Exact and Inexact Differentials

A Mathematical Digression

We have mentioned, from time to time, that the quantities, U, H, and so on, are state functions, but that q and w are not state functions. This has various consequences. One consequence is that we can write things like ΔU and ΔH, but we never write q or w with a Δ in front them. A more important consequence is that in a process ΔU and ΔH are independent of path. That is, ΔU and ΔH depend only on the initial and final states. However, q and w do depend on the path one takes to get from the initial to the final state.

Another consequence is that the differentials, dU and dH are mathematically different, in some sense, from dq and dw. Some writers write dq and dw with a line through the d to indicate this difference. We have not chosen to use such a specialized notation, but expect that we all will be able to just remember that dU and dH are mathematically different, in some sense, than dq and dw.

We must now consider in detail the nature of this difference.

Let's think, for the moment, in terms of functions of the variables x and y and consider the differential,

(1)
We ask the question, does there exist a function, f = f(x,y) such that,
(2)
In other words, does a function f(x,y) exist such that,
(3a, b)
Euler's test provides a way to see whether such a function, f(x,y) exists. Euler's test is based on the fact that for "nice" functions (and all of our functions are "nice") the mixed second derivatives must be equal. That is,
(4)
(On the left-hand side we take the derivative with respect to y first and then take the derivative of the result with respect to x, and vice versa on the right-hand side.

Let's try Euler's test on our differential, df. If f exists then the Equations (3a) and (3b) are correct. Use Equations (3a, b) to obtain the proposed second derivatives,

(5a, b)
The mixed second derivatives are equal. So we conclude that there exists a function, f(x,y) (actually x2y3) such that Equations (1) and (2) are equal.

The differential df, in Equation (1), is called an "exact differential" for the very reason that a function, f, exists such that Equation (2) can be used to calculate it.

Now, let's consider the differential,

(6)
Is dg an exact differential? Use Euler's test to find out. If dg is exact then the coefficients of dx and dy are the respective partial derivatives of g. Euler's test would then compare
(7)
These are not equal so that the putative second partial derivatives are not equal to each other. The differential, dg, is not exact and there does not exist a function, g(x,y), such that dg gives Equation (6).

Both of the differentials, df and dg can be integrated from, say, x1, y1 to x2, y2. The integral,

(8)
depends only on the initial and final points because df is exact and the function f exists.

The differential dg can be integrated, but there is no equivalent to Equation (8) for the integral of dg because there is no function, g(x,y) which gives Equation (6). The integral of dg would have to be carried out along some path and we would find that the value of the integral depends on the path as well as on the initial and final points.

So, what is the purpose of all this? We are getting ready to present the second law of thermodynamics. One of the consequences of the second law will be the demonstration that for a reversible process dq/T is exact.  dqrev is not exact, but dqrev/T is exact. That means that dqrev/T is the differential of some new function (a state function) whose integral is independent of path. We will call the new state function, S, and name it the "entropy."

The absolute temperature, T, is called an "integrating denominator" for dqrev. That is, when we divide the inexact differential, dqrev, by T  the resulting differential becomes exact. Notice that the inexact differential, dg above, has an integrating denominator. The variable, x, is an integrating denominator for dg. You can see this by noticing that dg/x = df.

WRS

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