In thermodynamics, the change in entropy in a reversible process which transforms the system from state 1 to state 2 is
where is the heat absorbed in the process. We can now ask if the entropy obtained starting from the microscopic description agrees with the standard thermodynamic definition. We will consider two types of processes as described below:
where is the heat absorbed as the state changes from 1 to 2. Now, from the first law of thermodynamics, the change in total internal energy of the system is
where is the work done on the system. Since, for the ideal gas,
and , and
The expansion/compression of the system gives rise to a change in pressure such that , where P(V) = NkT/V, is given by the equation of state (ideal gas law). Thus, the total work done on the system is
If we now use the statistical definition of entropy
the change in entropy is
where . Thus, we see that the two agree exactly.
and, from the first law,
However, for the ideal gas
Thus, the change in entropy is
From the statistical definition:
which agrees exactly with the thermodynamic entropy change.