Entropy as a State Variable

Equation D.9 shows that in the reversible process we can introduce an increment of the state variable S, called the entropy, for the heat flux dQby 

As an example, let us consider two integral paths 1 and 2 from the initial state A to the final state B as shown in Fig. D.5. For a reversible process we require 

This implies that the difference in S between the state B and the state A is independent of the integration path  

In conclusion, for a reversible process, the heat flux d Q input to the system is exchanged through a corresponding thermal internal energy increment T dS, and S is independent on the path of integration. This entropy increment can also be written as dS, since is exchangeable with the surroundings and reversible. Then (D.IO) can be written as 

The idea of exchangeable entropy is also supported by Boltzmann s theory based on statistical thermodynamics (see AppendixD.3). That is, Boltzmann s principle can be shown as 

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