Entropy changes accompanying heat flow

First we will consider the entropy changes accompanying chemical reactions that occur under conditions of constant temperature and pressure. As for the other types of processes we have considered, the entropy changes in the surroundings are determined by the heat flow that occurs as the reaction takes place. However, the entropy changes in the system (the reactants and products of the reaction) can be predicted by considering the change in positional probability. 

Two identical copper ingots, one at 500°C (773 K) and the other at 300°C (573 K) are placed in an insulated enclosure. Heat flows from the hot ingot to the cold ingot until the temperature of both ingots is the same (400°C). Calculate the total entropy change accompanying this process. 

To understand why this must occur, consider the entropy changes that would accompany heat transfer in the opposite direction. Suppose the burner is at 455 K and the water is at 373 K. We can calculate the entropy change that would occur if 100. J of heat flowed from the water to the burner. In this scenario, q for the burner is positive, so the burner gains entropy. For the water, q is negative, so the water loses entropy ... 

The infinitesimal change in entropy, dS, that accompanies an infinitesimal reversible heat flow, 8q e, is dS = Sq ev/T- Now imagine the change in a system from state 1 to state 2 is carried out in a series of such infinitesimal reversible steps. Summation of all these infinitesimal quantities through the calculus technique of integration yields AS. 

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