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So in my homework I was able to get this-

$\left(\frac{\partial U}{\partial V}\right)_T = T(\frac{\partial P}{\partial T})_V - P$

but now I'm trying to plug in $P = \frac{nRT}{V-nb}-a(\frac{n}{V})^2$ in the above equation. I'm struggling here, I got zero for $\left(\frac{\partial U}{\partial V}\right)_T$ but I'm then asked to evaluate $\left(\frac{\partial U}{\partial V}\right)_T$ in an integral from infinity to $V$ with respect to $V$ (i.e. $dV$).

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    – Poutnik
    Commented May 19, 2022 at 6:12
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    $\begingroup$ For $(\frac{\partial p}{\partial T})_V$, just take the derivative of your entire expression for p ($p=\frac{nRT}{V-nb}-a(\frac{n}{V})^2$ )with respect to T at constant V. Then multiply by T, and subtract the expression for p....exactly as your equation says. If you're getting zero, I suspect what you are doing is subtracting only the first of the two terms in the expression for p, rather than the whole thing. General suggestion: When you can't get the right answer, instead of trying to find the error in what you've done, start over from scratch; that way you're less likely to repeat it. $\endgroup$
    – theorist
    Commented May 19, 2022 at 6:12

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