Exact differential expression

Because these are exact differential expressions. Maxwell equations can be written by inspection. The two most useful ones are derived from equations 67 and 68 ... 

We have previously shown that the Pfaff differential <5pressure-volume work equation (2.43) is an inexact differential. It is easy to show that division of equation (2.43) by the absolute temperature T yields an exact differential expression. The division gives... 

Equations (13.33) and (13.34) are both exact differential expressions and the conditions of exactness can be applied. One expression, obtained from Equation (13.34), is... 

When F is a function of x and y, the right-hand side of Eq. (6.11) is an exact differential expression since Eq. (6.12) must then be satisfied, it serves as a criterion of exactness. 

Making use of the fact that Eq. (6.20) is an exact differential expression, show that... 

Since nG is a state function, the right side of tliis equation is an exact differential expression whence. 

For 1 mol of a homogeneous fluid of constant composition, Eqs. (4-6) and (4-11) through (4-13) simplify to Eqs. (4-14) through (4-17) of Table 4-1. Because these equations are exact differential expressions, application of the reciprocity relation for such expressions produces the common Maxwell relations as described in the subsection Multi-variable Calculus Applied to Thermodynamics in Sec. 3. These are Eqs. (4-18) through (4-21) of Table 4-1, in which the partial derivatives are taken with composition held constant. 

Hence equation (2.6) has been shown to be an exact differential expression, as we knew it must be because we derived it by differentiating the function V = RT/P. 

The rule in Eq. (9.14) follows from the fact that the differential expression M dx- - N dy is the total differential of some function / (x, y) that is, M dx + N dy is an exact differential expression. The converse is also true. For example, suppose that we have an expression of the form... 

Since each of the expressions on the right-hand side of these equations is an exact differential expression, it follows that the cross-derivatives are equal. From this we immediately obtain the four Maxwell relations ... 

Exact differential expressions and one coefficient for all four of these thermodynamic state functions are summarized below ... 

The path of integration is understood to be traversed in the counterclockwise sense. An informal proof can be based on the identification of f z) dz with an exact differential expression (see Section 10.6) ... 

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