Question

At what temperature does the$rms$speed of $\left(a\right)$a nitrogen molecule and $\left(b\right)$a hydrogen molecule equal the escape speed from the earth's surface? $\left(c\right)$You'll find that these temperatures are very high, so you might think that the earth's gravity could easily contain both gases. But not all molecules move with$V_{rms}$. There is a distribution of speeds, and a small percentage of molecules have speeds several times $V_{rms}$ . Bit by bit, a gas can slowly leak out of the atmosphere as its fastest molecules escape. A reasonable rule of thumb is that the earth's gravity can contain a gas only if the average translational kinetic energy per molecule is less than $1\%$of the kinetic energy needed to escape. Use this rule to show why the earth's atmosphere contains nitrogen but not hydrogen, even though hydrogen is the most abundant element in the universe.

Short answer

$\left(a\right)$. The temperature for nitrogen is$1.4\cdot {10}^5\mathrm{K}$.
$\left(b\right)$ The temperature for hydrogen is localid="1648535515937" ${10}^4\mathrm{K}$.

$\left(c\right)$ The temperature of the atmosphere is less than the maximum allowed temperature for nitrogen but higher than the maximum allowed temperatur for hydrogen.

Step-by-step solution

Step: 1 a Finding the Temperature for Nitrogen: (part a)

The $rms$speed formula is given by

$v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}$

The yield is

$v_{\mathrm{rms}}^2=\frac{3RT}{M}$

Solving for $T$we find

$T=\frac{M{v}_{\mathrm{rms}}^2}{3R}$

The molar mass of Nitrogen diatomic molecular gas $N_2$is $28\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$mol$}\right.$. The escape velocity for Earth is $11.2\raisebox{1ex}{$km$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$. Setting $V_{rms}$equal to this value we find for the temperature

$T=1.4\cdot {10}^5\mathrm{K}.$

Step: 2 b Finding the Temperature for Hydrogen: (part b)

The $rms$speed is given by

$v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}$

The yield is

$v_{\mathrm{rms}}^2=\frac{3RT}{M}$

Solving for we find

$T=\frac{M{v}_{\mathrm{rms}}^2}{3R}$

The molar mass of Hydrogen diatomic molecular gas $H_2$is $2\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$mol$}\right.$. The escape velocity for Earth is $11.2\raisebox{1ex}{$km$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$. Setting $V_{rms}$ equal to this value we find for the temperature

$T={10}^4\mathrm{K}$.

Step: 3 c Kinetic energy :

The average kinetic energy is proportional to temperature. As a result, the temperatures for Nitrogen and Hydrogen should be within a few degrees of each other. $1\%$of the values calculated in parts $a$. and $b$. i.e.

$T_{{\mathrm{N}}_2}\le 1400\mathrm{K},T_{{\mathrm{H}}_2}\le 100\mathrm{K}$

This criterion is met for nitrogen but not for hydrogen because the real temperature of the atmosphere is approximately.$300\mathrm{K}>100\mathrm{K}$.