Planck famously argued that if identical particles are considered indistinguishable this would
resolve Gibbs paradox by correcting for over-counting of the states. If the number of
possible arrangements, W, of the N particles of an ideal gas at volume, V, and temperatureT
with constant volume, heat capacity, C, is written as
W = T^(C/k) * V^NThen:
(a) Show, using Boltzmann's principle (S = k.lnW), that this leads to an expression for the entropy of an ideal classical gas that agrees with thermodynamics.
(b) define what is meant by indistinguishable in the context of plank's solutionand using the notaion of a homogeneous function of degree 1, describe what is meant by extensivity. Show mathematically why Plank's resolution appears to solve the paradox.
The Attempt at a Solution
Now for A it is clear that I should substitute W into the Bolzmann's principle. That is fine. But the resulting expression isn't like anything I have seen. All the entropy equations I have seen are more to do with heat exchange. I'm not entirely sure what I should be doing. Any help would be much appreciated.