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Pure Substances and Van der Waals

  • Thread starter nzyme
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1. Homework Statement

For any Pure Substance, use the relation (1) to show that for a Van der Waals gas, over a range of conditions where c[tex]_{v}[/tex] is effectively constant, that equation (2) gives equations (3) and (4)


2. Homework Equations

(1) ~ Tds = du + pdv

(2) ~ c[tex]_{v}[/tex]dT + T [tex]\frac{\partial p}{\partial t}[/tex][tex]_{v}[/tex]dv = 0

(3) ~ T(v-b)[tex]^{\alpha}[/tex] = constant

(4) ~ p + [tex]\frac{a}{v^{2}}[/tex][tex]\approx[/tex]T[tex]^{\frac{1+\alpha}{\alpha}}[/tex]

(5) ~ [tex]\alpha[/tex]=[tex]\frac{R}{c_{v}}[/tex]

(6) ~ (p + [tex]\frac{a}{v^{2}}[/tex])(v-b)=RT

3. The Attempt at a Solution

I really havent got a clue where to go with this one, this question obviously has something to do with c[tex]_{v}[/tex].

I was thinking for the first part since its equal to a constant then maybe integration or diferrentiation will come into this at some point. As for the second part i haven't got the slightest clue where to go so any tips would be nice.

I've written down all the equation i think could possibly come into play here, and have tried rearranging them but have not got anywhere with this yet. I can derive equation (2) from equation (1) but i can't see that being much use

I'm new round here and would like to apologise for any thing that i've done wrong

Thanks alot
 
Last edited:

Answers and Replies

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i have managed to show that

p+[tex]\frac{a}{v^{2}}[/tex]=[tex]\frac{RT}{v-b}[/tex]=T([tex]\frac{\partial p}{\partial T}[/tex])[tex]_{v}[/tex]

and that

[tex]\frac{a}{v^{2}}[/tex] = ([tex]\frac{\partial u}{\partial v}[/tex])[tex]_{T}[/tex]

but am now unsure where to go or if this is even the right path to take
 

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