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4.0 g of a gas occupies 22.4 litres at NTP. The specific heat capacity of the gas at constant volume is 5.0 $JK^{-1}mol^{-1}$. If the speed of sound in this gas at NTP is 952 $ms^{-1}$, then the heat capacity at constant pressure is (Take gas constant R = 8.3 $JK^{-1}mol^{-1}$)
A
8.5 $JK^{-1}mol^{-1}$
B
8.0 $JK^{-1}mol^{-1}$
C
7.5 $JK^{-1}mol^{-1}$
D
7.0 $JK^{-1}mol^{-1}$
Detailed Solution
Molar mass M = 4.0 g = $4\times10^{-3}$ kg/mol
$v = \sqrt{\frac{\gamma RT}{M}} \Rightarrow \gamma = \frac{Mv^2}{RT} = \frac{4\times10^{-3}\times 952^2}{8.3\times 273} \approx 1.6$
$C_p = \gamma C_v = 1.6\times 5.0 = 8.0$ $JK^{-1}mol^{-1}$
$v = \sqrt{\frac{\gamma RT}{M}} \Rightarrow \gamma = \frac{Mv^2}{RT} = \frac{4\times10^{-3}\times 952^2}{8.3\times 273} \approx 1.6$
$C_p = \gamma C_v = 1.6\times 5.0 = 8.0$ $JK^{-1}mol^{-1}$