Examples of Entropy Changes due to Irreversible Processes

To illustrate how entropy changes are related to irreversible processes, we shall consider some simple examples. All of them are discrete systems that consist 

Consider an isolated system which we assume (for simplicity) consists of two parts, each having a well-defined temperature, i.e. each part is locally in equilibrium. Let the temperatures of the two parts be Ti and T2 (Fig. 3.9), Ti T2- Let d 2 be the amount of heat flow from the hotter part to the colder part in a time dt. Since this isolated system does not exchange entropy with the exterior, d S — 0. And since the volume of each part is a constant, dW — 0. The energy change in each part is solely due to the flow of heat dUi = dQi, i = 1,2. In accordance with the First Law, the heat gained by one part is equal to the heat lost by the other. Therefore, —dQ = dQ2 = dQ. Both parts are locally in equilibrium with a well-defined temperature and entropy. The total change in entropy, d S, of the system is the sum of the changes of entropy in each part due to the flow of heat  

Since the heat flows irreversibly from the hotter part to the colder part, dQ is positive if T T2- Hence, 5 0. In expression (3.5.1), dQ and (1/T2 — 1/Ti) respectively correspond to dX and F in (3.4.6). In terms of the rate of flow of heat dQ/dt, the rate of entropy production can be written as 

Now the rate of heat flow or the heat current Jq = dQ/dt is given by the laws of heat conduction. For example according to the Fourier law of heat conduction, Jq = a(r 1 — T2), in which a is the coefficient of heat conductivity. Note that the thermod3mamic flow Jq is driven by the thermodynamic force F = (1/7 2 — 1/Ti). For the rate of entropy production we have from (3.5.2) that 

Due to the flow of heat from the hot part to the cold part, the temperatures eventually become equal, and the entropy production ceases. This is the state of equilibrium. The entropy production must vanish in the state of equilibrium, which implies that the force F and the corresponding flux Jq both vanish. In fact, we can deduce the properties of the equilibrium state by stipulating that all entropy production must vanish in that state. 

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