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  4. The temperature, at which the root mean square velocity o f hydrogen molecules equals their escape velocity from the earth, is closest to: [Boltzmann Constant kB = 1.38 × 10-23 : J/K Avogadro Number NA = 6.02 × 1026 /kg Radius of Earth: 6.4 × 106 m Gravitational acceleration on Earth = 10 ms-2]

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Q. The temperature, at which the root mean square velocity o f hydrogen molecules equals their escape velocity from the earth, is closest to : [Boltzmann Constant $k_B = 1.38 \times 10^{-23} \: J/K$
Avogadro Number $N_A = 6.02 \times 10^{26} /kg$
Radius of Earth : $6.4 \times 10^6 \; m $
Gravitational acceleration on Earth = $10\,ms^{-2}$]

JEE MainJEE Main 2019Kinetic Theory

Solution:

$v_{rms} = \sqrt{\frac{3RT}{m}} v_{escape} = \sqrt{2gR_{e}} $
$ v_{rms} = v_{escape } $
$\frac{3RT}{m} = 2gR_{e} $
$\frac{3\times1.38 \times10^{-23} \times6.02 \times10^{26}}{2} \times T $
$= 2\times10 \times6.4 \times10^{6} $
$T = \frac{4\times 10\times 6.4\times 10^{6}}{3\times 1.38\times 6.02\times 10^{3}} = 10\times 10^{3} = 10^{4} k $