2.9 The approach to equilibrium

Take-home message: Spontaneous changes maximise the entropy of an isolated system, but minimise the appropriate free energy of non-isolated systems.

We have seen that, for an isolated system, the entropy change will always be greater than or equal to zero. But more than that, we have seen in specific examples that if a change can take place which increases the entropy, it will. A hot and a cold block brought in to contact will exchange heat till they are at the same temperature, and not stop half way. Two gases which are allowed to mix will mix fully, and not just partially. In both cases the final state maximises the entropy with respect to the free parameter - the amount of heat exchanged, or the degree of mixing.

But what about non-isolated systems? Obviously the entropy of the universe increases. But it would be convenient if we could describe what happens by referring only to the system, and not the surroundings. In fact there is a way. If the external temperature is $T_0$ and pressure $P_0$, we can define the quantity

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle A=E-T_0 S+P_0 V,$ }}$

called the availability, and we find that it always decreases during a spontaneous change. (See here for details.)

For a process which starts and ends at the temperature and pressure of the surroundings, the initial and final availabilities are just the initial and final Gibbs free energy, so any such process minimises $G$.

For a process which starts and ends at the temperature of the surroundings and which is at constant volume, $\Delta V=0$ and so $\Delta A=\Delta F$. Such a process minimises the Helmholtz free energy $F$.

For an isolated system, both $V$ and $E$ must be constant, so a decrease of $A$ simply means an increase of $S$, as it should!

In all of the above, we are implicitly assuming that the system has unspecified internal degrees of freedom which are initially out of equilibrium, such as un-mixed gases or unreacted chemicals, or regions of different temperatures or pressures (for thermally- or mechanically-isolated systems, repectively). Otherwise the state of the system is fully specified and no evolution can happen.

References

• Mandl 4.2
• Bowley and Sánchez 2.9
• Adkins 10.2
• Zemansky 10.3 (Warning: Zemansky uses $A$ for the Helmholtz free energy!)