At what temperature $(K)$ would rms speed of $C O_{2}$ molecule be equal to $H_{2}$ at $27^{\circ} C$ .

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Solution

Since, the rms velocity is inversely proportional to the square root of the molar mass, rms velocity of $C O_{2}$ will be less than $H_{2}$ at the same temperature. In order for $C O_{2}$ to have the same rms velocity as that of $H_{2}$ , then the temperature must be increased.

${rms}_{C O_{2}} at T K = {rms}_{H_{2}} at 300 K$
$\Rightarrow {(\sqrt{\frac{3 R T}{M}})}_{C O_{2}} = {(\sqrt{\frac{3 R 300}{M}})}_{H_{2}}$
$\Rightarrow \frac{3 \times R \times T}{44 \times 10^{- 3}} = \frac{3 \times R \times 300}{2 \times 10^{- 3}}$
$\Rightarrow T = 6600 K$

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