DaTario said:
But what is the countable physical parameter in ∫dQ/T?
dQ is the change in the internal energy of the system due to heating. As the energy of the system is increased, the molecules can begin to occupy higher energy states. For example, as you raise the temperature a gas, you will find more molecules in high velocity states than before. Since more states become available (W increases, per Boltzmann definition), the entropy increases. To answer your question, the countable parameter is still W, the number of accessible microstates.
DaTario said:
Perhaps, an explanation restricted to the case of a 1 mol sample of ideal gas would be a very good start.
The canonical ensemble says that the probability of finding the gas in a microstate with energy E is proportional to ##e^{-E/kT}##. Since ##p(E) \propto e^{-E/kT}## and we know that probabilities have to sum to 1 (##\int p(E) = 1##), we know that ##p(E) = \frac{1}{Z} e^{-E/kT}## where ##Z = \int e^{-E/kT}##. For a single gas particle, we know the kinetic energy is ##\frac{1}{2} m v^2##, so $$Z = \int d^3 x \int d^3 v e^{-mv^2 / 2kT} = V\int d^3 v e^{-mv^2 / 2kT}$$ If you evaluate this integral, it gives ##Z = \frac{V}{\lambda^3}## where the constant ##\lambda = \sqrt{\frac{m}{2\pi kT}}## is the de Broglie wavelength of that particle. To get the partition function for N particles, take ##Z_N = Z^N = \left( \frac{V}{\lambda^3} \right) ^N##. Using some trickery from the canonical ensemble, we have $$S = \frac{\partial}{\partial T} (kT \ln Z_N) = k \ln Z_N + kT \frac{\partial \ln Z_N}{\partial T}$$ Since internal energy is given by ##U = -\frac{\partial }{\partial \beta} \ln Z_N = \frac{3}{2} NkT## and since ##\frac{\partial}{\partial \beta} = -kT^2 \frac{\partial}{\partial T}##, we have that $$S = k \ln Z_N + \frac{U}{T} = kN \left[\ln \left( \frac{V}{\lambda^3} \right) + \frac{3}{2} \right] \approx kN \ln \left( \frac{V}{\lambda^3} \right) $$
Notice that this last term is essentially the number of de Broglie wavelength-sized cubes you could stuff into a volume V, and this is essentially a number of microstates.
Note: all the ensembles have slight differences in the additive constant on the entropy. I forget which is the most accurate, but I would trust the
Sackur-Tetrode equation (from the microcanonical ensemble).