Thermodynamics involving escape velocity and Boltzmann factor

[bbrain]

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?

 

[rude man]

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 ...

 

[rude man]

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.

 

[bbrain]

That tells you the probability doesn't it? So if E is higher then the probability is lower, and vice versa?

 

[bbrain]

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?

 

[VantagePoint72]

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).

 

[VantagePoint72]

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)?

 

[rude man]

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?