When I first learned this concept in Vector Calculus and Ordinary Differential Equations, it seemed really pointless. Hardcore math geeks don't solve diff eq's with no pansy-ass Integrating Factors!
Then I took Thermodynamics.
Basically, in Physics they are important because they illustrate the idea of path independancy. The definition of an exact differential is as follows:
If there exists a function T(x,y) such that the partial of T with respect to x equals P(x,y) and the partial of T with respect to y equals Q(x,y), the differential equation formed by P(x,y) + Q(x,y) * y' = 0 is exact. This is because the function y(x) that satisfied the equation obtained by differentiating T(x,y) with respect to x by the chain rule. (Looks a hell of a lot like Green's Theorem).
Now, this just sounds more and more complicated and pointless. What's the point?
For equations that are exact differentials, there is a general solution. And general solutions, as everybody knows, are very good things.
To make a non-exact differential into an exact differential for easier solving involves an Integrating Factor. You multiply the whole equation by an integrating factor, which is either previously determined or just guess one that seems to 'fit' with the equation.
In thermo, these exact differentials are known as state functions because they depend only on the inital and final state of the system and not the path traversed.
Entropy is an exact differential. Heat is not.
I thought that was kind of cool. It turns out to be extremely logical because entropy is defined as a state of a system, similar to mass or volume, while heat is not. Therefore, the integrating factor to express heat is 1/T.
So the whole point of thermodynamics is to simply invent a set of functions which describe the system but have exact differentials with respect to variables P,T,V and S. |
