# Pure Substances and Van der Waals

## 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$$_{v}$$ is effectively constant, that equation (2) gives equations (3) and (4)

## Homework Equations

(1) ~ Tds = du + pdv

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

(3) ~ T(v-b)$$^{\alpha}$$ = constant

(4) ~ p + $$\frac{a}{v^{2}}$$$$\approx$$T$$^{\frac{1+\alpha}{\alpha}}$$

(5) ~ $$\alpha$$=$$\frac{R}{c_{v}}$$

(6) ~ (p + $$\frac{a}{v^{2}}$$)(v-b)=RT

## 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$$_{v}$$.

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

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i have managed to show that

p+$$\frac{a}{v^{2}}$$=$$\frac{RT}{v-b}$$=T($$\frac{\partial p}{\partial T}$$)$$_{v}$$

and that

$$\frac{a}{v^{2}}$$ = ($$\frac{\partial u}{\partial v}$$)$$_{T}$$

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