Why Do Two Objects at Different Temperatures Come to a Common Intermediate Temperature When Put in Contact? Entropy Is Maximized.

A classic problem in thermodynamics is to place two objects with different heat capacities and different temperatures in thermal contact and ask what is the final common temperature. Normally, this temperature is found using the first law of thermodynamics. A single, intermediate final temperature is an assumption (from experience) that is not required by the first law. Why is the final temperature of the two objects at equilibrium the same? It is shown that this outcome is a consequence of the second law of thermodynamics subject to the constraint of energy conservation (the first law). That is, the overall entropy of the universe is maximized when the two objects reach a common temperature. The analysis is extended to the case of two different samples of ideal gas at different pressures as well as different temperatures placed in mechanical (for example, by placing a moveable piston between the gases) and thermal contact. In this case, it is shown that the two gases at equilibrium reach a common tempera...