NEET Physics Question - Thermal Physics

$4.0 \text{ g}$ of a gas occupies $22.4 \text{ litres}$ at NTP. The specific heat capacity of the gas at constant volume is $5.0 \text{ JK}^{-1}\text{mol}^{-1}$. If the speed of sound in this gas at NTP is $952 \text{ ms}^{-1}$, then the heat capacity at constant pressure is (Take gas constant $R = 8.3 \text{ J/mol K}$): (2015 Re)

$8.5 \text{ J/K mol}$

$8.0 \text{ J/K mol}$

$7.5 \text{ J/K mol}$

$7.0 \text{ J/K mol}$

Solution:

The density $\rho = \frac{4 \times 10^{-3} \text{ kg}}{22.4 \times 10^{-3} \text{ m}^3}$. Using $v = \sqrt{\frac{\gamma P}{\rho}}$, we get $952 = \sqrt{\frac{\gamma \times 1.013 \times 10^5}{4/22.4}}$, yielding $\gamma \approx 1.6$. Since $\gamma = \frac{C_p}{C_v}$, $C_p = 1.6 \times 5.0 = 8.0 \text{ J K}^{-1} \text{mol}^{-1}$.

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