Mathematical Statement Derived from the Physical Statements

MATHEMATICAL STATEMENT DERIVED FROM THE PHYSICAL STATEMENTS 

In this section, we present two derivations of Caratheodory s principle from the physical statements of the second law one is based on physical statement a and the other is based on physical statement b. Then we derive the mathematical statement of the second law from Caratheodory s principle. 

The system under consideration is such that all its parts are at the same temperature. We assume that no phase transitions occur during the processes discussed in order that we may make assumptions about the continuity of the heat absorbed or liberated during these processes as a function of the variables of state. This restriction is easily removed. 

Proof I. We first consider a system, states of which are functions of the empirical temperature t and one parameter x (e.g., the volume). 

We assume that the heat absorbed in any process is a continuous function of X along the isotherms. 

---