To solve this problem it is necessary to apply the concept related to root mean square velocity, which can be expressed as
[tex]v_{rms} = \sqrt{\frac{3RT}{n}}[/tex]
Where,
T = Temperature
R = Gas ideal constant
n = Number of moles in grams.
Our values are given as
[tex]v_e =11.2km/s = 11200m/s[/tex]
The temperature is
[tex]T = 30\°C = 30+273 = 303K[/tex]
Therefore the root mean square velocity would be
[tex]v_{rms} = \sqrt{\frac{3(8.314)(303)}{0.002}}[/tex]
[tex]v_{rms} = 1943.9m/s[/tex]
The fraction of velocity then can be calculated between the escape velocity and the root mean square velocity
[tex]\alpha = \frac{v_{rms}}{v_e}[/tex]
[tex]\alpha = \frac{1943.9}{11200}[/tex]
[tex]\alpha = 0.1736[/tex]
Therefore the fraction of the scape velocity on the earth for molecula hydrogen is 0.1736
