Question

In a certain gas, the ratio of the velocity of sound and root mean square velocity of gas molecules is $\sqrt{\frac{5}{9}}$. If the gas obeys polytopic process $PT=$ constant, then the molar heat capacity of the gas is

(where $R=$ universal gas constant)

• A. $\frac{R}{2}$
• B. $\frac{3R}{2}$
• C. $\frac{5R}{2}$
• D. $\frac{7R}{2}$

Answer 1

The correct option is D $\frac{7R}{2}$

Velocity of sound is $v_{\text{sound}}=\sqrt{\frac{\gamma P}{\rho }}\dots \dots \left(1\right)$
We know that,
$P=\frac{1}{3}{v}_{\text{rms}}^2$
$v_{\text{rms}}^2=\frac{3P}{\rho }$
$v_{\text{rms}}=\sqrt{\frac{3P}{\rho }}\dots \dots \left(2\right)$
From equation (1) and (2),
$\frac{v_{\text{sound}}}{v_{\text{rms}}}=\sqrt{\frac{\gamma P}{P}}\times \sqrt{\frac{\rho }{3\rho }}=\sqrt{\frac{5}{9}}$
$\sqrt{\frac{5}{9}}=\sqrt{\frac{\gamma }{3}}$
$\gamma =\frac{5}{3}$
Given that,
$PT=\text{constant=K}\dots \dots \left(1\right)$
Ideal gas $PV=nRT$
$P=\frac{nRT}{V}$
Put the value of $P$ in equation (1),
$\frac{nR{T}^2}{V}=K$ $\dots \dots \left(2\right)$
Differentiate it -
$2nRT\Delta T=K\Delta V$
$2mRT\Delta T=PT\Delta V$
$2nR\Delta T=P\Delta V\dots \dots \left(3\right)$
From thermodynamics first law,
$dQ=dU+dW$
$C\Delta T=C_v\Delta T+P\Delta V$
$C\Delta T=C_v\Delta T+2nR\Delta T$
Here $n=1$
$C=C_v+2R$
$C=\frac{R}{\gamma -1}+2R$
$C=\frac{R}{\frac{5}{3}-1}+2R$
$C=\frac{R}{\frac{5-3}{3}}+2R$
$C=\frac{3R}{2}+2R=\frac{7R}{2}$