Helium gas flows out of a thermally insulated vessel into a vacuum through a small hole. The flow ra

Question

Helium gas flows out of a thermally insulated vessel into a vacuum through a small hole. The flow rate of the gas inside the vessel is assumed to be negligible under these conditions. The temperature of the helium in the vessel is $T = 1000 	ext{ K}$. Assuming the gas undergoes adiabatic expansion into the vacuum, find the rate $v$ (velocity of efflux) with which the helium flows out. (Enter the value in km/s, rounded to two decimal places. Take $R = 8.314 	ext{ J/(mol K)}$ and molar mass of Helium $M = 4 	ext{ g/mol}$).

Professor PC Thomas Classes · Answer

By the steady flow energy equation (or Bernoulli's equation for compressible adiabatic flow), the sum of enthalpy and kinetic energy per mole is constant: $H_1 + \frac{1}{2}Mv_1^2 = H_2 + \frac{1}{2}Mv_2^2$. Inside the vessel, the flow velocity is negligible so $v_1 \approx 0$, and the temperature is $T_1 = 1000 	ext{ K}$. For free expansion into a vacuum, the gas expands fully, converting all its initial enthalpy into the kinetic energy of the flow, meaning $T_2 	o 0$ and $H_2 	o 0$. Thus, $H_1 = \frac{1}{2}Mv^2$. The enthalpy of an ideal gas per mole is $H = C_p T = \frac{\gamma R T}{\gamma - 1}$. Helium is a monatomic gas, so its adiabatic exponent is $\gamma = 5/3$. Thus, $C_p = \frac{5}{2}R$. Equating the enthalpy to kinetic energy: $\frac{5}{2} R T = \frac{1}{2} M v^2 \implies v = \sqrt{\frac{5 R T}{M}}$. Substituting the given values: $v = \sqrt{\frac{5 	imes 8.314 	imes 1000}{0.004}} = \sqrt{10392500} \approx 3223.7 	ext{ m/s} = 3.22 	ext{ km/s}.$