Exact differentials

Let x and y be two specified properties. Denoting one of the dependent properties by z, the relation between the three is written as z(x,y). The total differential of z is

The partial derivatives depend on x and y as expressed by the functions M and N in the second equality of Eq (6.1):

Total differentials have the mathematical characteristic of being exact or inexact. The distinction is determined by comparison of the mixed second derivatives of z:

The partial derivatives depend on x and y as expressed by the functions M and N in the second equality of Eq (6.1):

The total differential is exact if the order of forming the mixed second derivative is immaterial, or if:

A mathematical corollary of the exactness property of a total differential is that its integral z2 - zi from state 1 to state 2 [i.e., for the process from (xi,yi) to (x2,y2)] is independent of the path (or constraint) y = F(x). For example, if x and y denote p and T, the path could be a combination of an isothermal step and an isobaric step. Or, x and y could vary in a continuous fashion as shown by the curve y = F(x) on the x - y plane in Fig. 6.1. This curve generates a corresponding path on the z(x,y) surface, as shown in the diagram.

Inexact Differential
Fig. 6.1 The surface z(x,y) and a path y = F(x)) from state 1 to state 2

If y is eliminated from Eq (6.1) using y = F(x), the total differential can be integrated from state i to state 2 to yield:

If the total differential is exact, Eq (6.4) is satisfied and Eq (6.5) gives the same value of z2 - z1 for all paths F(x). Problem 6.1 provides a purely mathematical exercise to help understand the distinction between exact and inexact differentials.

The relevance of the above mathematical prelude to thermodynamic calculations stems from the independence of thermodynamic property changes on the path of a process. This characteristic implies that all thermodynamic functions must satisfy Eq (6.4), whether M and N are partial derivatives of particular properties (as in Eq (6.2)) or, more generally, when they are any functions of thermodynamic properties.

Getting Started With Solar

Getting Started With Solar

Do we really want the one thing that gives us its resources unconditionally to suffer even more than it is suffering now? Nature, is a part of our being from the earliest human days. We respect Nature and it gives us its bounty, but in the recent past greedy money hungry corporations have made us all so destructive, so wasteful.

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