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## Equilibrium constant

Current time:0:00Total duration:15:50

# Keq derivation intuition

## Video transcript

I've already made one video on
it, and at least it attempts to give you the intuition behind
how the equilibrium constant formula is derived
or where it comes from. From maybe the probabilities
of different molecules interacting, if they're
in some small volume. But I think I was a little
hand wavy with it, and it might not have been clear
how probabilities and concentrations relate. So what I thought I would do
in this video is kind of do the same exercise, but do it
with real numbers and a real reaction, so just a's,
b's and c's. So what I wrote here is
the Haber process. This is how we get ammonia in
the world and feed everyone. Ammonia is a very important
fertilizer, but that's beside the point. The Haber process, which you
see right here, is in equilibrium, which doesn't mean
that the concentrations are the same. In fact, this is an equilibrium
concentration that I worked out before starting
this video. Notice, the concentrations of
nitrogen and hydrogen are very different than the concentration
of ammonia, which is much less. What equilibrium tells us is
that once we get to this concentration of nitrogen and
hydrogen, the rate of reaction of going in the rightward
direction is the same as the rate of reaction of going in the
leftward direction, when we have this much ammonia. So let's just think about what
that rate of reaction means. And then I'll tell you
how I think about it. At least how I think about it is
that if you have some small volume-- Let's call that dV. You could kind of arbitrarily
pick how small. So dV. And the way it works in my head
is that if you pick some small volume in this solution--
we don't know how large of a solution we actually
have. We just have the concentrations-- that in
this volume, you're just as likely to have a reaction going
into that direction as you are to have a reaction
going in the backwards direction. So let's think about what the
probability is of having a reaction in this volume. So the probability, let's say
of the forward reaction. Probability of N2 plus
3H2, going in this forward direction. And whatever I do for this
direction, then you just have to use the same logic for
the backward direction. I just want to give you the
intuition that it's equal to some constant related to
their concentration. So the probability of going in
that direction to 2NH3 in our little box, I claim-- and I
think this will hopefully make some sense-- it's equal to
first, the probability that they react given in the box. So if you know that if you
have the constituent particles, you have one nitrogen
molecule-- which has two nitrogen atoms in it-- and
three hydrogen molecules. If you know you have those,
there's some probability that they're going to react based
on their configurations and their kinetic energy and how
they're approaching each other and all of these different
types of things. So this is the probability they
react given that they're in this little box of dV. And then, of course, you're
going to have to multiply that times the probability in the
box, that you have the constituent particles
in the box. Now, my claim is that
this piece right here, this is a constant. If you know at a certain
temperature, the Haber process, these concentrations,
this would happen at 300 degrees Celsius-- I just
looked that up. No need to memorize something
like this. But equilibrium constants hold
at a certain temperature. So I'm claiming that if I give
you a temperature-- say, 300 degrees Celsius-- and if I
tell you that I have one nitrogen and three molecules of
hydrogen in your box, that there's some constant
probability that they react. I mean, it depends on their
configuration and all of that. So I'll just call this the
constant-- I'll just make up a constant probability, whatever
it is, or in the box. I could write anything there. So what we should be concerned
with is what is the probability that we have those
four molecules-- three molecules of hydrogen
and two molecules of nitrogen-- in the box. So this is equal to
some constant. I'll call it the constant of
probability of react-- or let me say react. That's a good one: react. The constant of reaction-- if
you have it in the box-- times the probability that
they're in the box. Let me draw the box. So we want to know the
probability, where this box is just some volume, that I have
three hydrogen molecules. So one, two, three. And one nitrogen molecule. And we should pick a box that's
small enough so that that would be indicative of how
close the molecules need to get to actually react. So I'm just going to pick my
dV to be-- I don't know. Let's pick my dV to be-- I
looked up the diameter of an ammonia molecule. It was about 1/10
of a nanometer. If this was a nanometer box,
you could put 10 in each direction, so you can almost fit
1,000 if you packed them really tightly. So let's make this half a
nanometer in each direction. So if I pick my dV-- and
remember, I don't know if this is the right distance. I'm just trying to give you
the intuition behind the equilibrium formula. But if I pick this as being
0.5 nanometers by 0.5 nanometers by 0.5 nanometers,
what is my volume? So my little volume is going
to be 0.5 times 10 to the minus 1/9 meters-- that's a
nanometer-- to the third power, because we're dealing
with cubic meters. So this is equal to 0.5
to the 1/3 power. That's what? 0.5 times 0.5 is 0.25
times 0.5 is 0.125. I want to do the math right, so
let me just make sure I got that right. 0.5 to the 1/3 power. Right, 0.125 times-- negative
9 to the 1/3 power is minus 27-- 10 to the minus
27 meters cubed. So that's my volume. Now, we know the
concentration. Let's figure out what's
the probability. So this is the probability
in the box, right? That's what we're concerned
with, the probability in the box. Well, the probability in the
box, that's the probability that I have one hydrogen in the
box, times the probability that I have another hydrogen
in the box, times the probability that I have another
hydrogen in the box-- these are all in-the-box
probabilities-- times the probability that I have
a nitrogen in the box. I'll do the nitrogen in a
different color just to ease-- oh, I should've done these in
the orange because those are the color of the molecules
up there. And I'll do this
one in purple. What's the probability of having
hydrogen in the box? Well, we know hydrogen's
concentration at equilibrium is 2 Molar. So concentration of hydrogen,
we know hydrogen's concentration is equal to 2
Molar, which is 2 Moles per liter, which is equal to-- 2
Moles is just 2 times 6 times 10 to the twenty-third power--
Moles is just a number-- divided by liters. So 1 liter is-- we could write
it in meters cubed, or we could just make the
conversion. Actually, let me just
do this for you. 1 liter is equal to 1 times 10
to the minus 3 meters cubed. If you actually take a meter
cubed, you can actually put 1,000 liters in there. So the other way you could say
this is 1 times 10 to the minus 3 meters cubed, and then
if we want to figure out our dV times-- how many dV's do
we have per meter cubed? Or how many meter cubes
are their per dV? So we know that already, so
it's 0.125 times 10 to the minus 27 meters cubed per
our volume, right? I just got that from up here,
that I have a small fraction of a meter cubed
per my volume. And now, I just have
to do some math. So let's see, I can cancel out
some things first, because there's a lot of
exponents here. So let's see, if I take the
twenty-third-- so let me write it out here. So my hydrogen per box-- So my
concentration of hydrogen per dV, is equal to 12 times
10-- whoops! That's not helping when
my pen malfunctions. Let me get that right. 12 times 10 to the twenty-third
power times 0.125 times 10 to the minus
twenty-seventh power. All of that divided by 10
to the minus 3, right? That's 1 times 10
to the minus 3. So let's cancel out
some exponents. If we get rid of the minus 3
here, you divide by minus 3, then this becomes minus 24. And then the minus 24 and the
minus-- so this is equal to-- what's 12 times 1.25? So times 12 is equal to 1.5. So the 12 times the 1.25 is
equal to 1.5 times-- and then 10 to the twenty-third times 10
to the minus twenty-fourth is equal to 10 to the
minus 1, right? So it's just divided by 10. So on average, your
concentration of hydrogen in a little cube that's half a
nanometer in each direction is equal to 0.15 molecules-- not
Moles anymore-- of hydrogen molecule per my little
dV, my little box. And so this is a probability,
right? This is a probability, because
obviously I can't have 0.15 molecules in every box. This is just saying, on average,
there's a 0.15 chance that I have a hydrogen
molecule in my box. So if I want to go back here to
this, this is 0.15, this is 0.15, this is 0.15. But how did we get this 0.15? We multiplied the concentration
of hydrogen, which was this right here. That's the concentration of
the hydrogen-- I should've written it in a more vibrant
color-- times just a bunch of scaling factors, right? We could just say that, well,
this was just equal to the concentration of hydrogen times,
based on how I picked my dV, I had to do all
of this scaling. But it was times some constant
of scaling, scaling to my appropriate factor. So if we want to figure out each
of these, this is just the concentration of hydrogen
times some scaling factor. And this is going to
be the same thing. We could do the same exercise
right here. We figured out the exact value
with the hydrogen, but you could do the same thing
with the nitrogen. In fact, nitrogen's
concentration is just half of the hydrogen, so we know it. It's going to be half of that
0.15, so it's going to be 0.075, which is just equal to
the concentration of nitrogen times some scaling factor. It's actually going to be
the same scaling factor. So let's go back to our
original problem. So our probability that the
forward reaction is going to occur in the box is going to be
equal to some probability that is going to react-- given
that you're on the box, that's some constant value-- times the
probability that they're in the box. And I'm making the claim that
that's equal to all of these things multiplied
by each other. So that's the concentration of
hydrogen times some scaling factor, some other scaling
factor-- I'll call it K sub s-- times the concentration of
hydrogen times some scaling factor, times the concentration
of hydrogen times some scaling factor, times
the concentration of nitrogen times some
scaling factor. And what is that equal to? Well, if you combine all the
constants, a bunch of scaling constants times the constant
out here, that all just becomes a constant. So you get the probability of
the forward reaction in the box is going to be equal to
just some constant-- let's just call it constant
forward-- times the concentration of the hydrogen
to the third power-- I multiplied it three
times-- times the concentration of nitrogen. Now, if you wanted to go in
the reverse direction, probability of reverse, you
could use the exact same argument that I just used, and
I'm not going to do it just for the sake of time, but
it'll be some constant. This is the constant that the
ammonia will react in the reverse direction on its own,
times the scaling factor, and all of that. But it's the same exact idea. So, times the reverse, which is
just going to be-- How many Moles of ammonia do we have? Or how many molecules? What's its stoichiometric
coefficient? It's 2. So the reverse direction is
going to be concentration of ammonia to the second power. And when we're in equilibrium,
these two things, the probability of having a forward
reaction in the box, is going to be equal to the
probability of a reverse reaction the box. So these two things are going
to equal each other. So this is going to equal-- if
I could just copy and paste it-- that up there. There you go. Then if you set the constants
equal to each other, and then you could pick what the-- you
normally put the products on the right-hand side
of the equation. So I'll take these and divide
them into this, and I'll divide that into that, and
you're left with KF/KR is equal to the concentration of
ammonia to the second power, divided by the concentration
of hydrogen to the third power, times the concentration
of nitrogen. And you could call that the
equilibrium constant. And there you have it. A pseudo-derived formula for
the equilibrium constant. It's all, at least in my mind,
coming from common sense, from the probability that if you have
a small volume, things are actually going to react.