141.

In the calculation of the entropy of an ideal gas and of a system of resonators, as carried out in the preceding chapters, we proceeded in both cases, by first determining the entropy for an arbitrarily given state, then introducing the special condition of thermodynamic equilibrium, i.e., of the maximum of entropy, and then deducing for this special case an expression for the entropy.

If the problem is only the determination of the entropy in the case of thermodynamic equilibrium, this method is a roundabout one, inasmuch as it requires a number of calculations, namely, the determination of the separate distribution densities $w_1$, $w_2$, $w_3,\dots$ which do not enter separately into the final result. It is therefore useful to have a method which leads directly to the expression for the entropy of a system in the state of thermodynamic equilibrium, without requiring any consideration of the state of thermodynamic equilibrium. This method is based on an important general property of the thermodynamic probability of a state of equilibrium.

We know that there exists between the entropy $S$ and the thermodynamic probability $W$ in any state whatever the general relation [eqn:(164)] (164). In the state of thermodynamic equilibrium both quantities have maximum values; hence, if we denote the maximum values by a suitable index: $S_m=k\log W_m.\mathrm{\backslash Label}[eqn](229)\text{upshape (229)}$ It follows from the two equations that: $\frac{W_m}{W}=e^{\frac{S_m-S}{k}}.$ Now, when the deviation from thermodynamic equilibrium is at all appreciable, ${\displaystyle \frac{S_m-S}{k}}$ is certainly a very large number. Accordingly

$W_m$ is not only large but of a very high order large, compared with $W$, that is to say: The thermodynamic probability of the state of equilibrium is enormously large compared with the thermodynamic probability of all states which, in the course of time, change into the state of equilibrium.

This proposition leads to the possibility of calculating $W_m$ with an accuracy quite sufficient for the determination of $S_m$, without the necessity of introducing the special condition of equilibrium. According to [sect:123.] Sec. 123, et seq., $W_m$ is equal to the number of all different complexions possible in the state of thermodynamic equilibrium. This number is so enormously large compared with the number of complexions of all states deviating from equilibrium that we commit no appreciable error if we think of the number of complexions of all states, which as time goes on change into the state of equilibrium, i.e., all states which are at all possible under the given external conditions, as being included in this number. The total number of all possible complexions may be calculated much more readily and directly than the number of complexions referring to the state of equilibrium only.