Question

An ideal gas with adiabatic exponent $\gamma$ is expanded according to the law $P=\alpha V$ where $\alpha$ is a constant. The volume is $)V_0$. As a result of expansion, the volume increases $\eta$ times. Then

• A. Increment in internal energy of gas is $\frac{V_0^2\alpha ({\eta }^2-1)}{(\gamma -1)}$
• B. Increment in internal energy of gas is $\frac{V_0^2\alpha ({\eta }^2-1)}{2(\gamma -1)}$
• C. Molar heat capacity of the gas in the process is $\frac{R(\gamma +1)}{(\gamma -1)}$
• D. Molar heat capacity of the gas in the process is $\frac{R(\gamma +1)}{2(\gamma -1)}$

Answer 1

Answer: (A), (D)

The correct options are
A Increment in internal energy of gas is $\frac{V_0^2\alpha ({\eta }^2-1)}{(\gamma -1)}$
D Molar heat capacity of the gas in the process is $\frac{R(\gamma +1)}{2(\gamma -1)}$

$P=\alpha V=\frac{nRT}{V}\Rightarrow T=\frac{\alpha V_0^2}{nR}\Rightarrow \Delta T=\frac{V_0^2\alpha }{nR}({\eta }^2-1)\Delta U=n{C}_v\Delta T\Delta U=\frac{nR}{\gamma -1}\times \frac{V_0^2\alpha }{nR}({\eta }^2-1)=\frac{v_0^2\alpha ({\eta }^2-1)}{\gamma -1}Also,RT=\alpha V^2\Rightarrow RdT=2\alpha VdVCdT=CVdT+PdV\Rightarrow C=C_v+\frac{PR}{2V\alpha }=C_v+\frac{R}{2}=\frac{R(\gamma +1)}{2(\gamma -1)}$