For instance, the entropy of a perfect gas changes with its volume V according to the relationship

where the subscripts f and i denote the final and initial states. Note that if Vf > V (i.e. if the gas expands into a larger volume) the logarithmic (ln) term will be positive and the equation predicts an increase of entropy. This is expected since expansion of a gas is a spontaneous process and will be accompanied by an increase in the disorder because the molecules are now moving in a greater volume.

Similarly, increasing the temperature of a system should increase the entropy because at higher temperature the molecular motion is more vigorous and hence the system more chaotic. The equation which relates entropy change to temperature change is

where Cv is the molar heat capacity at constant volume. Inspection of equation (3.8) shows that AS will be positive when Tf > Tl, as predicted.

The entropy of a substance will also change when it undergoes a phase transition, since this too leads to a change in the order. For example, when a crystalline solid melts, it changes from an ordered lattice to a more chaotic liquid (see Fig. 3.1) and consequently an increase in entropy is expected. The entropy change accompanying the melting of a solid is given by

By a 'reversible process' we mean one in which the changes are carried out infini-tesimally slowly, so that the system is always in equilibrium with its surroundings. In this case we infer that the temperature of the surroundings is infinitesimally higher than that of the system, so that the heat changes are occurring at an infinitely slow rate, so that the heat transfer is smooth and uniform.

We can see the link between entropy and disorder by considering some specific examples.

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