To escape the Earth's gravity, an object near the top of the atmosphere at an altitude of 100 km must travel away from Earth at 11.1 km/s. This speed is called the escape velocity. The temperature at which gas molecules attain the rms speed, which is equal to the escape velocity, can be estimated by using the equation for the average kinetic energy of the gas molecules. According to the kinetic theory of gas, the average kinetic energy of the gas molecules is proportional to its temperature. The higher the temperature, the higher the rms speed of the gas molecules.
For hydrogen molecules, the rms speed equal to the escape velocity is attained at a temperature of 99.7 x 102 K. This temperature is a few orders of magnitude higher compared to the temperature at an altitude of 100 km away from the Earth's surface. At such altitudes, the temperature is roughly 250 K. This simply means that the probability of hydrogen molecules escaping the Earth's atmosphere is negligible. However, on the contrary, very few hydrogen molecules are left in the Earth's atmosphere. The hydrogen loss occurs because a few molecules have speeds higher than the Earth's escape velocity, even at normal temperatures. The speed of a hydrogen molecule changes from one collision to the next; hence, at any instant, there is a small but finite chance that the molecule's speed is greater than the escape velocity. The chance is high enough that over the lifetime of Earth, almost all the hydrogen molecules in the atmosphere have reached the escape velocity at high altitudes and escaped from the Earth's gravitational pull.
However, heavier molecules, such as oxygen, nitrogen, and water, have lower rms speeds, so it is much less likely that any of them will have speeds greater than the escape velocity. The likelihood is so small that billions of years are required to lose significant amounts of heavier molecules from the atmosphere. The moon's lack of atmosphere can be explained using the same concept. Since the Moon's gravitational pull is much weaker, it has lost almost its entire atmosphere.
