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### Course: Physics library>Unit 10

Lesson 1: Temperature, kinetic theory, and the ideal gas law

# The Maxwell–Boltzmann distribution

The Maxwell–Boltzmann distribution describes the distribution of speeds among the particles in a sample of gas at a given temperature. The distribution is often represented graphically, with particle speed on the x-axis and relative number of particles on the y-axis. Created by Sal Khan.

## Want to join the conversation?

• If some air particles are moving faster than sound, wouldn't they perform a tiny sonic boom? Or is that just possible for objects with greater mass? This made me curious...
• Wow, mind blown! Cool question and I think the answer is "no sonicboom":
Particles in air will be able to travel at speeds greater than the speed of sound (300m/s) but I don't think this will cause a sonic boom because think about what sound is: sound is caused by the movement (compression and rarefractions) of molecules in the air. Sonic booms do not exist in space because there is no air and sound can therefore not travel. For the tiniest molecule in air, there is practically no such thing as air for sonic boom: the air molecule is floating in a vacuum with other air molecules but it does not have anything to "cause" a boom through any medium.
• If the particles around us are traveling so quickly, why don't we hear sonic booms? I understand that the particles are INCREDIBLY small, yes, but when you sum enough tiny things up, you get something of decent size...
• its a nice question.

the sonic boom is due to motion and pressure between air particles so the movement of the particles themselves would have nothing to 'boom' against.
• How often do air particles collide/interact? Wouldn't the kinetic energy of each particle approach the same velocity over time because faster particles would impart their momentum to slower particles?
• Its an interesting thought.

Now I am thinking you could be right. Intuitively it kind of makes sense.
On the other hand, I think that may only be the case if some energy is lost each time there is a collision. But, according to the theory, the total amount of ke remains constant. So your slow particle simply becomes the faster particle.

Can I suggest a thought experiment: How about thinking about just two partilces in a box. and considering the energy changes as they collide. What would be the mechanism by which they would tend towards equal energies over time??

I will chew on it further but let us know your thoughts

IM
• At Sal says, " If we have the same no. of molecules, the areas under the curves need to be the same".
Why is that?
• Because the area under the curve represents the number of molecules. This is why when you change the temperature, the graph changes shape but the area will always remain constant because the number of molecules does not change.
• At Sal says that the area under the graphs has to be the same because the number of molecules is the same. But the green system has about 30 percent more energy. Should the green system have 30 percent more area?
• The area under the curve in this case refers to the mass (number of molecules) in the system, not the energy they have. More energy does not make molecules randomly appear in an ideal gas, so it would not affect the mass, or the area under the curve.
The number of molecules/mass here is the same, so the areas under the curves must be the same.
The "speed" on the x-axis just shows the speed which the different molecules fall into, but the area itself refers to the number of molecules, on the y-axis.
• At 6.02, I didn't really understand why the area of A must be equal to the area of B (In the coordinate plane).
• The two graphs are for the same container of gas at two different temperatures. The graphs is of the number of molecules at the various speeds. The total area of the graph is the number of molecules in the container so since it is the same container of gas the total number of molecules is the same so the areas have to be the same.
• Why does the area under the curve represent the number of molecules?
• because it is a distribution graph. so if you drew, straight down, a line from every point of the curve, the length of each line is the number of particles with a given speed. add up all the lines and you get the area under the curve which, then, is the same as the number of particles.
• I'm confused, it seems on the graph as if A and B have different number of molecules, but I thought they both have 10 molecules?
Even if B is supposed to have a narrower graph, but shouldn't A and B have the same number of molecules? They only differ in speed and temperature correct?
• Indeed, they have both 10 molecules. The areas under the graphs should be identical (although the drawing is not perfect, I agree). They indeed only differ in speed and temperature (and resulting from that kinetic energy as well).
(1 vote)
• Why the distribution is a curve ?
(1 vote)
• The Boltzmann distribution shows the number of particles with an energy and displays this as an average. Not all particles will have the exact same amount of energy. The same way that when you have water at 30C, then 30C is technically the average temperature of all the water. Individual water molecules may have more or less energy and therefore be at different temperatures.