The temperature at which the r.m.s. speed of | vedclass

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(D) The escape velocity from the Earth's surface is $v_{e} = 11.2 \, km/s = 11.2 	imes 10^3 \, m/s$.The $r.m.s.$ speed of gas molecules is given by $

Vedclass · Accepted Answer

$10063$

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(D) The escape velocity from the Earth's surface is $v_{e} = 11.2 \, km/s = 11.2 	imes 10^3 \, m/s$.The $r.m.s.$ speed of gas molecules is given by $v_{rms} = \sqrt{\frac{3RT}{M}}$.Equating the two,we have $\sqrt{\frac{3RT}{M}} = v_{e}$.Squaring both sides,$\frac{3RT}{M} = v_{e}^2$,which implies $T = \frac{M v_{e}^2}{3R}$.For hydrogen gas $(H_2)$,the molar mass $M = 2 	imes 10^{-3} \, kg/mol$ and the universal gas constant $R = 8.31 \, J/(mol \cdot K)$.Substituting the values: $T = \frac{(2 	imes 10^{-3}) 	imes (11.2 	imes 10^3)^2}{3 	imes 8.31}$.$T = \frac{2 	imes 10^{-3} 	imes 125.44 	imes 10^6}{24.93} = \frac{250880}{24.93} \approx 10063 \, K$.

The temperature at which the $r.m.s.$ speed of hydrogen molecules is equal to the escape velocity on the Earth's surface will be ...... $K$.

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