Define Entropy Change

Define Entropy Change :

In this section we define the entropy changes s that occurs when a closed system changes from a well-defined final state by a process, that we can describe as reversible.

One mole of ideal gas is confined to the left-hand side (as drawn) of a thermally isolated container, and occupies a volume Vo. The right hand side of the container, also containing a volume Vo, is evacuated. The tap (solid line) between the two halves of the container is then suddenly opened and the gas fills the entire container, of volume 2Vo.

We propose that both the previous and new temperature and pressure follow the Ideal Gas Law, so that initially we have PiVi=RTi and then, when the tap is opened, we have PfVf=RTf, where R is the ideal gas constant.

As the system is thermally isolated, it cannot exchange heat with its surroundings. Also, since the system's volume is kept constant, the system does not do work on its surroundings. As a result, the change in internal energy 0, and because the internal energy is a function of temperature only for the ideal gas, we know that Thus the pressure halves; i.e.

For an ideal monatomic gas, the entropy as a function of the internal energy.

This result is also valid if the gas is not monatomic, as the volume dependence of an ideal gas in the dilute limit (in which classical statistical mechanics correctly describes the translational degrees of freedom) is the same as for a monatomic gas.

One can also evaluate the entropy change using purely thermodynamic methods. It is necessary to then take another route from the initial state to the final state, such that all the intermediary states are in thermal equilibrium.

This means that such a route can only be realized in the limit where the changes happen infinitely slowly. Such routes are also referred to as quasistatic routes.

In some books one demands that a quasistatic route has to be reversible, here we don't add this extra condition.

The net entropy change from the initial state to the final state is independent of the particular choice of the quasistatic route, as the entropy is a function of state.