Collision theory and the Maxwell–Boltzmann distribution | Kinetics | AP Chemistry | Khan Academy

Collision theory can be related to Maxwell-Boltzmann distributions. First, we'll start with collision theory. Collision theory says that particles must collide in the proper orientation and with enough kinetic energy to overcome the activation energy barrier.

So let's look at the reaction where A reacts with B and C to form AB plus C. On an energy profile, we have the reactants over here on the left. So A (atom A) is colored red, and we have molecule BC over here. These two particles must collide for the reaction to occur, and they must collide with enough energy to overcome the activation energy barrier.

The activation energy on an energy profile is the difference in energy between the peak here, which is the transition state, and the energy of the reactants. This energy here is our activation energy; the minimum amount of energy necessary for the reaction to occur. So, if these particles collide with enough energy, we can just get over this activation energy barrier, and the reactants can turn into our two products.

If our reactant particles don't hit each other with enough energy, they simply bounce off of each other, and our reaction never occurs. We never overcome this activation energy barrier.

As an analogy, let's think about hitting a golf ball. Imagine we have a hill, and on the right side of the hill somewhere is the hole down here. The left side of the hill is our golf ball. We know we have to hit this golf ball with enough force to give it enough kinetic energy for it to reach the top of the hill and to roll over the hill into the hole. We can imagine this hill as being a hill of potential energy, and this golf ball needs to have enough kinetic energy to turn into potential energy to go over the hill.

If we don't hit our golf ball hard enough, it might not have enough energy to go over the hill. So, if we hit it softly, it might just roll halfway up the hill and roll back down again. Kinetic energy is equal to one half mv squared, so v would be the velocity. We have to hit it with enough force so it has a high enough velocity to have a high enough kinetic energy to get over the hill.

Let's apply collision theory to a Maxwell-Boltzmann distribution. Usually, the Maxwell-Boltzmann distribution has the fraction of particles or relative numbers of particles on the y-axis and particle speed on the x-axis. A Maxwell-Boltzmann distribution shows us the range of speeds available to the particles in a sample of gas.

Let's say we have a particulate diagram over here. Let’s say we have a sample of gas at a particular temperature T. These particles aren't traveling at the same speed; there's a range of speeds available to them. One particle might be traveling really slowly, so we'll draw a very short arrow here. A few more might be traveling a little faster, so we'll draw the arrow longer to indicate a faster speed.

Maybe one particle is traveling the fastest, so we'll give this particle the longest arrow. We can think about the area under the curve for the Maxwell-Boltzmann distribution as representing all of the particles in our sample. We had this one particle here moving very slowly, and if we look at our curve and think about the area under the curve that's at a low particle speed, this area is smaller than other parts of the curve.

This is represented here by only this one particle moving very slowly. If we think about the next part of the curve, most of this is a large amount of area in here, and these particles are traveling at a higher speed. Maybe these three particles here would represent the particles moving at a higher speed. Finally, we had this one particle here that we drew an arrow longer than the others, indicating that this particle is traveling faster than the others.

We know from collision theory that particles have to have enough kinetic energy to overcome the activation energy for a reaction to occur. We can draw a line representing the activation energy on a Maxwell-Boltzmann distribution. If I draw this line, this dotted line represents my activation energy. Instead of particle speed, you could think about the x-axis as being kinetic energy if you want.

So, the higher the faster a particle is traveling, the higher its kinetic energy. The area of the curve under the curve to the right of this dashed line represents all of the particles that have enough kinetic energy for this reaction to occur.

Next, let's think about what happens to the particles in our sample when we increase the temperature. When we increase the temperature, the Maxwell-Boltzmann distribution changes. The peak height drops, and our Maxwell-Boltzmann distribution curve gets broader. It looks something like this at a higher temperature.

We still have some particles traveling at relatively low speeds. Remember, it's the area under the curve. Maybe that's represented by this one particle here. Now let's think about the area to the left of this dashed line for EA. Let's make these particles green here. We have some particles traveling at a little faster speed, so let me go ahead and draw these arrows a little bit longer.

Notice what happens to the right of this dashed line. We think about the area under the curve for the magenta curve. Notice how the area is bigger than in the previous example. Maybe this time we have these two particles here traveling at a faster speed, so I'll draw these arrows longer to indicate they're traveling at a faster speed.

Since they're to the right of this dashed line, both of these particles have enough kinetic energy to overcome the activation energy for our reaction. So we can see when you increase the temperature, you increase the number of particles that have enough kinetic energy to overcome the activation energy.

It's important to point out that since the number of particles hasn't changed, all we've done is increase the temperature here. The area under the curve remains the same. The area under the curve for the curve in yellow is the same as the area under the curve for the one drawn in magenta. The difference, of course, is the one in magenta is at a higher temperature, and therefore there are more particles with enough energy to overcome the activation energy.

So increasing the temperature increases the rate of reaction.