Why Can Gas Molecules Escape the Atmosphere Despite Low Average Speeds?

[UrbanXrisis]

I'm reading in my physics book that:

"A rule of thumb used by astrohysicists is that a gas will escape from a planet's atmosphere in 10^8 years if the average speed of its molecules is 1/6th of the escape velocity."

they said this statement without any derivatins and I was windering why this is true? I mean, the molecules is not even near excape speed so how is it possible that the molecules can excape the Earth's atmosphere?

 

[chroot]

The average speed is 1/6th of the escape velocity, but there is a distribution of speeds. Some of the molecules at the top of the atmosphere have velocities in excess of escape velocity, and thus escape. Over 10^8 years, essentially all of them will have escaped.

- Warren

 

[UrbanXrisis]

so eventually, the molecules that are on the bottom of the atomsphere will eventually rise to the top and then slowly gain speeds that are above excape velocity?

what causes these gases to rise? and why do they gain speed at the top of the atmosphere?

 

[dav2008]

Let's set the basics down first. The average kinetic energy of a gas is proportional to its temperature. It follows that the average velocity of the gas is proportional to the square root of temperature.

This does not mean that in a sample of gas at a temperature T all of the molecules will be traveling at a certain velocity. There is a distribution called the Maxwell-Boltzmann distribution that (basically) gives the probability of a molecule having a certain speed, from 0 to infinity.(Or I guess Since the molecules are constantly colliding it follows that a certain molecule will have its velocity and speed change frequently.

Here is a chart of different gasses and their distributions to give you an idea:
http://en.wikipedia.org/wiki/Image:MaxwellBoltzmann.gif

Here is a very good program to visualize this:
http://intro.chem.okstate.edu/1314f00/laboratory/GLP.htm

The red marking follows one particular molecule. If you increase the number of molecules in the sample you'll see the speed of the red molecule frequently changing. You can also change the temperature to see what effect it has on the distribution.

 

[UrbanXrisis]

so... molecules collide so that it transfer's their KE to other molecules where they reach excape velocity. I'm guessing that because of the changing seasons, during the summer, the molecules that transferred their KE to give other molecules excape velocity will be rembursed with speed because of the increase in temperature and so the cycle continues. is this the correct way of thinking?

 

[dav2008]

Not quite...I think you're missing the point here.

Take the atmoshphere. According to your first post, the average speed of the gas particles is 1/6th that of the escape velocity. That doesn't mean that all of the particles are traveling at 1/6th the escape velocity. Some are traveling slower while others are traveling faster as shown by the distribution I posted in my previous post.

The point is that there is a certain, albeit small, probability of a particle having the enough speed to escape the atmosphere.