91Ó°ÊÓ

We are given that,

escape velocity $=11km/s$

Temperature,$T=1000k$

The probability density function for velocity in a gas is given by the maxwell Boltzmann distribution

$f\left(v\right)={\left(\frac{m}{2\mathrm{Ï€¸é°Õ}}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$$}\right.2}4{\mathrm{Ï€\pm ¹}}^2{\mathrm{e}}^{\raisebox{1ex}{$-{\mathrm{mv}}^2$}\!\left/ \!\raisebox{-1ex}{$$}\right.}\mathrm{RT}$

therefore probability of velocity being greater than $11km/s$is

$P(v>11km/s)={∫}_{11,000}^{∞}f\left(v\right)dv$

$P={∫}_{11,000}^{∞}{\left(\frac{m}{2\mathrm{Ï€¸é°Õ}}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}4{\mathrm{Ï€\pm ¹}}^2\mathrm{e}\frac{-{\mathrm{mv}}^2}{\mathrm{RT}}dv$

Replacing $v↔x\&a=\sqrt{\frac{RT}{2m}}$

we get,

$P={∫}_{11,000}^{∞}\sqrt{\frac{2}{\mathrm{π}}}\frac{x^{2}e{\displaystyle \raisebox{1ex}{$-x^2$}\!\left/ \!\raisebox{-1ex}{$2a^2$}\right.}}{a^3}dx$

This, integral is not simple & we will need to use error function.

$P=erfc\left(\frac{x}{\sqrt{2}a}\right)+\sqrt{\frac{2}{\mathrm{Ï€}}}\frac{xe{\displaystyle \raisebox{1ex}{$-x^2$}\!\left/ \!\raisebox{-1ex}{$2a^2$}\right.}}{a}$

where, $erfc$is the error function complement

$\left\{Here,erfc\left(x\right)=\frac{1}{\sqrt{\mathrm{π}}}{∫}_{-x}^x{e}^{-t^2}dt\right\}$

not possible by hand calculation, so using web tool

$a=\sqrt{\frac{RT}{2m}}$

where $m=$molecular mass of gases$=28amu$

$a=\sqrt{\frac{1.38×{10}^{-23}×1000}{2×28×1.67×{10}^{-27}}}$

$a=385.3$

$\frac{x}{\sqrt{2}a}=\frac{11000}{\sqrt{2}×385.3}=20.188$

$erfc(20.188)≈0.0$

$P=0.0+2e-176$

$P≈0$thus, the $N_2$molecule wont be escaping at all.

We are given that,

For$H_2$,

$m=2v$energy thing else is same.

$m=2×1.67×{10}^{-27}$

Now we put the values in formula,

$a=1441.63$

$\frac{x}{\sqrt{2}a}=5.39$

localid="1650109697287" $erfc\left(\frac{x}{\sqrt{2}a}\right)≈0$

$P=0+1.4e^{-12}$

$Pâ‰\dots 1.4e^{-12}≈0$

thus, probability is still very small of velocity $>11km/s$.

Since the mass of helium lies in between that of hydrogen and $N_2$,the velocity $>11km/s$is also$≈0$.

Thus all these gas molecules are not able to escape the atmosphere.

We are given that,

Since, the escape velocity of moon is $2.4km/s$

The molecule of atmospheric gases on surface of moon can move with thermal velocities greater than the velocities of moon,

Thus, all gas molecules escape and there is no atmosphere left on moon.