A) \[2.303{{p}_{1}}{{V}_{1}}\log \left( \frac{{{V}_{2}}}{{{V}_{1}}} \right)\]
B) \[R{{T}_{1}}\log \left( \frac{{{V}_{2}}}{{{V}_{1}}} \right)\]
C) \[\frac{p_{1}^{2}\,V_{1}^{2}-p_{2}^{2}\,V_{2}^{2}}{R({{T}_{2}}-{{T}_{1}})}\]
D) \[{{C}_{V}}({{T}_{1}}-{{T}_{2}})\]
E) \[R\,{{C}_{V}}\left( \frac{{{T}_{1}}+{{T}_{2}}}{2} \right)\]
Correct Answer: D
Solution :
Work done during adiabatic expansion \[W=\frac{R\,({{T}_{1}}-{{T}_{2}})}{(\gamma -1)}\] Since, \[\gamma =\frac{{{C}_{p}}}{{{C}_{V}}}\] and \[{{C}_{p}}-{{C}_{V}}=R\] \[\therefore \] \[W=\frac{({{C}_{p}}-{{C}_{V}})\,({{T}_{1}}-{{T}_{2}})}{({{C}_{p}}-{{C}_{V}})/{{C}_{V}}}\] \[\Rightarrow \] \[W={{C}_{V}}({{T}_{1}}-{{T}_{2}})\]
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