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 $(V - \beta n)\,\left( {P + \frac{{\alpha {n^2}}}{V}} \right) = nRT$
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(a)According to given Vander Waal’s equation
$P = \frac{{nRT}}{{V - n\beta }} - \frac{{\alpha {n^2}}}{{{V^2}}}$
Work done, $W = \int_{{V_1}}^{{V_2}} {PdV} = nRT\int_{{V_1}}^{{V_2}} {\frac{{dV}}{{V - n\beta }}} - \alpha {n^2}\int_{{V_1}}^{{V_2}} {\frac{{dV}}{{{V^2}}}} $
$ = nRT\,\left[ {{{\log }_e}(V - n\beta )} \right]\,_{{V_1}}^{{V_2}} + \alpha {n^2}\left[ {\frac{1}{V}} \right]_{{V_1}}^{{V_2}}$
$ = nRT{\log _e}\frac{{{V_2} - n\beta }}{{{V_1} - n\beta }} + \alpha {n^2}\left( {\frac{{{V_1} - {V_2}}}{{{V_1}{V_2}}}} \right)$
$P = \frac{{nRT}}{{V - n\beta }} - \frac{{\alpha {n^2}}}{{{V^2}}}$
Work done, $W = \int_{{V_1}}^{{V_2}} {PdV} = nRT\int_{{V_1}}^{{V_2}} {\frac{{dV}}{{V - n\beta }}} - \alpha {n^2}\int_{{V_1}}^{{V_2}} {\frac{{dV}}{{{V^2}}}} $
$ = nRT\,\left[ {{{\log }_e}(V - n\beta )} \right]\,_{{V_1}}^{{V_2}} + \alpha {n^2}\left[ {\frac{1}{V}} \right]_{{V_1}}^{{V_2}}$
$ = nRT{\log _e}\frac{{{V_2} - n\beta }}{{{V_1} - n\beta }} + \alpha {n^2}\left( {\frac{{{V_1} - {V_2}}}{{{V_1}{V_2}}}} \right)$
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