Internal heat flow in an isolated system

Suppose the system is a solid body whose temperature initially is nonuniform. Provided there are no internal adiabatic partitions, the initial state is a nonequilibrium state lacking internal thermal equilibrium. If the system is surrounded by thermal insulation, and volume changes are negligible, this is an isolated system. There will be a spontaneous, irreversible internal redistribution of thermal energy that eventually brings the system to a final equilibrium state of uniform temperature. 

In order to be able to specify internal temperatures at any instant, we treat the system as an assembly of phases, each having a uniform temperature that can vary with time. To describe a region that has a continuous temperature gradient, we approximate the region with a very large number of very small phases, each having a temperature infinitesimally different from its neighbors. 

We use Greek letters to label the phases. The temperature of phase a at any given instant is T . We can treat each phase as a subsystem with a boundary across which there can be energy transfer in the form of heat. Let represent an infinitesimal quantity of heat transferred during an infinitesimal interval of time to phase a from phase p. The heat transfer, if any, is to the cooler from the warmer phase. If phases a and P are in thermal contact and is less than T, then d otp is positive if the phases are in thermal contact and r is greater than T, d otp is negative and if neither of these conditions is satisfied, d oip is zero. 

To evaluate the entropy change, we need a reversible path from the initial to the final state. The net quantity of heat transferred to phase a during an infinitesimal time interval is dg = The entropy change of phase a is the same as it would be for the 

There is also the condition of quantitative energy transfer, = — d oip. which we use to rewrite Eq. 4.6.7 in the form 

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