A) \[nRT{{\log }_{e}}\left( \frac{{{V}_{2}}-n\beta }{{{V}_{1}}-n\beta } \right)+\alpha \,{{n}^{2}}\,\left( \frac{{{V}_{1}}-{{V}_{2}}}{{{V}_{1}}{{V}_{2}}} \right)\]
B) \[nRT{{\log }_{10}}\left( \frac{{{V}_{2}}-\alpha \beta }{{{V}_{1}}-\alpha \beta } \right)+\alpha \,{{n}^{2}}\,\left( \frac{{{V}_{1}}-{{V}_{2}}}{{{V}_{1}}{{V}_{2}}} \right)\]
C) \[nRT{{\log }_{e}}\left( \frac{{{V}_{2}}-n\alpha }{{{V}_{1}}-n\alpha } \right)+\beta \,{{n}^{2}}\,\left( \frac{{{V}_{1}}-{{V}_{2}}}{{{V}_{1}}{{V}_{2}}} \right)\]
D) \[nRT{{\log }_{e}}\left( \frac{{{V}_{1}}-n\beta }{{{V}_{2}}-n\beta } \right)+\alpha \,{{n}^{2}}\,\left( \frac{{{V}_{1}}{{V}_{2}}}{{{V}_{1}}-{{V}_{2}}} \right)\]
Correct Answer: A
Solution :
According to given Vander Waal?s equation \[P=\frac{nRT}{V-n\beta }-\frac{\alpha {{n}^{2}}}{{{V}^{2}}}\] Work done, \[W=\int_{{{V}_{1}}}^{{{V}_{2}}}{PdV}=nRT\int_{{{V}_{1}}}^{{{V}_{2}}}{\frac{dV}{V-n\beta }}-\alpha {{n}^{2}}\int_{{{V}_{1}}}^{{{V}_{2}}}{\frac{dV}{{{V}^{2}}}}\] \[=nRT\,\left[ {{\log }_{e}}(V-n\beta ) \right]\,_{{{V}_{1}}}^{{{V}_{2}}}+\alpha {{n}^{2}}\left[ \frac{1}{V} \right]_{{{V}_{1}}}^{{{V}_{2}}}\] \[=nRT{{\log }_{e}}\frac{{{V}_{2}}-n\beta }{{{V}_{1}}-n\beta }+\alpha {{n}^{2}}\left( \frac{{{V}_{1}}-{{V}_{2}}}{{{V}_{1}}{{V}_{2}}} \right)\]
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