$4.0 ext{ g}$ of a gas occupies $22.4 ext{ L}$ at NTP. The specific heat capacity of the gas at constant v

Question

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

Accepted Answer

1. **Find Molar Mass ($M$):**
   At NTP (Normal Temperature and Pressure, $T = 273 	ext{ K}, P = 1 	ext{ atm}$), $1 	ext{ mole}$ of an ideal gas occupies $22.4 	ext{ L}$. 
   Given that $4.0 	ext{ g}$ of the gas occupies $22.4 	ext{ L}$, the molar mass of the gas is:
   $$M = 4.0 	ext{ g mol}^{-1} = 4.0 	imes 10^{-3} 	ext{ kg mol}^{-1}$$
2. **Calculate the Ratio of Specific Heats ($\gamma$):**
   The speed of sound $v$ in a gas is given by Laplace's formula :
   $$v = \sqrt{\frac{\gamma R T}{M}}$$
   Squaring both sides and solving for $\gamma$:
   $$\gamma = \frac{v^2 M}{R T}$$
   Substitute the given values ($v = 952 	ext{ ms}^{-1}$, $M = 4.0 	imes 10^{-3} 	ext{ kg mol}^{-1}$, $R = 8.3 	ext{ J K}^{-1}	ext{mol}^{-1}$, $T = 273 	ext{ K}$):
   $$\gamma = \frac{(952)^2 	imes 4.0 	imes 10^{-3}}{8.3 	imes 273} = \frac{906304 	imes 0.004}{2265.9} = \frac{3625.216}{2265.9} \approx 1.6$$
3. **Calculate Heat Capacity at Constant Pressure ($C_p$):**
   We know that the ratio of molar heat capacities is $\gamma = \frac{C_p}{C_v}$ .
   Given $C_v = 5.0 	ext{ J K}^{-1}	ext{mol}^{-1}$, we can find $C_p$:
   $$C_p = \gamma 	imes C_v = 1.6 	imes 5.0 = 8.0 	ext{ J K}^{-1}	ext{mol}^{-1}$$

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