Thermodynamics, 5th Edition

Chapter 9

Reprise to the Second Law. Mathematical Proof of the Carathéodory Theorem and Resulting Interpretations

9.1 Fundamentals

In Chapter 1, we had introduced functions of state that are mathematical analytic functions of the general form

$R=R(x_1,\dots ,x_i,\dots ,x_n),$ (9.1.1)

(9.1.1)

in which the function R is uniquely specified by the values of the n independent variables x_i for the thermodynamic properties of the system. The corresponding linear differential form is given by

$dR=\sum _{i=1}^n{(\partial R/\partial x_i)}_{x_{j\ne i}}d{x}_i,$ (9.1.2)

(9.1.2)

and in Chapter 1 was labeled an