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

- [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.