A) \[\frac{5}{4}\]
B) \[\frac{5}{3}\]
C) \[\frac{5}{2}\]
D) \[\frac{3}{5}\]
Correct Answer: C
Solution :
Poisson?s equation for adiabatic process is given by \[{{\operatorname{PV}}^{\gamma }}=constant\] For adiabatic process, Poisson?s equation is given by \[{{\operatorname{PV}}^{\gamma }}=constant\] [a] Ideal gas relation is \[PV\,\,=\,\,\,\,RT\] \[\Rightarrow \,\,\,\,\,V=\frac{RT}{P}\] [b] From Eqs. [a] and [2], we get \[P{{\left( \frac{RT}{P} \right)}^{\gamma }}=cons\tan t\] \[\Rightarrow \,\,\,\,\,\frac{T}{{{P}^{\gamma -1}}}=cons\tan t\] [c] where is the ratio of specific heats of the gas. Given, \[P\,\,\propto \,\,{{T}^{C}}\] [d] On comparing with Eq. [c], we have \[C=\frac{\gamma }{\gamma -1}\] For a monoatomic gas \[\gamma =\frac{5}{3}\] \[\therefore \] We have \[C=\frac{\frac{5}{3}}{\frac{5}{3}-1}=\frac{5}{2}\,\,\,\,\,\,\,\,\,\]
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