Question

At what temperature would the average speed of helium atoms equal the escape speed from the Earth, 1.12 x 10^4 m/s, the mass of a helium atom being = 6.64310227 kg

Answer 1

The average speed of helium atoms can be found from the formula:

$$v_{avg,He}=\sqrt{\dfrac{8k_BT}{\pi m}}.$$

In order to find the temperature at which the average speed of helium atoms equals the escape speed from the Earth we must equate $v_{avg,He}$ and $v_{esc}$:

$$v_{esc}=\sqrt{\dfrac{8k_BT}{\pi m}},$$ $$T=\dfrac{\pi mv_{esc}^2}{8k_B},$$ $$T=\dfrac{\pi\cdot6.64\cdot10^{-27}\ kg\cdot(1.12\cdot10^4\ \dfrac{m}{s})^2}{8\cdot1.38\cdot10^{-23}\ \dfrac{J}{K}}=2.37\cdot10^4\ K.$$