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# 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 23rd 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 subs -- 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.