Question

At what temperature, hydrogen molecules will escape from the earth's surface? (Take mass of hydrogen molecule $=0.34\times 10^{-26}kg,$ Boltzmann constant $-1.38\times 10^{-23}J{K}^{-1},$ Radius of earth $=6.4\times 10^{6}m$ and acceleration due to gravity $=9.8m{s}^{-2}$ )

• A. 10 K
• B. $10^{2}K$
• C. $10^{3}K$
• D. $10^{4}K$

Answer 1

Answer: (D)

The root mean square velocity of gas is $v_{ms}=\sqrt{\frac{3kT}{m}}$ .....(i) Escape velocity of gas molecules is $v_{cs}=\sqrt{2g{R}_e}$ .....(ii) As the root mean square velocity of gas molecules must be equal to the escape velocity. = From Eqs. (i) and (ii), we get $\sqrt{\frac{3kT}{m}}=\sqrt{2g{R}_e}$ $\Rightarrow T=\frac{2g{R}_{e}m}{3k}$ $\Rightarrow T=\frac{2\times 9.8\times 6.4\times 10^6\times 0.34\times 10^{-26}}{3(1.38\times 10^{-23})}$ $=10^{4}K$ Therefore, $10^{4}K$ is the temperature at which hydrogen molecules will escape from earth's surface.