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The Boltzmann Distribution: Temperature and Kinetic Energy of Gases

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  • 0:02 Gas Particles Are in…
  • 1:28 Predicting Particle Velocity
  • 4:11 Changes in Probabilities
  • 5:38 Lesson Summary
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Lesson Transcript
Instructor: Sarah Friedl

Sarah has two Master's, one in Zoology and one in GIS, a Bachelor's in Biology, and has taught college level Physical Science and Biology.

Gas particles are always moving around at random speeds and in random directions. This makes it difficult to determine what any one particle is doing at a given time. Luckily, the Maxwell-Boltzmann distribution provides some help with this.

Gas Particles Are in Constant Motion

Places like Siberia and Antarctica are really, really cold. But they're no match for absolute zero; this is the lowest possible temperature. Absolute zero is a temperature of 0 Kelvin (or about -460 °F)! This is so cold that scientists have not yet been able to reach this temperature, but they sure have gotten close. This is a state where the motion of particles comes to a near stop because there is so little energy left in the substance.

Thankfully, we don't live in a world of absolute zero, which means that things are not that cold and that particles of objects have energy and are in motion. For some objects, like solids, this motion is more like vibration. This is why solid objects stay the same shape - their particles like to hang out together in close proximity.

Gases are quite different, though. Gaseous particles really like their personal space and are in a constant state of random motion. They fly around in random directions, filling the entire space of the container they occupy. This random motion doesn't mean that they fly around all willy-nilly, though. The particles fly in a straight-line path until they collide with something else - either the boundary of the container or other gas particles that are also flying around randomly. When these collisions occur, they cause kinetic energy to be transferred, and this changes the particle's direction of motion.

Predicting Particle Velocity

But gaseous particles are also traveling at random speeds as well as in random directions. Some particles move quite slowly, while others are traveling very fast, and the direction and speed of the particles is changing all the time. So it wouldn't make sense to measure all of the particles' velocities because there would be so much variation, and it would be difficult to pinpoint what one gas particle is doing at any given time.

What we can do instead is look at a probability distribution of the speed of any given particle of gas in the container. Like any probability distribution, it tells us which situation is more likely to occur. This distribution is called the Maxwell-Boltzmann distribution, named for James Clerk Maxwell and Ludwig Boltzmann.

If we pick any random gas particle from our container it will fall somewhere on this curve, though it is more likely to be a certain speed than others. This is because as one particle collides with another, it will either gain kinetic energy (and therefore gain speed) from the particle it collided with or lose kinetic energy to that particle (and therefore slow down). The distribution of speeds stays the same because there is no change in the system, just a shuffling of change through the individual particles.

It's like this - say you're at a party where everyone carpooled to get there. You might have a total of 20 people at this party, and they all arrived in a total of five cars but with different numbers of people in each car. So one car might have brought only one person, while other cars brought two, three or even five people. At the end of the night, if everyone drives home with different people than they came with you might have five people leaving in a car that originally brought only one person to the party and two people leaving in a car that originally brought four people. The same number of people and cars are coming and going from the party, but each car has a different number of people.

Now imagine that the people are kinetic energy and speed and the cars are gas particles and you can apply this same idea to gases in closed systems! When gas particles collide, some particles gain kinetic energy and speed but others then have to lose energy and speed to keep the total energy of the system the same.

The distribution of these speeds is represented under the curve of probabilities. What it shows us is that some speeds are more likely than others for any given particle because of this shuffling. Most particles fall within a certain range of speeds, but for that small number that has very high speeds, there are a similar number of particles traveling at very low speeds. In this way, they essentially balance each other out in the overall system.

Changes in Probabilities

The shape of the Maxwell-Boltzmann distribution can change depending on a couple of factors. First, the average speed of the gas molecules is affected by temperature. Higher temperatures mean more kinetic energy (faster speeds), while lower temperatures mean less kinetic energy (slower speeds). So when the temperature of the system is increased, the curve flattens out and spreads over a wider area.

It's like adding more cars to the carpool party. There are more ride options than before so the distribution spreads out. But it also means the probability of any one person getting in a specific car is less than before, so the curve flattens a bit too.

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