Some Mathematical Theorems
We can derive some useful relations among groups of four variables, with two still being independent.
- If we write dz = Mdx Ndy , then the following important relations can be derived
- The two relations above were based on z = f (x, y) , which represents a dependence of one variable on two others. We can also derive some useful relations among groups of four variables, with two still being independent (e.g. P, v, T, and s). Let w, x, y, and z be our variables, and start with the relations x = f (w, y) , and y = f (w, z) .
Writing the differentials, we have
Now, substitute dy from the second equation into the first,
Compare this to the the differential for x = f (w, z) :
Equating the coefficients on dw yields:
and equating the coefficients on dz gives
Now we have four general equations relating derivatives of thermodynamic properties: