Kinetic Theory of Gases & Escape Speed

[emperrotta]

Homework Statement

In the first 2 parts of the problem, I calculated at what temperatures the rms speeds v_rms of H₂ molecules and O₂ molecules would equal the escape speed of Earth (11200 m/s). The answers I calculated for this were 10000 K for H₂ and 160000 K for O₂. Those answers are correct according to the back of the textbook.

The last part of the question states: Considering the answers to parts (a) and (b), should there be much hydrogen and oxygen high in the Earth's upper atmosphere, where the temperature is about 1000 K?Attempt at a solution
According to the back of the textbook, the answer is no for H₂ and yes for O₂.

My thought process was the answer is yes for both because the temperature of 1000 K high in the Earth's upper atmosphere is small relative to both the temperatures I calculated (10000 K for hydrogen and 160000 K for oxygen). Thus the temperatures of either gas molecules will not reach a high enough temperature to have a large fraction of the molecules of either gas reach the escape speed of Earth.

How is the answer no for hydrogen? What am I missing here?

 

[Gear300]

I'm not sure how high up in the atmosphere 1000 K would correspond to, but the escape speed taken in respect to that elevation should be less than when taken at the surface of Earth.

Just a side question...the atmosphere would be 1000 K for when it is facing the sun, right (that seems a bit high otherwise)?

 

[baba89]

I would like to clarify that the answer is technically correct according to the calculations, but it may not be a complete explanation of the situation. The Kinetic Theory of Gases states that the average kinetic energy of gas molecules is directly proportional to the temperature. This means that at higher temperatures, gas molecules have higher average speeds.

In this case, the temperatures calculated for H2 and O2 are much higher than the temperature of the Earth's upper atmosphere (1000 K). This means that at this temperature, both H2 and O2 molecules will have average speeds much lower than the escape speed of Earth. However, the question is asking about the likelihood of these molecules reaching the escape speed, not just their average speeds.

At lower temperatures, the distribution of molecular speeds shifts towards lower speeds, with a smaller fraction of molecules having very high speeds. This means that at the temperature of 1000 K, the fraction of H2 molecules with speeds equal to or higher than the escape speed of Earth is much lower than the fraction of O2 molecules with speeds equal to or higher than the escape speed.

Therefore, while it is still unlikely for either H2 or O2 molecules to reach the escape speed at 1000 K, it is even less likely for H2 molecules compared to O2 molecules. This is why the answer is no for H2 and yes for O2.