state functions, and the first law
We said earlier that state functions are those that depend only on the present state of a system. Another way of expressing this is to say that state functions are path independent. Indeed, path independence may be used as a test of whether a variable is a state function or not. This is to say that if Y is a state function, then for any process that results in a change Y1 →Y2, the net change in Y, ΔY, is independent of how one gets from Y1 to Y2. Furthermore, if Y is a state function, then the differential dY is said to be mathematically exact.
Let's explore what is meant by an exact differential. An exact differential is the familiar kind, the kind we would obtain by differentiating the function u with respect to x and y, and also the kind we can integrate. But not all differential equations are exact. Let's first consider the mathematical definition of an exact differential, then consider some thermodynamic examples of exact and inexact differentials.
Consider the first order differential expression:
containing variables M and N, which may or may not be functions of x and y. Equation 2.25 is said to be an exact differential if there exists some function u of x and y relating them such that the expression:
is the total differential of u:
Let's consider what this implies. Comparing 2.26 and 2.27, we see that:
(2.28)
A necessary, but not sufficient, condition for 2.25 to be an exact differential is that M and N must be functions of x and y.
A general property of partial differentials is the reciprocity relation or cross-differentiation identity, which states that the order of differentiation does not matter, so that:
(2.29)
(The reciprocity relation is an important and useful property in thermodynamics, as we shall see at the end of this chapter.) If eqn. 2.26 is the total differential of u, it follows that:
which is equivalent to:
Equation 2.31 is a necessary and sufficient condition for 2.25 to be an exact differential; that is, if the cross-differentials are equal, then the differential expression is exact.
Exact differentials have the property that they can be integrated and an exact value obtained. This is true because they depend only on the initial and final values of the independent variables (e.g., x and y in eqn. 2.27).
Now let's consider some thermodynamic examples. Volume is a state function and we can express it as an exact differential in terms of other state functions, as in eqn. 2.17:
Substituting the coefficient of thermal expansion and compressibility for ∂V/∂T and ∂V/∂P respectively, equation 2.30 becomes equal to eqn. 2.18:
According to eqn. 2.31, if V is a state function, then:
You should satisfy yourself that eqn. 2.32 indeed holds for ideal gases and therefore that V is a state variable.
Work is not a state function, that is, the work done does not depend only on the initial and final states of a system. We would expect then that dW is not an exact differential, and indeed, this is easily shown for an ideal gas.
For PV work, dW = –PdV. Substituting eqn. 2.17 for dV and rearranging, we have:
(2.33)
Evaluating ∂V/∂T and ∂V/∂P for the ideal gas equation and multiplying through by P, this becomes:
(2.34)
but
(2.35)
We cannot integrate eqn. 2.34 and obtain a value for the work done without additional knowledge of the variation of T and P because the amount of work done does not depend only on the initial and final values of T and P; it depends on the path taken. Heat is also not a state function, not an exact differential, and is also path dependent. Path dependent functions always have inexact differentials; path independent functions always have exact differentials.
On a less mathematical level, let's consider
