# Sound travels with a speed $v$ in hydrogen gas at NTP. At what temperatures does carbon-di-oxide have a rms (root mean square) velocity of $2v$ (Note: $T_0 = 273K$ and $\gamma$ for $H_2 = \large\frac{7}{5}$)

(A) $4T_0$ (B) $41T_0$ (C) $44T_0$ (D) $444T_0$

Toolbox:
• Molar mass of $CO_2$ is 44.00964 ± 0.00003 g/mol
• Molar mass of $H_2$ is 2.015894 ± 0.000002 g/mol
• $v_{rms} = \large \sqrt \frac{3k_BT}{M}$, where $k_B$ is the Boltzman's constant and M is the molar mass of the gas and T is the temperature.
• $v_{sound} = \large \sqrt \frac{\gamma k_BT}{M}$, where $k_B$ is the Boltzman's constant and M is the molar mass of the gas, $\gamma$ is the adiabatic index and T is the temperature.
• $\gamma$ for $H_2 = \large\frac{7}{5}$
$v_{rms} = \large \sqrt \frac{3k_BT}{M}$, where $k_B$ is the Boltzman's constant and M is the molar mass of the gas and T is the temperature.
$v_{rms} CO_2 = \large \sqrt \frac{3k_BT_1}{44}$
$v_{sound} = \large \sqrt \frac{\gamma k_BT}{M}$, where $k_B$ is the Boltzman's constant and M is the molar mass of the gas, $\gamma$ is the adiabatic index and T is the temperature.
$v_{sound}H_2 = \large \sqrt \frac{7 k_BT_0}{5M}$
$2 \times v_{sound} H_2 = v_{rms} CO_2 \rightarrow 2 \large \sqrt \frac{7k_B\times T_0}{5\times2.01} $$= \large \sqrt \frac{3k_BT_1}{44} T_1 = \large\frac{4 \times 7\times T_0 \times 44}{5 \times 2 }$$\approx 41 T_0$
edited Jan 24, 2014