Why do lighter atoms and molecules diffuse upwards?

Question

If a relatively light atom such as helium is released in the middle of a room, it will tend to diffuse or random-walk upwards. If a relatively heavy atom such as argon is released in the middle of a room, it will tend to diffuse or random walk downward. Can anyone provide a simple explanation from a microscopic point of view of why this should be so?

I'm imagining a helium atom that is suddenly released in the middle of a room as being like a small ping-pong ball that is interacting with and bouncing off the (heavier) ping-pong balls representing the nitrogen and oxygen air molecules around it. Why would there be a net random-walk or diffusional movement of the helium atom in the upward direction and of an argon atom in the opposite direction?

Answer 1

It's a statistical walk, but its biased. Consider the energy in the system containing all of the molecules. One factor is the potential energy due to gravity for all molecules, and the other factor is the velocity of the particles which is in a random direction. Due to thermodynamics, we know that the system will proceed to the system which has the highest entropy, which in this case means preferring the configurations which have less potential energy.

The actual mechanic which biases this walk is that the path of the molecules is not straight. It is a more ballistic trajectory due to gravity. The effect of this gravitation term is tiny on the order of the free path lengths of the atoms (62nm for atmospheric pressure), but it's there. This means that a particle which would see equal collisions from all directions sees that the collisions from below are slightly lower velocity than the ones from above. This means the collisions from below will balance the conservation of momentum and conservation of energy differently because one is dependent on $v$, while the other is dependent on $v^2$. The heavier molecules will slowly proceed towards lower altitudes, while the lighter ones get bounced upwards by this series of uneven collisions at higher velocities than they are bounced downwards. Think of it like a stacked ball drop. If, in the collision, the heavier ball is on the bottom, the conservation of energy and momentum will send the small ball skyrocketing. If the smaller ball is on the bottom, the bigger ball doesn't go very far at all. This effect will be occurring in all directions in a gas, but because the collision velocities from above and below are different, the effect will have a tendency to send smaller molecules higher more often than it will send them lower.

Answer 2

Let us consider a 5 km tall air tank where cool air is at atmospheric pressure. At the bottom of the tank free path lengths of the atoms are 62nm. At the top of the tank free path lengths of the atoms are 124nm.

Now let us consider an atom that diffuses from the bottom to the top. The free path length of that atom increases by 62nm from 62nm to 124nm.

Let's say the average speed of the atom is 100m/s and it took million seconds to move from bottom to top.

So the atom traveled 100 million meters, which consists of, let's say 100 million meters / 62nm paths, that is 1.6*10^15 free paths.

Now let's consider an upwards directed free path, let's say the free path length increases during this free path by 62nm * (62nm / 5km) , that is by 7.7* 10^-19 meters.

Conversely during a downwards directed free path free path length shortens by 7.7* 10^-19 meters.

From the above we can conclude that upwards directed free paths are about 7.7* 10^-19 meters longer that downwards directed free paths.

So now let's say an atom moves 0.8*10^15 times a distance of 7.7 10^-19 meters, well that is only 0.000616 meters total distance. From that we can conclude that the million seconds time was much too short. It takes 8116883 times more time for an atom to bounce from the bottom to the top. That is 257384 years.

To answer the question: An atom tends to float upwards, because upwards directed free paths are very slightly longer than downwards directed free paths. (The atom floats upwards if its weigh is low enough or its speed is high enough)

Upwards directed free paths are longer than downwards directed free paths, because free path are longer at upper areas than lower areas.

And free paths are longer at upper areas, because gases are thinner at upper areas.

If a helium atom is released in the middle of a room, after a minute it will have moved downwards at probability 0.499999 and upwards at probability 0.500001.