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- [Instructor] Collision theory

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can be related to
Maxwell-Boltzmann distributions.

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And first we'll start
with collision theory.

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Collision theory says that
particles must collide

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in the proper orientation and
with enough kinetic energy

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to overcome the activation energy barrier.

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So let's look at the reaction
where A reacts with B and C

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to form AB plus C.

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On an energy profile, we
have the reactants over here

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in the left.

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So A, atom A is colored red,

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and we have molecule BC over here,

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So these two particles must collide

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for the reaction to occur,

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and they must collide with enough energy

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to overcome the activation energy barrier.

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So the activation energy
on an energy profile

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is the difference in energy

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between the peak here, which
is the transition state

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and the energy of the reactants.

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So this energy here is
our activation energy.

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The minimum amount of energy necessary

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for the reaction to occur.

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So if these particles
collide with enough energy,

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we can just get over this
activation energy barrier

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and the reactions can turn
into our two products.

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If our reactant particles
don't hit each other

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with enough energy, they
simply bounce off of each other

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and our reaction never occurs.

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We never overcome this
activation energy barrier.

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As an analogy, let's think
about hitting a golf ball.

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So let's imagine we have a hill,

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and on the right side of the hill,

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somewhere is the hole down here,

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and the left side of the
hill is our golf ball.

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So we know we have to hit this
golf ball with enough force

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to give it enough kinetic energy

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for it to reach the top of the hill

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and to roll over the hill
and go into the hole.

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So we can imagine this hill

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as being a hill of potential energy.

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And this golf ball needs to
have enough kinetic energy

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to turn into enough potential
energy to go over the hill.

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If we don't hit our golf ball hard enough,

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it might not have enough
energy to go over the hill.

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So if we hit it softly,
it might just roll halfway

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up the hill and roll back down again.

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Kinetic energy is equal to 1/2 MV squared.

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And so M would be the
mass of the golf ball

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and V would be the velocity.

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So we have to hit it with enough force

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so it has enough as a high enough velocity

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to have a high enough kinetic energy

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to get over the hill.

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Let's apply collision theory

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to a Maxwell-Boltzmann distribution.

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Usually a Maxwell-Boltzmann distribution

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has fractional particles or
relative numbers of particles

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on the y-axis and particle
speed on the x-axis.

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And a Maxwell-Boltzmann distribution

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shows us the range of speeds
available to the particles

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in a sample of gas.

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So let's say we have,

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here's a particulate diagram over here.

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Let's say we have a sample of gas

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at a particular temperature T.

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These particles aren't
traveling at the same speed,

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there's a range of
speeds available to them.

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So one particle might be
traveling really slowly

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so we'll draw a very short arrow here.

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A few more might be
traveling a little faster,

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so we'll draw the arrow longer
to indicate a faster speed.

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And maybe one particle
is traveling the fastest.

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So we'll give this
particle the longest arrow.

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We can think about the
area under the curve

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for a Maxwell-Boltzmann distribution

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as representing all of the
particles in our sample.

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So we had this one particle
here moving very slowly,

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and so if we look at
our curve and we think

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about the area under the curve

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that's at a low particle speed,

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this area is smaller than
other parts of the curve.

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So that's represented here
by only this one particle

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moving very slowly.

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We think about this
next part of the curve,

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most, this is a large
amount of area in here

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and these particles are
traveling at a higher speed.

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So maybe these three
particles here would represent

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the particles moving at a higher speed.

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And then finally, we had
this one particle here,

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We drew this arrow longer than the others.

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So this particle's traveling
faster than the other one.

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So maybe this area under the curve up here

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is represented by that one particle.

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We know from collision theory,

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that particles have to
have enough kinetic energy

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to overcome the activation
energy for a reaction to occur.

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So we can draw a line
representing the activation energy

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on a Maxwell-Boltzmann distribution.

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So if I draw this line,
this dotted line right here,

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this represents my activation energy.

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And instead of particle
speed, you could think

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about the x-axis as being
kinetic energy if you want.

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So the faster a particle is traveling,

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the higher its kinetic energy.

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And so the area under the curve

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to the right of this dash line,

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this represents all of the particles

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that have enough kinetic energy
for this reaction to occur.

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Next, let's think about what
happens to the particles

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in our sample when we
increase the temperature.

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So when we increase the temperature,

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the Maxwell-Boltzmann
distribution changes.

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So what happens is the peak height drops

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and our Maxwell-Boltzmann
distribution curve gets broader.

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So it looks something like
this at a higher temperature.

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So we still have some particles traveling

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at relatively low speeds, right?

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Remember it's the area under the curve.

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So maybe that's represented
by this one particle here,

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and next, let's think about the area

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to the left of this dash line for Ea.

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So we want to make these
particles green here

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as we have some particles traveling

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a little bit of faster speeds.

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So let me go ahead and draw
these arrows a little bit longer

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but notice what happens to
the right of this dash line.

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We think about the area under the curve

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for the magenta curve.

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Notice how the area is bigger
than in the previous example.

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So maybe this time we have
these two particles here

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traveling at a faster speed.

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So I'm gonna draw these arrows longer

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to indicate they're
traveling at a faster speed.

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And since they're to the
right of this dash line here,

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both of these particles
have enough kinetic energy

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to overcome the activation
energy for our reaction.

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So we can see when you
increase the temperature,

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you increase the number of particles

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that have enough kinetic energy

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to overcome the activation energy.

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It's important to point out

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that since the number of
particles hasn't changed,

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all we've done is increase
the temperature here,

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the area under the curve remains the same.

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So the area under the curve
for the curve in yellow,

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is the same as the area under the curve

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for the one drawn in magenta.

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The difference of course
is the one in magenta

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is at a higher temperature,

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and therefore there are more
particles with enough energy

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to overcome the activation energy.

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So increasing the temperature

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increases the rate of reaction.