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(D) The root mean square speed $(u_{rms})$ of a gas is given by the formula: $u_{rms} = \sqrt{\frac{3RT}{M}}$.Here,$R$ is the gas constant,$T$ is the

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$H_2$ at $-50\,^{\circ}C$

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(D) The root mean square speed $(u_{rms})$ of a gas is given by the formula: $u_{rms} = \sqrt{\frac{3RT}{M}}$.Here,$R$ is the gas constant,$T$ is the temperature in Kelvin,and $M$ is the molar mass of the gas.To maximize $u_{rms}$,we need a high temperature $(T)$ and a low molar mass $(M)$.Let's calculate the relative values for each:$A$: $O_2$ $(M=32)$,$T=273\,K \implies \frac{T}{M} = \frac{273}{32} \approx 8.53$$B$: $N_2$ $(M=28)$,$T=1273\,K \implies \frac{T}{M} = \frac{1273}{28} \approx 45.46$$C$: $CH_4$ $(M=16)$,$T=298\,K \implies \frac{T}{M} = \frac{298}{16} \approx 18.63$$D$: $H_2$ $(M=2)$,$T=223\,K \implies \frac{T}{M} = \frac{223}{2} = 111.5$Comparing the values of $\frac{T}{M}$,the gas with the highest value is $H_2$ at $-50\,^{\circ}C$.

Which of the following gases has the highest root mean square speed $(u_{rms})$ at the given conditions?

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