Modern Thermodynamics

- Chapter 1

43

The same is valid for all higher derivatives such as

∂3U

∂T2∂V

, i.e. the order of differentiation does not

matter.

Basic Identities

Consider three variables x, y and z, each of which can be expressed as a function of the other two

variables, x=x(y,z), y=y(z,x) and z=z(x,y). (p,V and T in the ideal gas equation pV=NRT is an example).

Then the following identities are valid:

∂x

∂yz

=

1

∂y

∂xz

(A1.1.6)

∂x

∂yz

∂y

∂z

x

∂z

∂xy

= −1

(A1.1.7)

Consider a functions of x and y, f=f(x,y), other than z. Then:

∂f

∂xz

=

∂f

∂xy

+

∂f

∂yx∂xz

∂y

(A1.1.8)