Temperature and Transfer of Heat

The aim of this section is to justify the quite formal definition of the (absolute) temperature T by 

In order to present the essential point of our considerations as clearly as possible, let us start with a very simple situation in which two systems (1) and (2), each of which is in internal equilibrium, are brought into thermal contact. The combined system (1) + (2) as a whole is assumed to be isolated from any further energetic interaction with the surroundings. By thermal contact we mean that there may be energetic exchange between systems (1) and (2) only by transfer of heat, but not by transfer of any other form of energy. Without further specifications, systems (1) and (2) apparently need not be in equilibrium with respect to each other or, in 

Our first problem is to find a condition for the equilibrium of the combined system by making use of the first and second law of thermodynamics. 

The second law tells us that subsystems (1) and (2) are each characterized by their equilibrium entropies and as functions of their internal energies and 

We have dropped the variables V, 2. .. since their values are kept constant for each of the subsystems (1) and (2). As entropy is an extensive variable, the entropy S of the combined system is given as 

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