physical chemistry - Why don't gases escape Earth's atmosphere?

Some gases are lighter than others and rise. Why don't they continue going up, leave the atmosphere, and then enter outer space?

Answer

The atmosphere actually loses gases to outer space.

The average velocity $\bar v$ of gas molecules is determined by temperature $T$. However, not all the molecules travel with the same velocity. The probability of finding a molecule with a velocity near $v$ is described by the Maxwell distribution of speeds

$$\begin{align} f{\left(v\right)}&=4\pi\sqrt{{\left(\frac m{2\pi kT}\right)}^3}v^2\exp\left(-\frac{m{v^2}}{2kT}\right)\\[6pt] &=4\pi\sqrt{{\left(\frac M{2\pi RT}\right)}^3}v^2\exp\left(-\frac{M{v^2}}{2RT}\right) \end{align}$$ where $m$ is the mass of the molecule, $k$ is the Boltzmann constant, $M$ is the molar mass of the gas, and $R$ is the molar gas constant.

Individual molecules may reach escape velocity $v_\mathrm e$ and thus be able to leave the atmosphere.

Escape velocity is the minimum velocity that is sufficient for an object to escape from the gravitational attraction of a massive body. For a planet, the escape velocity may be estimated by using the formula

$${v_\mathrm e}=\sqrt{\frac{2Gm_\text{planet}}r}$$ where $G$ is the gravitational constant, $m_\text{planet}$ is the mass of the planet, and $r$ is the distance from the centre of mass of the planet.

Therefore, atmospheric escape depends on the mass of the planet, the temperature of the atmosphere, and the molar mass of the gas.