Escape Velocities of Gases

Escape velocity is the minimum speed an object must travel to break free from a planet's gravitational pull. For Earth, this speed is 11.1 km/s at an altitude of 100 km. Understanding escape velocity helps explain why lighter gases like hydrogen gradually leave Earth's atmosphere while heavier gases remain.

According to kinetic theory, hydrogen molecules attain an rms speed equal to Earth's escape velocity at approximately 99,700 K. This temperature is several orders of magnitude higher than the 250 K found at 100 km altitude. Since actual atmospheric temperatures are far lower, hydrogen molecules cannot escape through thermal motion alone.

Although average hydrogen molecules lack escape velocity, the distribution of molecular speeds means some molecules move faster than others at any instant. During collisions, individual hydrogen molecules occasionally reach speeds exceeding escape velocity, allowing them to escape. Over billions of years, this process has removed nearly all hydrogen from Earth's atmosphere.

Heavier molecules have lower rms speeds than hydrogen at the same temperature. The probability that oxygen or nitrogen molecules reach escape velocity is extremely small, requiring billions of years to lose significant amounts. Their greater mass makes atmospheric escape far less likely than for lightweight hydrogen.

The kinetic theory of an ideal gas relates average kinetic energy directly to temperature. By equating the rms kinetic energy formula to the escape velocity energy and solving for temperature, scientists determine the thermal conditions required for gas molecules to escape. This calculation uses the gas molecule's mass and Avogadro's number.

The Moon's gravitational pull is much weaker than Earth's, resulting in a lower escape velocity. Consequently, gas molecules on the Moon reach escape velocity more easily, and the Moon has lost nearly its entire atmosphere over time. The same escape velocity principles apply to both bodies, but the Moon's weaker gravity accelerates atmospheric loss.

Molecular collisions continuously redistribute kinetic energy among gas molecules, causing individual speeds to fluctuate. At each collision, a molecule's speed changes, creating a finite probability that it temporarily exceeds escape velocity. Over geological timescales, these rare high-speed events allow lighter gases to gradually escape despite average temperatures being far too low.