**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 Δ*U* and
Δ*H*, but we never write *q* or *w*
with a Δ in front them. A more important consequence
is that in a process Δ*U* and Δ*H*
are independent of path. That is, Δ*U*
and Δ*H* 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,

(2)In other words, does a function

(3a, b)Euler's test provides a way to see whether such a function,

(4)(On the left-hand side we take the derivative with respect to

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,

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

(7)These are not equal so that the putative second partial derivatives are not equal to each other. The differential,

Both of the differentials, *df* and *dg* can be integrated
from, say, *x*_{1}, *y*_{1} to *x*_{2},
*y*_{2}. The integral,

(8)depends only on the initial and final points because

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. *dq*_{rev} is not exact, but *dq*_{rev}/*T*
is exact. That means that *dq*_{rev}/*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 *dq*_{rev}. That is, when we divide the inexact differential,
*dq*_{rev}, 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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