Thermodynamics involving escape velocity and Boltzmann factor

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Homework Help Overview

The discussion revolves around thermodynamics, specifically the escape velocity of gases like helium and nitrogen from Earth, and the implications of the Boltzmann factor in atmospheric retention on celestial bodies like Titan and the Moon.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore why helium atoms can escape Earth's gravity despite their lower root mean square speed compared to escape velocity, questioning the role of temperature and the Boltzmann factor.
  • There is a discussion about the atmospheric retention of Titan compared to the Moon, with participants considering density and temperature differences.
  • Some participants analyze the implications of the Boltzmann factor on the probability of helium molecules achieving escape energy.

Discussion Status

The discussion is active, with participants raising questions about temperature effects on gas behavior and the implications of the Boltzmann factor. There is no clear consensus, but various interpretations and lines of reasoning are being explored.

Contextual Notes

Participants note that the temperature decreases with altitude in the atmosphere, but also mention the thermosphere where temperatures can rise significantly. There is an acknowledgment of the complexity of the problem and suggestions that it may belong in a more advanced forum.

bbrain
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At 293K, helium atoms have a root mean square speed of about 1.35 km/s, whereas escape velocity at the Earth's surface is about 11km/s. Explain why is it nevertheless possible for helium atoms to escape from the Earth into space.

Is it because near the top of the atmosphere the temperature is higher?



Nitrogen does not escape from the Earth nearly as readily as helium. Refer to the Boltzmann factor in your answer.

Not sure about this one.



Saturn's moon Titan is similar in size to the Earth's moon; yet titan has been able to retain a significant atmosphere whereas the Moon has almost none. Refer to the Boltzmann factor in your answer.

Titan is more dense than the Moon?
 
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bbrain said:
Saturn's moon Titan is similar in size to the Earth's moon; yet titan has been able to retain a significant atmosphere whereas the Moon has almost none. Refer to the Boltzmann factor in your answer.

Titan is more dense than the Moon?


Brrr, it sure is chilly here on Titian ...
 
bbrain said:
At 293K, helium atoms have a root mean square speed of about 1.35 km/s, whereas escape velocity at the Earth's surface is about 11km/s. Explain why is it nevertheless possible for helium atoms to escape from the Earth into space.

Is it because near the top of the atmosphere the temperature is higher?
No, it gets colder as you go higher ...

p(E) = (const)exp(-βE) where β = 1/kT is the probability that a He molecule has energy E. This formula enables you to derive the average energy of a helium molecule, but what else does it tell you?

PS this problem belongs in the advanced physics forum in my opinion.
 
That tells you the probability doesn't it? So if E is higher then the probability is lower, and vice versa?
 
rude man said:
No, it gets colder as you go higher ...

p(E) = (const)exp(-βE) where β = 1/kT is the probability that a He molecule has energy E. This formula enables you to derive the average energy of a helium molecule, but what else does it tell you?

PS this problem belongs in the advanced physics forum in my opinion.

That tells you the probability doesn't it? So if T is low the probability is also low?
 
rude man said:
No, it gets colder as you go higher ...

Until you get to the thermosphere, where the temperature goes up to 2000°C due to solar UV and X-ray excitation. Though the important bit is, as you alluded to, the fact that the RMS speed is only root mean squared (since even at 2000°C the RMS speed only increases to 3.76 km/s).
 
Last edited:
bbrain said:
Saturn's moon Titan is similar in size to the Earth's moon; yet titan has been able to retain a significant atmosphere whereas the Moon has almost none. Refer to the Boltzmann factor in your answer.

Titan is more dense than the Moon?

How does the temperature of titan compare to the Moon's (during their respective days)?
 
bbrain said:
That tells you the probability doesn't it? So if T is low the probability is also low?

Right. So is there a finite probability of a molecule being in a high enough state to escape from the Earth?
 

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