MCQ
Work done by a system under isothermal change from a volume $V_1$ to $V_2$ for a gas which obeys Vander Waal's equation $(V-\beta n)\left(P+\frac{\alpha n^2}{V}\right)=n R T$
  • $n R T \log _e\left(\frac{V_2-n \beta}{V_1-n \beta}\right)+\alpha n^2\left(\frac{V_1-V_2}{V_1 V_2}\right)$
  • B
    $n R T \log _{10}\left(\frac{V_2-\alpha \beta}{V_1-\alpha \beta}\right)+\alpha n^2\left(\frac{V_1-V_2}{V_1 V_2}\right)$
  • C
    $n R T \log _e\left(\frac{V_2-n \alpha}{V_1-n \alpha}\right)+\beta n^2\left(\frac{V_1-V_2}{V_1 V_2}\right)$
  • D
    $n R T \log _e\left(\frac{V_1-n \beta}{V_2-n \beta}\right)+\alpha n^2\left(\frac{V_1 V_2}{V_1-V_2}\right)$

Answer

Correct option: A.
$n R T \log _e\left(\frac{V_2-n \beta}{V_1-n \beta}\right)+\alpha n^2\left(\frac{V_1-V_2}{V_1 V_2}\right)$
(a) According to given Vander Waal's equation
$P=\frac{n R T}{V-n \beta}-\frac{c n^2}{V^2} $
$\text { Work done, } W=\int_{V_1}^{V_2} P d V=n R T \int_{V_1}^{V_2} \frac{d V}{V-n \beta}-c n^2 \int_{V_1}^{V_2} \frac{d V}{V^2} $
$=n R T\left[\log _e(V-n \beta)\right]_{V_1}^{V_2}+o n^2\left[\frac{1}{V}\right]_{V_1}^{V_2} $
$=n R T \log _e \frac{V_2-n \beta}{V_1-n \beta}+c n^2\left(\frac{V_1-V_2}{V_1 V_2}\right) $

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