vmelkon
National Hazard
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Earth's atmosphere
I was wondering, how can you show mathematically why Earth has an atmosphere. Is the gravitational force strong enough to prevent molecules from
flying off into space?
I'm guessing I can use this equation
v = v_initial - a * t
We can imagine a molecule with a certain speed, let's say 2000 m/s going straight up. Gravity slows it down to 0 m/s and then it falls back to Earth.
0=2000 m/s - 9.8 m/s/s * t
solve for t:
t = 204.08 sec (a molecule going straight up reaches a speed of 0 m/s in 204 seconds)
The distance covered by the molecule is
distance = 2000 m/s * t - 1/2*9.8*t*t
distance = 204081 meters
Does that make sense? That distance seems insanely high.
This isn't homework.
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smaerd
International Hazard
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Well I don't think its safe to assume a single molecule alone travels that far that fast. Recall gas interactions are 'chaotic' and 'randomly' hit one
another transferring kinetic energy and other-wise. To assume that a molecule is shot straight up and then shot straight back down is a bit of an over
reduction in my opinion. It appears you are using a basic kinetics velocity equation without enough 'information'. Also one should assume that not
every molecule or radical going say "up" will come back down  . I don't think
gravity is the best 'lense' for this thought experiment.
There is an 'escape velocity' for the earth's 'atmosphere'. This is where I would start: http://en.wikipedia.org/wiki/Escape_velocity .
Now as I said before if we want to treat molecules in such a way we probably shouldn't look at them the way you attempt too. We should likely seek the
"Kinetic molecular theory" and see what we can come up with. There are some useful derivations here, http://cnx.org/content/m42217/latest/?collection=col11406/la... Even an example of how helium is 'leaking' from the atmosphere. Like I said not
all of them are going to come back "down", in-fact a straight path for a gas would I think require a perfect vacuum or electro-magnetic field and no
other molecules to collide with(sure a physics buff could smack that statement down)? It's probably also important to note that not all of the kinetic
energy in a molecule is going to go towards motion iirc. Many things to take into account such as degrees of freedom intramolecular interactions etc.
In general these things can be omitted to achieve a good enough understanding.
It appears that a useful factor here is the temperature which gives rise to kinetic energy(of course) as well as mass of said particle.
I guess if you want to really over simplify things you're not going to get good results. For example usually when considering velocity of gasseous
molecules and to be simple we use the Vrms and not just velocity.
Also if we are to consider gravity it might be best to actually consider it as a gradient, as the gravity at the top of the atmosphere is not the same
as the 9.8 constant, it does change. A good means of solving for gravitation problems would be the gaussian approach(http://en.wikipedia.org/wiki/Gauss%27s_law_for_gravity). The effect of gravity is quiet small for a molecule anyways as they have such little
mass, electrostatic interactions dominate at the molecular level. Don't get me wrong with enough "g" forces we can separate mixture such as how a
centrifuge works(centrifugal force), but the scale is a big issue when it comes to Earth vs a single solitary molecule.
[Edited on 22-3-2013 by smaerd]
edit but actually the earths atmosphere is about 480km(according too http://www.enchantedlearning.com/subjects/astronomy/planets/...), so your result falls within that range. Not that I'd really consider it a good
means but for a physics 1 student its not bad. Once you get to physics two(if you haven't already) these types of ideas should become more of a
reality.
[Edited on 22-3-2013 by smaerd]
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macckone
Dispenser of practical lab wisdom
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To get our atmosphere you have to show that molecules lighter than N2 escape while N2 and heavier do not. And that has to be for all but a small
number of particles.
The thermodynamic temperature is going to play into that as well as gravity and temperature/pressure gradient.
http://en.wikipedia.org/wiki/Thermodynamic_temperature
http://apollo.lsc.vsc.edu/classes/met130/notes/chapter1/vert...
Obviously not a class assignment because proving that is more of a masters thesis.
Oh and don't forget tidal effects of the moon which are thought to have a very large role in formation of our current atmosphere.
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