Equilibrium of Constant-Temperature Constant-Pressure System

The analysis of the equilibrium conditions for system follows similar lines to the that in the previous section. The entropy, , of the combined system satisfies the condition that in any spontaneous process

If absorbs heat from in this process then . Furthermore, the first law of thermodynamics yields

(9.20) |

where is the work done by against the constant pressure, , of the reservoir, , and where denotes any other work done by in the process. (For example, might refer to electrical work.) Thus, we can write

(9.21) |

or

Here, we have made use of the fact that and are constants. We have also introduced the definition

(9.23) |

where becomes the Gibbs free energy, , of system when the temperature and pressure of this system are both equal to those of the reservoir, .

The fundamental condition (9.19) can be combined with Equation (9.22) to give

(assuming that is positive). This relation implies that the maximum work (other than the work done on the pressure reservoir) that can be done by system is . (Incidentally, this is the reason that is also called a ``free energy''.) The maximum work corresponds to the equality sign in the preceding equation, and is obtained when the process used is quasi-static (so that is always in equilibrium with , and , ).

If the external parameters of system (except its volume) are held constant then , and Equation (9.24) yields the condition

Thus, we arrive at the following statement:

If a system is in contact with a reservoir at constant temperature and pressure then the stable equilibrium state is such that

The preceding statement can again be phrased in more explicit statistical terms. Suppose that the external parameters of (except its volume) are fixed, so that . Furthermore, let be described by some parameter . If changes from some standard value then the corresponding change in the total entropy of is

But, in an equilibrium state, the probability, , that the parameter lies between and is given by

Now, from Equation (9.26),

(9.28) |

However, because is just some arbitrary constant, the corresponding constant terms can be absorbed into the constant of proportionality in Equation (9.27), which then becomes

This equation shows explicitly that the most probable state is one in which attains a minimum value, and also allows us to determine the relative probability of fluctuations about this state.