A) \[1+\frac{2}{n}\]
B) \[1-\frac{2}{n}\]
C) \[1+\frac{1}{n}\]
D) \[2-\frac{1}{n}\]
Correct Answer: A
Solution :
Key Idea: Mayor's formula is\[{{C}_{P}}-{{C}_{V}}=R\]. Internal energy of a gram mole of a perfect gas having n degrees of freedom is \[U=N\left( n\cdot \frac{1}{2}kt \right)=\frac{n}{2}RT\] \[{{C}_{V}}=\frac{dU}{dt}=\frac{d}{dt}\left( \frac{n}{2}RT \right)=\frac{n}{2}R\] From Mayor?s formula \[{{C}_{P}}-{{C}_{V}}=R\] \[\Rightarrow \] \[{{C}_{P}}=R+{{C}_{V}}\] \[\Rightarrow \] \[{{C}_{P}}=R+\frac{n}{2}R=\left( \frac{n}{2}+1 \right)R\] \[\therefore \] \[\gamma =\frac{{{C}_{P}}}{{{C}_{V}}}=\frac{\left( \frac{n}{2}+1 \right)R}{\frac{n}{2}R}=1+\frac{2}{n}\]
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