Work done by a system under isothermal change from volume $V_{1}$ to $V_{2}$ for a gas which obeys Vander Waal's equation,
$(V - β n) (P + \frac{α n^{2}}{V})$ = nRT is :

n R T l o g_{e} (\frac{V_{2} - n β}{V_{1} - n β}) + α n^{2} (\frac{V_{1} - V_{2}}{V_{1} V_{2}})

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n R T l o g_{10} (\frac{V_{2} - n β}{V_{1} - n β}) + α n^{2} (\frac{V_{1} - V_{2}}{V_{1} V_{2}})

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n R T l o g_{e} (\frac{V_{2} - n β}{V_{1} - n β}) + β n^{2} (\frac{V_{1} - V_{2}}{V_{1} V_{2}})

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n R T l o g_{e} (\frac{V_{1} - n β}{V_{2} - n β}) + α n^{2} (\frac{V_{1} V_{2}}{V_{1} - V_{2}})

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Solution

The correct option is A $n R T l o g_{e} (\frac{V_{2} - n β}{V_{1} - n β}) + α n^{2} (\frac{V_{1} - V_{2}}{V_{1} V_{2}})$
According to Vander Waal's equation
$P$ = $\frac{n R T}{V - n β} - \frac{α n^{2}}{V^{2}}$
Work done,
$W$ = $\int_{V_{1}}^{V_{2}} P d V$
$= n R T \int_{V_{1}}^{V_{2}} \frac{d V}{V - n β} - α n^{2} \int_{V_{1}}^{V_{2}} \frac{d V}{V^{2}}$
$= n R T {[l o g_{e} (V - n β)]}_{e}^{V_{2}} + α n^{2} {[\frac{1}{V}]}_{V_{1}}^{V_{2}}$
$= n R T l o g_{e} [\frac{V_{2} - n β}{V_{1} - n β}] + α n^{2} [\frac{V_{1} - V_{2}}{V_{1} V_{2}}]$

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Work done by a system under isothermal change from volume V1 to V2 for a gas which obeys Vander Waal's equation, (V−βn)(P+ αn2V) = nRT is :

The correct option is A nRTloge( V2−nβV1−nβ)+αn2( V1−V2V1V2)According to Vander Waal's equationP = nRTV−nβ−αn2V2Work done,W = ∫V2V1PdV=nRT ∫V2V1 dVV−nβ−αn2 ∫V2V1 dVV2=nRT[loge(V−nβ)]V2V1+αn2[ 1V]V2V1=nRTloge[ V2−nβV1−nβ]+αn2[ V1−V2V1V2]

Work done by a system under isothermal change from volume $V_{1}$ to $V_{2}$ for a gas which obeys Vander Waal's equation,
$(V - β n) (P + \frac{α n^{2}}{V})$ = nRT is :