KVPY Sample Paper KVPY Stream-SX Model Paper-11

For an ideal gas if molar heat capacity varies on \[C={{C}_{v}}+3a{{T}^{2}}.\]equation of process:

A) \[Ve\left( \frac{3a}{2R} \right){{T}^{2}}=cons\tan t\]

B) \[Ve\left( \frac{-3a}{2R} \right){{T}^{2}}=cons\tan t\]

C) \[T{{V}^{2}}=cons\tan t\]

D) \[V{{T}^{2}}=cons\tan t\]

Correct Answer: B

Solution :

given \[C={{C}_{V}}+3a{{T}^{2}}\]

From of thermodynamics

\[Q=\Delta U+W\]

\[or\,C\Delta T={{C}_{V}}\Delta T+P(\Delta V)\]

Or \[C={{C}_{V}}+P\left( \frac{\Delta V}{\Delta T} \right);\] or \[C={{C}_{V}}+P\left( \frac{dV}{dT} \right)\]

So\[P\left( \frac{dV}{dt} \right)=3a{{T}^{2}}\]

Also \[P=\frac{RT}{V},\]\[So\frac{RT}{V}\left( \frac{dV}{dT} \right)=3a{{T}^{2}}\]

Or \[\int{\frac{dV}{V}=\frac{3a}{R}\int{TdT;}}\] \[or\,\ln \,V=\frac{3a}{2R}{{T}^{2}}\]

Or \[V={{e}^{3a{{T}^{2}}/2R}}\]

Or \[V{{e}^{\frac{-3a}{2R}{{T}^{2}}}}=Cons\tan t.\]