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## Q. 9.3

Compute the mean molecular speed $\bar{v}$ in the light gas hydrogen ($H _{2}$) and the heavy gas radon (Rn), both at room temperature 293 K. (Use the longest-lived radon isotope, which has a mass of 222 u.) Compare the results.

Strategy The mean speed is given by Equation (9.17) as

$\bar{v}=\frac{4}{\sqrt{2 \pi}} \sqrt{\frac{k T}{m}}$ (9.17)

With the temperatures the same, we need to use the appropriate molecular masses to complete the computation. The lighter mass (hydrogen) will have the higher average speed.

## Verified Solution

The mass of the hydrogen molecule is twice that of a hydrogen atom (neglecting the small binding energy), or 2(1.008 u) = 2.02 u. Thus the average molecular speed of hydrogen is

\begin{aligned}\bar{v} &=\frac{4}{\sqrt{2 \pi}} \sqrt{\frac{k T}{m}} \\&=\frac{4}{\sqrt{2 \pi}} \sqrt{\frac{\left(1.38 \times 10^{-23} J / K \right)(293 K )}{(2.02 u )\left(1.66 \times 10^{-27} kg / u \right)}}=1750 m / s\end{aligned}

The average molecular speed of radon is

\begin{aligned}\bar{v} &=\frac{4}{\sqrt{2 \pi}} \sqrt{\frac{k T}{m}} \\&=\frac{4}{\sqrt{2 \pi}} \sqrt{\frac{\left(1.38 \times 10^{-23} J / K \right)(293 K )}{(222 u )\left(1.66 \times 10^{-27} kg / u \right)}}=167 m / s\end{aligned}

The hydrogen molecule is more than 10 times faster, on average. That’s to be expected, because its mass is more than 100 times lighter. Most other gases have molecular masses that fall between these two extremes, so their mean speeds should be between the two values we computed here.