What is entropy change?

Question

There is huge number N of file cabinets,
each cabinet contains huge number K of sealed drawers.

Each cabinet contains exactly one penny in
one of the drawers, and K-1 empty drawers.
All pennies are indistinguishibale.

During earthquake all drawers fell on the floor,
and janitor replaced all the scattered drawers
back into cabinets, each drawer into arbitrary
cabinet, because it was impossible to say
which drawer fell from which cabinet.

In ensemble of identical pennies, what is increase
of entropy per penny?

Answer 1

This is considering the state to be defined by the locations of the N pennies (not the drawers).

The total number of accessible states is originally K^N. After the rearrangement, any N of the NK total drawers can be occupied, so the new number of states is (NK)!/((NK-N)!N!). Using the approximation:

ln(n!) ~ n ln(n) - n

the logarithm of the latter is:

(NK)ln(NK) - NK - N ln(N) + N - (NK-N)ln(NK-N) + NK - N
= NK ln(K) - (NK-N)(ln(K-1))
~ N ln(K) + N

whereas the logarithm of the original number of states was N ln(K) so the increase is N. Per penny that's 1, or in calories/degree multiply by k (Boltzmann's constant) giving k.

Answer 2

FREE ENERGY