Mathematics of Exact Differentials in Thermodynamics

Our considerations of equilibrium have produced some useful results--these results are usually in the form of some differential expression between quantities.

We will now consider some the useful mathematical relations that follow from the combined form of first and second law.

First

$$\input{equations/15-1A}$$ (21-1)

This differential implies

$$\input{equations/15-1B}$$ (21-2)

We can think of this equation as defining a ``rolling tangent plane''

Figure 21-1: Illustration of a rolling tangent plane in one dimension. Evaluated at some arbitrary point $x_o$, the slope of the plane is the conjugate force to that variable, $F(x_o)$. Using the equation of the tangent plane, we can define a new function. $\Phi (F)$, which is the the intercept of the tangent plane with slope $F$ and the internal energy. The value of the slopes of the tangent surface to $U$ are related to the conjugate forces--the values of these forces are in turn functions of where the tangent plane is placed. In other words, the conjugate forces are also functions of the extensive variables.