A) \[-{{w}_{vdw}}=nRT\] In \[\left( \frac{{{V}_{2}}}{{{V}_{1}}} \right)+a{{n}^{2}}\left( \frac{1}{{{V}_{2}}}-\frac{1}{{{V}_{1}}} \right)\]
B) \[-{{w}_{vdw}}=nRT\] In \[\left( \frac{{{V}_{1}}}{{{V}_{2}}} \right)+a{{n}^{2}}\left( \frac{1}{{{V}_{2}}}-\frac{1}{{{V}_{1}}} \right)\]
C) \[-{{w}_{vdw}}=nRT\] In \[\left( \frac{{{V}_{2}}}{{{V}_{1}}} \right)+{{a}^{2}}\left( \frac{1}{{{V}_{2}}}-\frac{1}{{{V}_{1}}} \right)\]
D) \[-{{w}_{vdw}}=nRT\] In \[\left( \frac{{{V}_{1}}}{{{V}_{2}}} \right)+{{a}^{2}}\left( \frac{1}{{{V}_{2}}}-\frac{1}{{{V}_{1}}} \right)\]
Correct Answer: A
Solution :
[a] \[-dw=PdV\] \[-w=\int\limits_{{{V}_{2}}}^{{{V}_{1}}}{PdV}\] For an vander Waals gas, \[\left( P+\frac{a{{n}^{2}}}{{{V}^{2}}} \right)\,(V-nb)=nRT\] \[\therefore \,\,\,\,\,\,P=\frac{nRT}{V-nb}-\frac{a{{n}^{2}}}{{{V}^{2}}}\] Hence, \[-{{w}_{vdw}}=\int\limits_{{{V}_{1}}}^{{{V}_{2}}}{\left( \frac{nRT}{V-nb}-\frac{a{{n}^{2}}}{{{V}_{2}}} \right)}dV\] \[=\int\limits_{{{V}_{1}}}^{{{V}_{2}}}{\frac{nRT}{V-nb}}\,dV-\frac{a{{n}^{2}}}{{{V}_{2}}}dV\] \[=nRT\] In \[\left( \frac{{{V}_{2}}-nb}{{{V}_{1}}-nb} \right)+a{{n}^{2}}\left[ \frac{1}{{{V}_{2}}}-\frac{1}{{{V}_{1}}} \right]\] When\[V>>nb\]. \[-{{w}_{vdw}}=nRT\] In \[\left( \frac{{{V}_{2}}}{{{V}_{1}}} \right)+a{{n}^{2}}\left[ \frac{1}{{{V}_{2}}}-\frac{1}{{{V}_{1}}} \right]\]
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