logarithm of the volume in phase space, where we’ve modded out by the permutations of the $N$ particles.

Despite having had to gloss over a couple of steps where a little calculus was required, I’m rather proud of this “elementary” derivation. I don’t think I’ve seen anything even remotely resembling a satisfactory explanation in any of the elementary textbooks (even the calculus-based ones).