Total Differential of a Dependent Variable

This chapter begins with a discussion of mathematical properties of the total differential of a dependent variable. Three extensive state functions with dimensions of energy are introduced enthalpy, Helmholtz energy, and Gibbs energy. These functions, together with internal energy, are called thermodynamic potentials. Some formal mathematical manipulations of the four thermodynamic potentials are described that lead to expressions for heat capacities, surface work, and criteria for spontaneity in closed systems. 

Recall from Sec. 2.4.1 that the state of the system at each instant is defined by a certain minimum number of state functions, the independent variables. State functions not treated as independent variables are dependent variables. Infinitesimal changes in any of the independent variables will, in general, cause an infinitesimal change in each dependent variable. 

A dependent variable is a function of the independent variables. The total differential of a dependent variable is an expression for the infinitesinial change of the variable in terms of the infinitesimal changes of the independent variables. As explained in Sec. F.2 of Appendix F, the expression can be written as a sum of terms, one for each independent variable. Each term is the product of a partial derivative with respect to one of the independent variables and the infinitesimal change of that independent variable. For example, if the system has two independent variables, and we take these to be T and V, the expression for the total differential of the pressme is 

in the case of a fixed amount of an ideal gas with pressure given hy p = nRT/ V, the total differential of the pressme can be written 

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If the coefficients of the total differential of a dependent variable are known as functions of the independent variables, the expression for the total differential may be integrated to obtain an expression for the dependent variable as a function of the independent variables. 

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