Examine the data in Tables 42.1 and 42.2, and consider that...

Examine the data in Tables 42.1 and $42.2,$ and consider that the average speed of a molecule in a gas with temperature $T$ is approximately \[ V_{m o l}=0.15 \mathrm{km} / \mathrm{sec} \times \sqrt{T / m} \] where $T$ is the temperature in Kelvin and $m$ is the mass of the molecule in atomic mass units; so $m\left(\mathrm{H}_{2}\right)=2,$ and $m\left(\mathrm{CO}_{2}\right)=$ 44. About how many times smaller than the escape velocity does $V_{m d}$ have to be for a planet to retain a gas?

Examine the data in Tables 42.1 and $42.2,$ and consider that the average speed of a molecule in a gas with temperature $T$ is approximately
\[
V_{m o l}=0.15 \mathrm{km} / \mathrm{sec} \times \sqrt{T / m}
\]
where $T$ is the temperature in Kelvin and $m$ is the mass of the molecule in atomic mass units; so $m\left(\mathrm{H}_{2}\right)=2,$ and $m\left(\mathrm{CO}_{2}\right)=$
44. About how many times smaller than the escape velocity does $V_{m d}$ have to be for a planet to retain a gas?

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Key Concepts

Kinetic Theory of Gases

This theory connects the motion of individual gas molecules with macroscopic observables like temperature and pressure. It explains that higher temperatures lead to faster molecular speeds, and that the speed depends inversely on the square root of molecular mass, as indicated by the formula Vmol = 0.15 km/sec × ?(T/m).

Escape Velocity

Escape velocity is the minimum speed needed for an object to overcome a planet’s gravitational pull without further propulsion. It is a key factor in determining whether gas molecules can remain in a planet's atmosphere or escape into space.

Atmospheric Retention Criteria

For a planet to retain its gas, the average molecular speed of the gas must be significantly lower than the escape velocity. A common rule-of-thumb is that the molecular speed should be about one-sixth as fast as the escape velocity to minimize the likelihood that gas molecules will achieve enough speed to escape.

Effect of Molecular Mass on Molecular Speed

The mass of gas molecules influences their thermal speed; lighter molecules move faster than heavier ones at the same temperature. This effect is critical in planetary science because it helps determine which gases a planet can hold onto, with lighter gases like hydrogen being more prone to escape compared to heavier gases such as carbon dioxide.

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