Question

Justify the identification of the statistical entropy with the thermodynamic entropy.

Answer 1

The main term on which the definition of statistical entropy is based is probability of number of atoms or the number of equivalent microstates.

Boltzman defined entropy as: S= Kln​​​​​​, where K is a proportionality constant and is the number of equivalent microstates. Replacing k by the number of moles (i.e N/Na=n; where Na is avogadros number) and rate constant (R) the equation becomes

S= nRln

If we take change in entropy (S) and replace the number of microstates (​​​​​​) by change in volume (V2-V1) the equation becomes

S= nRln(V2/V1).

Now the main term that defines THERMODYNAMIC ENTROPY is change in heat in a reversible system.

q(rev) = nRT ln(V2/V1)

Dividing the whole equation by T, we get

q(rev) /T = nR ln(V2/V1)

(i. e) S = nR ln (V2/V1)