Question

The escape velocity on earth is 11.2 km/s. What fraction of the escape velocity is the rms speed of H2 at a temperature of 31.0 degrees Celsius on the earth? Note that virtually all the molecules will have escaped the earth's atmosphere if this fraction exceeds 0.15.

Answer 1

To solve this problem it is necessary to apply the concept related to root mean square velocity, which can be expressed as

$v_{rms} = \sqrt{\frac{3RT}{n}}$

Where,

T = Temperature

R = Gas ideal constant

n = Number of moles in grams.

Our values are given as

$v_e =11.2km/s = 11200m/s$

The temperature is

$T = 30\°C = 30+273 = 303K$

Therefore the root mean square velocity would be

$v_{rms} = \sqrt{\frac{3(8.314)(303)}{0.002}}$

$v_{rms} = 1943.9m/s$

The fraction of velocity then can be calculated between the escape velocity and the root mean square velocity

$\alpha = \frac{v_{rms}}{v_e}$

$\alpha = \frac{1943.9}{11200}$

$\alpha = 0.1736$

Therefore the fraction of the scape velocity on the earth for molecula hydrogen is 0.1736