More Mathematical Thermodynamics: Homogeneous Functions

Consider $U(S,V,N_i)$, if I scale all the extensive variables by multiplying each of the extensive variables with the same ``scale factor'' $\lambda$ then

$$\input{equations/U-homo}$$ (24-9)

Functions that have the property of Equation 24-9, like $U$, are called ``homogeneous degree one'' (HD1) function of their variables.

Notice that $G$ is not a completely homogeneous function:

$$\input{equations/G-homo-false}$$ (24-10)

i.e., increasing the pressure is not like changing an extensive variable.

However,

$$\input{equations/G-homo-true}$$ (24-11)

$G$ is HD1 only in the $N_i$.

Notice that (here lies a common mistake!)

$$\input{equations/16-6A}$$ (24-12)

$\ensuremath{\overline{G}}$ is a different function than $G$.

Consider carefully, what can be deduced from Equation 24-11.

Taking the derivative with respect to $\lambda$

$$\input{equations/16-6B}$$ (24-13)

We get the following very important equation:

$$\input{equations/gibbs-rel}$$ (24-14)

This corresponds to what has been discussed about the relation of the Gibbs free energy. It corresponds to the internal degrees of freedom.