The change in the internal energy of the mass of gas, when the volume changes from ' V ' to ' 2 V ' at…

The change in the internal energy of the mass of gas, when the volume changes from ' V ' to ' 2 V ' at constant pressure ' P ' is $(\gamma$ is the ratio of specific heat of gas at constant pressure to specific heat at constant volume)
  1. $\frac{\mathrm{PV}}{\gamma-1}$
  2. $\frac{\mathrm{PV}}{\gamma+1}$
  3. $\frac{\gamma-1}{\mathrm{PV}}$
  4. $\frac{\gamma+1}{\mathrm{PV}}$

Solution

Change in internal energy is given by, $\Delta \mathrm{U}=\mathrm{n} \mathrm{C}_{\mathrm{v}} \Delta \mathrm{~T}...(i)$
Given, $\frac{C_P}{C_v}=\gamma$ and $C_P-C_v=R$ $1+\frac{\mathrm{R}}{\mathrm{C}_{\mathrm{v}}}=\gamma$ $\begin{aligned} C_V & =\frac{R}{(\gamma-1)} \\ \therefore \quad \Delta U & =n\left(\frac{R}{\gamma-1}\right) \Delta T \end{aligned}$ ...[From(i)] Using $\mathrm{P} \Delta \mathrm{v}=\mathrm{nR} \Delta \mathrm{T}$ under constant P , $\Delta U=\frac{P \Delta v}{\gamma-1}=\frac{P(2 v-v)}{\gamma-1}=\frac{P v}{(\gamma-1)}$

Asked in: MHT CET 2024 (04 May Shift 2)

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