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# Keq derivation intuition

## Video transcript

I've already made one video on and at least attempts to give you the intuition behind how the equilibrium constant formula is derived or where it comes from from you know 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 and it might have not been clear how 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 so what I did and a real reaction not just with A's B's and C's so what I wrote here is the Haber process 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 are you see right here it's in equilibrium which means that it 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 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 problem or 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 you know how I think about it so at least how I think about it is that if you have some small volume some small volume let's call that DV it doesn't I mean you could kind of arbitrarily pick how small so DV though in 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 are 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 the forward reaction probability of n 2 + 3 H 2 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 - - and nh3 in in our in our little box right I claim and I think this this will hopefully make some sense it's equal to you know first the probability probability that they react given in the box so if you know that if you have the Constituent particles you have one or or 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 that you react given that they're in this little box of DV and then of course you're gonna have to multiply that times times the probability in the Box in the box that you have the Constituent particles in the Box now my claim is is that this piece right here this is a constant if you know at a certain temperature you know the Haber process this these concentrations these this would happen at 300 degrees Celsius I just look 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 nitrogen one one nitrogen and three molecules of hydrogen in your box that that there's some constant probability that they react right I mean you know it depends on their configuration and all of that so I also call this the 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 what is the probability that we have those those four molecules right three molecules of hydrogen and two molecules of nitrogen in the box so we want we want to figure out so this is equal to some constant I'll call it the constant of probability of react or let me say react that's good one react the constant of reaction if you have 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 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 you know that that that would be indicative of how close the molecules need to get to actually react so I'm just going to pick my my DV to be I don't know let's pick my DV to be I looked up the the diameter or the the diameter of a of an ammonia molecule it was about a tenth of a of a nanometer so you could put ten of these if you this this was a nanometer box you could put ten in each direction so you can almost fit a thousand I guess if you packed them really tightly so I don't know let's make this I'll say half a nanometer in each direction so if I pick my DV and remember I'm just I don't know if this is the right 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 what is my volume so my my little volume is going to be 0.5 times 10 to the minus ninth meters right that's a nanometer to the third power because we're dealing with cubic meters so this is equal to 0.5 to the third power that's what 0.5 times 0.5 is 0.25 times 0.5 is one point one two five nine one two five I want to do the math right so let me just make sure I got that right point five to the third power right point one two five times negative nine to the third power is minus 27 10 to the minus 27 meters cube so that's my volume now we know the concentration let's let's figure out what's the probability so this is the probability in the box right that's what we are 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 have done these in the orange those was 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 two molar so concentration of hydrogen we know hydrogen concentration is equal to two molar which is two moles per liter which is equal to 2 moles just 2 times 6 times 10 to the 23rd power that's moles is just a number divided by liters so liters is one liter is we could write this as well we could write it in meters cubed or we could just make the conversion 1 Li actually let me just do this for you one liter is equal to 1 times 10 to the minus 3 meters cubed if you actually take a meter cubed you can actually put a thousand liters in there so we could say so the other way you could you could say this is this is well I'll do it I'll just say it so 1 times 10 to the minus 3 meters cubed and then if we want to figure out our DV times how many DVS do you have per per meter cubed or how many meter cubes are there 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 if I multiply let's see I can cancel out some things first there's a lot of exponents here so let's see if I take the 23rd so let me write it out here so hydrogen so my hydrogen per box so my concentration of hydrogen per DV is equal to 12 let me do that 12 times 10 whoops it's not helping with my pen malfunctions let me get that right 12 times 10 to the 23rd power times 0.125 times 10 to the minus twenty seventh power all of that divided by 10 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 my so this is equal to what's 12 times 1.25 so so x times 12 is equal to 1.5 so the 12 times the 0.125 is equal to 1.5 times and then 10 to the 23rd times 10 to the minus 24 is equal to 10 to the minus 1 right so it's just divided by 10 so your hydrogen per on average your concentration of hydrogen in a little cube that's half a nanometer in each direction is equal to 0.15 molecules molecules 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 point 1 5 molecules in every box this is just saying on average there's a point 1 5 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 point one five we multiplied the concentration where is it we multiplied the concentration of hydrogen which was this right here that's the concentration of the hydrogen this is concentration of a hydrogen I shoot in a more vibrant color times just a bunch of scaling factors right we could just say that well this was just equal to sum' the concentration of hydrogen times you know 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 this each of these this is just the concentration of hydrogen times some scaling factor and this is going to be the same thing I mean 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 nitrogens concentration is just half of the hydrogen so we know it it's going to be half of that point one five 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 original problem so our probability that we're going to 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 come scaling factor times the concentration of nitrogen times some scaling factor and what is that equal to well if you if you combined all the constants right a bunch of con scaling constants times constant I hear that all just becomes a constant so you get the probability of the forward reaction forward in the box is going to be equal to just some constant let's call it constant forward times the concentration of the hydrogen to the third power right I multiplied it three times times the concentration of nitrogen now if you wanted to go in the reverse direction 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 good for the sake of time but it'll be some constant this is the constant that then 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 we have or how many molecules or the what's its stoichiometric coefficient it's two so two 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 equal to the probability of a reverse reaction in the Box so these two things are going to equal each other so this is going to equal I could just copy and paste it that up there there you go and then if you set the constants equal to each other if you set the constants ended you know you could pick what the you normally put the products on the right hand side of the equation so I'll take I'll take these and divide them into this and I'll divide that into that and you're left with you're left with KF over 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 this is just you could call that the equilibrium constant and there you have it the 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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