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Finding the equations of state via the fundamental relation

  1. Sep 12, 2017 #1
    [​IMG] 1. The problem statement, all variables and given/known data

    ques.png
    [​IMG]
    [​IMG]
    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: Sep 12, 2017
  2. jcsd
  3. Sep 12, 2017 #2
    I can't figure out why the image won't load, so I have to put it in a reply to the thread.
    View attachment 210922
     
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