Finding the equations of state via the fundamental relation

[hijasonno]

Homework Statement

Homework Equations

$$\frac{\partial S}{\partial u}\Bigr|_{v} = \frac{1}{T}$$
$$\frac{\partial S}{\partial v}\Bigr|_{u} = \frac{-P}{T}$$[/B]

The Attempt at a Solution

a.) $$\frac{S}{R} = \frac{UV}{N} - \frac{N^3}{UV}$$ $$\frac{S}{R} = \frac{UV}{N} - \frac{N}{uv}$$ $$ \frac{S}{NR} = \frac{UV}{N^2} - \frac{1}{uv} $$ $$\frac{s}{R} = uv - \frac{1}{uv}$$ where $s = \frac{S}{N}$, $u = \frac{U}{N}$, and $v = \frac{V}{N}$.

Equations of state(in entropy representation):

$$\frac{\partial S}{\partial u}\Bigr|_{v} = \frac{1}{T} = R(v + \frac{1}{uv^2}) $$

$$\frac{\partial S}{\partial v}\Bigr|_{u} = \frac{P}{T} = R(u + \frac{1}{vu^2}) $$

And this is where I get stuck; when I try and multiply $u$ and $v$ by a constant, i.e. $\lambda v + \frac{1}{\lambda u (\lambda v)^2}$, this is obviously not homogeneous of order zero. I have an idea about how to continue the rest of the parts, but I want to be sure that the first one is correct be for I move on.
My values for $P$ and $T$ in the entropy representation seem right, maybe I'm not quite understanding what a homogeneous function actually is. Any help is appreciated.[/B]

 

[hijasonno]

I can't figure out why the image won't load, so I have to put it in a reply to the thread.
https://www.physicsforums.com/attachments/210922