First slide

Rms speed

Question

At what temperature will the rms speed of oxygen molecules become just sufficient for escaping from the Earth's atmosphere? (Given : Mass of oxygen molecule $(m) = 2.76 \times 10^{- 26} kg, Boltzmann's constant k_{B} = 1.38 \times 10^{- 23} {JK}^{- 1})$

Moderate

A

$2.508 \times 10^{4} K$
B

$8.360 \times 10^{4} K$
C

$5.016 \times 10^{4} K$
D

$1.254 \times 10^{4} K$

Solution

Escape velocity from the Earth's surface is $v_{escape} = 11200 m s^{- 1}$
Say at temperature T, oxygen molecule attains escape velocity.
$So, v_{escape} = \sqrt{\frac{3 k_{B} T}{m_{O_{2}}}} \Rightarrow 11200 = \sqrt{\frac{3 \times 1.38 \times 10^{- 23} \times T}{2.76 \times 10^{- 26}}}$
$On solving, T = 8.360 \times 10^{4} K$

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