How to Derive Van der Waals Equations from Thermodynamic Relations?

[nzyme]

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}}\approxT^{\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 haven't 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 a lot

 

[nzyme]

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