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## Chemistry library

### Unit 17: Lesson 3

Arrhenius equation and reaction mechanisms- Collision theory
- The Arrhenius equation
- Forms of the Arrhenius equation
- Using the Arrhenius equation
- Collision theory and the Maxwell–Boltzmann distribution
- Elementary reactions
- Reaction mechanism and rate law
- Reaction mechanism and rate law
- The pre-equilibrium approximation
- Multistep reaction energy profiles
- Catalysts
- Types of catalysts
- Types of catalysts

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# Collision theory and the Maxwell–Boltzmann distribution

AP.Chem:

TRA‑4 (EU)

, TRA‑4.B (LO)

, TRA‑4.B.1 (EK)

, TRA‑4.B.2 (EK)

, TRA‑4.B.3 (EK)

Collision theory states that in order for a reaction to occur, reactant particles must collide with enough kinetic energy to overcome the activation energy barrier. A Maxwell–Boltzmann distribution shows the distribution of particle energies at a given temperature and allows for a qualitative estimation of the fraction of particles with sufficient energy to react at that temperature. Created by Jay.

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## Video transcript

- [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.