Calculating Temperature: Escape Velocity of Helium Atoms

Identify the knowns and unknowns and determine which equations to use to solve the problem.

1. Identify the knowns: $v$ is the escape velocity, 11.1 km/s.

2. Identify the unknowns: We need to solve for temperature, $T$. We also need to solve for the mass $m$ of the helium atom.

3. Determine which equations are needed.

• To solve for mass $m$ of the helium atom, we can use information from the periodic table:
  $$m=\frac{\text{molar mass}}{\text{number of atoms per mole}}.$$

  13.62
• To solve for temperature $T$, we can rearrange either
  $$\overline{\text{KE}}=\frac{1}{2}m\overline{v^2}=\frac{3}{2}\text{kT}$$

  13.63

  or

  $$\sqrt{\overline{v^2}}=v_{\text{rms}}=\sqrt{\frac{3\text{kT}}{m}}$$

  13.64

  to yield

  $$T=\frac{m\overline{v^2}}{\text{3}k},$$

  13.65
  where $k$ is the Boltzmann constant and $m$ is the mass of a helium atom.

4. Plug the known values into the equations and solve for the unknowns.

This temperature is much higher than atmospheric temperature, which is approximately 250 K $(–\text{25}\text{º}\text{C}$ or $–\text{10}\text{º}\text{F})$ at high altitude. Very few helium atoms are left in the atmosphere, but there were many when the atmosphere was formed. The reason for the loss of helium atoms is that there are a small number of helium atoms with speeds higher than Earth’s escape velocity even at normal temperatures. The speed of a helium atom changes from one instant to the next, so that at any instant, there is a small, but nonzero chance that the speed is greater than the escape speed and the molecule escapes from Earth’s gravitational pull. Heavier molecules, such as oxygen, nitrogen, and water (very little of which reach a very high altitude), have smaller rms speeds, and so it is much less likely that any of them will have speeds greater than the escape velocity. In fact, so few have speeds above the escape velocity that billions of years are required to lose significant amounts of the atmosphere. Figure 13.25 shows the impact of a lack of an atmosphere on the Moon. Because the gravitational pull of the Moon is much weaker, it has lost almost its entire atmosphere. The comparison between Earth and the Moon is discussed in this chapter’s Problems and Exercises.