How do collisions between gas molecules result in equal average kinetic energy?

[chemistry1]

Hi,

I was wondering, how is it possible that different gases can have the same average kinetic energy?(at same temperature) Can anybody give me a SIMPLE explanation of this? Knowing that some particles will move faster than others, I wonder how this can be possible :/

It would really help ! Thank you

 

[Yanick]

From Kinetic Theory of Gases we get the relation:

KE_avg = (3/2)kT, also KE_avg = (1/2)m(v_avg)². Where k is the Boltzmann constant, T is temperature, m is mass, v is velocity and KE is kinetic energy.

The derivation is slightly annoying so you can believe it or look for the derivation yourself to convince yourself its valid.

You can see from the first expression above that the average KE of a gas is dependent only on its temperature. So when you have a mixture of two (ideal) gases at the same temperature they must have the same average KE. In order to satisfy the second relation (KE_avg(v_avg)) smaller gases, with smaller mass, must have a higher average velocity than the larger gas molecules and vice versa.

Keep in mind that these are all averages and that the entire system of molecules will have a distribution of velocities and kinetic energies governed by the Maxwell distribution. We are allowed to derive things by using the concepts of averages and such because even a tiny volume will have a huge amount of molecules contained within (~10¹⁹ molecules of an ideal gas in a cubic centimeter of volume at STP).

I hope this helps.

 

[Ygggdrasil]

It basically works out this way because collisions between gas molecules will tend to equalize the average kinetic energy of gas particles in the system.

Consider for example, a system that consists of a large number of identical gas molecules, half of whom are moving very slowly and half are moving very rapidly. Will this situation last? No, the fast gas molecule will eventually collide with the slow molecules, transfering kinetic energy from the fast molecules to the slow molecules until eventually, all molecules are moving on average at the same speed (sometimes collisions will slow the molecules down, and sometimes they'll be moving faster than the average because of collisions, but one molecules over time the average kinetic energy will be the same across the entire population).

Now consider a population of two gasses, one heavy and one light. What will the effect of these random collisions be? Will they all move at the same velocity? No, because when a heavy molecule collides with a light molecule, the larger molecule's momentum will transfer more kinetic energy to the light molecule than if a light molecule hits a heavy molecule. Thus, when things reach an equilibrium, the lighter molecules should be moving faster than the heavy molecules. If you do the math (probably the thing to look to for a formal proof is something called the equipartition theorem), you'll see that the situation where gas particles have equal kinetic energies is what results from the random collision of gas molecules.