Thermodynamics involving escape velocity and Boltzmann factor

In summary, despite the lower root mean square speed of helium atoms at 293K compared to the escape velocity at the Earth's surface, it is still possible for helium atoms to escape into space due to the decreasing temperature at higher altitudes. This is supported by the Boltzmann factor, which shows that the probability of a molecule having enough energy to escape decreases as the temperature decreases. This also explains why nitrogen does not escape as readily as helium. Additionally, the higher temperature near the top of the atmosphere plays a role in allowing helium atoms to escape. This is evident in the case of Saturn's moon Titan, which has a similar size to the Earth's moon but is able to retain a significant atmosphere due to its higher density and warmer
  • #1
bbrain
3
0
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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  • #2
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 ...
 
  • #3
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.
 
  • #4
That tells you the probability doesn't it? So if E is higher then the probability is lower, and vice versa?
 
  • #5
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?
 
  • #6
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).
 
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  • #7
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)?
 
  • #8
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?
 

Related to Thermodynamics involving escape velocity and Boltzmann factor

1. What is escape velocity?

Escape velocity is the minimum speed needed for an object to escape the gravitational pull of a celestial body, such as a planet or moon. It is dependent on the mass and radius of the body.

2. How is escape velocity related to thermodynamics?

In thermodynamics, escape velocity is used to calculate the minimum energy required for a particle to leave a system. This is important in understanding the behavior of gases and other particles in a system.

3. What is the Boltzmann factor?

The Boltzmann factor is a mathematical expression used in thermodynamics to describe the probability of a particle having a certain energy level. It is dependent on the temperature and energy of the system.

4. How is the Boltzmann factor related to escape velocity?

The Boltzmann factor is used in the equation for escape velocity, as it describes the probability of a particle having enough energy to escape a system. The higher the temperature and energy of the system, the higher the probability of particles having enough energy to escape.

5. Can the Boltzmann factor be used to predict the escape of particles from a system?

Yes, the Boltzmann factor can be used to calculate the probability of a particle escaping from a system. The higher the energy and temperature of the system, the higher the probability of particles escaping. However, there are other factors such as the mass and radius of the system that also play a role in determining the escape of particles.

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