Finding the equations of state via the fundamental relation

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1. Sep 12, 2017

hijasonno

1. The problem statement, all variables and given/known data

2. Relevant equations

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

3. 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.

Last edited by a moderator: Sep 12, 2017
2. Sep 12, 2017

hijasonno

I can't figure out why the image won't load, so I have to put it in a reply to the thread.
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