Relation to thermodynamic entropy
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Relation to thermodynamic entropy

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:

I.
Isothermal expansion/compression of the system from volume, to . In an isothermal process, the temperature, T, does not change. Thus, the entropy relation can be integrated immediately to yield

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

Thus,

and

If we now use the statistical definition of entropy

the change in entropy is

where . Thus, we see that the two agree exactly.

II.
Isochoric heating/cooling from temperature to . In an isochoric process, the volume remains constant. Hence,

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.

These two examples illustrate that the statistical approach agrees exactly with the standard thermodynamic definition of entropy.

Mark Tuckerman
Thu Feb 20 00:47:55 EST 2003