Question

Molar heat capacity of an ideal gas in the process $P{V}^x$= constant, is given by:$C=\frac{R}{\gamma -1}+\frac{R}{1-x}.$ An ideal diatomic gas with $C_v=\frac{5R}{2}$ occupies a volume $V_1$ at pressure $P_1$. If this gas undergoes a process in which the pressure is proportional to the volume and it is found that the rms speed of the gas molecules has doubled from its initial value at the end of the process. Then

• A. The molar heat capacity of the gas in the given process is 3R.
• B. The molar heat capacity of the gas in the given process is 2R
• C. Heat supplied to the gas in the given process is $8P_1{V}_1$.
• D. Heat supplied to the gas in the given process is $9P_1{V}_1$.

Answer 1

Answer: (A), (D)

The correct options are
A The molar heat capacity of the gas in the given process is 3R.
D Heat supplied to the gas in the given process is $9P_1{V}_1$.

$P\propto V\Rightarrow P{V}^{-1}=const.\Rightarrow x=-1Here,C=Cv+\frac{R}{2}=\frac{5R}{2}+\frac{R}{2}=3R{V}_{rms}\alpha \sqrt{T}as{V}_{rms}doubles,finaltemperaturewillbe4TSo,\Delta T=3TQ=nC\Delta T=n\left(3R\right)\Delta T=n\left(3R\right)3T=9nRT$
And it is given in the question $P_1{V}_1$= nRT