Collision theory and the Maxwell–Boltzmann distribution

- [Instructor] Collision theory can be related to Maxwell-Boltzmann distributions. And 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 in the left. So A, atom A is colored red, and we have molecule BC over here, So these two particles must collide for the reaction to occur, and they must collide with enough energy to overcome the activation energy barrier. So 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. So 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 reactions 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. So let's imagine we have a hill, and on the right side of the hill, somewhere is the hole down here, and the left side of the hill is our golf ball. So 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 and go into the hole. So 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 enough 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 1/2 MV squared. And so M would be the mass of the golf ball and V would be the velocity. So we have to hit it with enough force so it has enough as 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 a Maxwell-Boltzmann distribution has fractional particles or relative numbers of particles on the y-axis and particle speed on the x-axis. And a Maxwell-Boltzmann distribution shows us the range of speeds available to the particles in a sample of gas. So let's say we have, here's 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. So 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. And 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 a Maxwell-Boltzmann distribution as representing all of the particles in our sample. So we had this one particle here moving very slowly, and so if we look at our curve and we think about the area under the curve that's at a low particle speed, this area is smaller than other parts of the curve. So that's represented here by only this one particle moving very slowly. We think about this next part of the curve, most, this is a large amount of area in here and these particles are traveling at a higher speed. So maybe these three particles here would represent the particles moving at a higher speed. And then finally, we had this one particle here, We drew this arrow longer than the others. So this particle's traveling faster than the other one. So maybe this area under the curve up here is represented by that one particle. We know from collision theory, that particles have to have enough kinetic energy to overcome the activation energy for a reaction to occur. So we can draw a line representing the activation energy on a Maxwell-Boltzmann distribution. So if I draw this line, this dotted line right here, this represents my activation energy. And instead of particle speed, you could think about the x-axis as being kinetic energy if you want. So the faster a particle is traveling, the higher its kinetic energy. And so the area under the curve to the right of this dash line, this 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. So when we increase the temperature, the Maxwell-Boltzmann distribution changes. So what happens is the peak height drops and our Maxwell-Boltzmann distribution curve gets broader. So it looks something like this at a higher temperature. So we still have some particles traveling at relatively low speeds, right? Remember it's the area under the curve. So maybe that's represented by this one particle here, and next, let's think about the area to the left of this dash line for Ea. So we want to make these particles green here as we have some particles traveling a little bit of faster speeds. So let me go ahead and draw these arrows a little bit longer but notice what happens to the right of this dash line. We think about the area under the curve for the magenta curve. Notice how the area is bigger than in the previous example. So maybe this time we have these two particles here traveling at a faster speed. So I'm gonna draw these arrows longer to indicate they're traveling at a faster speed. And since they're to the right of this dash line here, 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. So 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.