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11.3.4: Equilibrium

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    The partition model leads to interesting results when the system and its environment are allowed to come into contact after having been isolated. In thermodynamics this process is known as the total configuration coming into equilibrium.

    Let us suppose that the system and the environment have been in isolation and therefore are characterized by different, unrelated, values of energy, entropy, and other quantities. Then suppose they are allowed to interact, using a model of interaction in which the total energy is unchanged. Energy may flow from the system to the environment or vice versa because of mixing, and this flow of energy is called heat. As a result, the probabilities of occupancy will change, although the descriptions of the states and their properties, including their energies, do not change.

    We have developed general formulas that relate small changes in probabilities, and in \(E, S, \alpha\), and \(\beta\) which can be used now. If the energy of the system is assumed to change somewhat (because of mixing), that fact could be incorporated into a new application of the Principle of Maximum Entropy to the system that would result in modified probabilities, \(E, S, \alpha\), and \(\beta\). In particular, we saw earlier that the signs of \(dE\) and \(d\beta\) are opposite, so that if \(E\) goes up, \(\beta\) then goes down, and vice versa.

    Soon, the transfer of energy between the system and the environment may result in our not knowing the energy of each separately, but only the total energy (which does not change as a result of the mixing). In that case, it would be appropriate to use the Principle of Maximum Entropy on the total combination of system and environment considered together. When that is done, there will be a new single value of \(\beta\) and a new total entropy. What can be said about these values?

    First, the new entropy is the sum of the new entropy of the system and the new entropy of the environment, because entropy is an extensive quantity. Furthermore, the old total entropy (at the time the interaction started) is the sum of the old system entropy and the old environment entropy, for the same reason. However, what is interesting is the new total entropy compared with the old total entropy.

    The new entropy, because it is evaluated with the probability distribution that comes from the Principle of Maximum Entropy, is the largest value consistent with the total energy. Any other probability distribution consistent with the same total energy would lead to a smaller (or possibly equal) entropy. One such probability distribution is the distribution prior to the mixing, the one that led to the old entropy value. Therefore the total entropy has increased (or at best stayed constant) as a result of the interaction between the system and the environment. It may be that the entropy of the system alone has gone down, but if so then the entropy of the environment must have gone up at least as much.

    The energies of the system and the environment have changed, and as a result the values of \(\beta_s\) and \(\beta_e\) have changed, in opposite directions. Their new values are the same (each is equal to \(\beta_t\)), and therefore this new value lies between the two old values.

    This page titled 11.3.4: Equilibrium is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Paul Penfield, Jr. (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.