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# Keq derivation intuition (can skip; bit mathy): A more concrete attempt at showing how the probabilities of molecules reacting is related to their concentration.

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- 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.

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