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Let's see if we can develop some intuition as to why the

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equilibrium constant equation looks the way it does.

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Just as review, this is it: equilibrium constant.

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It would be the concentration of our molecule Y raised to

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its coefficient power or, if we're thinking in moles,

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raised to the number of moles.

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If we think of these as kind of the mole ratios, or the

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molar ratios-- or we could just view them as the

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molecular ratios, either way-- times the concentration of our

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molecule Z.

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Now, we're not doing some calculus here.

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d is just the number of moles we need of Z for every c moles

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of Y, b moles of X, and a moles of V.

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So it's Z to the d power divided by the concentration

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of V to the a power and X to the b power.

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So it's a nice, little, clean equation, but why does

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it look this way?

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And I actually made a video earlier today where I started

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exploring this with natural logs.

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And I think I got someplace, but that one

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started to break down.

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And I think I've come up with a much simpler reason why this

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looks this way.

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So I've deleted that video, and I think I've come up with

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a much more intuitive one that explains more of why this

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works and actually some of the other things we're going to

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learn about the equilibrium constants in future videos.

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So what makes a reaction happen?

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Or what does equilibrium mean?

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It means the rate at which the forward reaction is happening.

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So that means that the rate of this happening, of V plus X

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turning into Y plus Z-- I can't forget the

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coefficients-- is going to be equal to the reverse reaction,

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is equal to the rate of the reverse reaction.

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So our c moles of Y plus d moles of Z turning and going

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the other way, turning into the V and the X

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with certain ratios.

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It doesn't mean necessarily that the concentrations are

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the same, because we could have one where we end up

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heavily favoring the forward reaction.

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Where we end up with much higher concentrations of Y and

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Z, or we might heavily favor the backwards reactions where

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we have more V and X.

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But when we're in equilibrium, we're saying that our

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concentrations have reached a stability point, which implies

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that the rate of going in this direction is equal to the rate

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going into that direction.

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So let's just think a little bit about what drives these

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rates, what drives these rates of reactions.

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In order for this forward reaction to happen that I drew

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in purple, what needs to happen?

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We have to have a molecules of V roughly.

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And let's say in any volume of space, we have to have some V

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molecules, and preferably a V molecules, being in the

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vicinity of b X molecules.

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So there's got to be b of these X molecules, and they

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have to be in the right configuration and in the right

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place and kind of close enough in order for

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the reaction to happen.

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So the reaction is really going to be driven by, if you

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think about it, the probability of finding a V

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molecules and b molecules all within close enough confines

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that they can actually react.

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So you could say that the rate is going to be driven by--

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maybe it's going to be proportional.

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Let's say it's just equal to-- let's say some constant that

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takes into account things like temperature and how the

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molecules are actually configured.

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Because it's not dependent just on them being there.

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You have to have worry about their kinetic energies.

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You have to worry about their shape, because some shapes are

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going to be more conducive to reaction than others.

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So let's just let that be taken into account with a K.

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And we're talking about the forward reaction, right?

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So in order for the forward reaction to happen, let's call

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that K plus for the forward reaction.

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We have to have a molecules of V there and b molecules of X.

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So what's the probability of having a molecules of X?

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Or what's a rough approximation of the

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probability?

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Well, the concentration.

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Let's think about this a second.

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When we write the concentration of the molecule

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V, which I think when I did this was the blue one right

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here, what is that given in?

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That is given in moles per liter.

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Moles is just a number, so this tells us, look, in any

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given volume, roughly how many of the molecules do

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you expect to find?

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That's what concentration is.

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So if I wanted to figure out the probability of finding a

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of these molecules, because that's how many I need, I need

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to multiply this by itself a times,

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because I need a of them.

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The probability of having just one molecule in just some

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small fraction, you would just use the concentration once.

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But you're going to use it a times, because you want a of

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those molecules there, right?

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You could look at it like what's the probability of

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having five heads?

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Well you would multiply the probability of

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one head five times.

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So the forward reaction probability is going to be the

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concentration of V to the a power, and, of course, that's

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not enough to have the reaction happen.

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You also need to have b of the X molecules there.

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So you have the concentration of X to the b power.

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And I want to make sure you understand this.

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My claim is that this is approximation-- or actually

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it's a pretty good way of calculating-- the probability.

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So let me write it this way.

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The rate is equal to some constant that takes into

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account the temperature and the molecular configurations

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times the probability of having a V molecules and b X

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molecules in a sufficiently small area

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all at the same time.

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And the best way to approximate that is with their

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concentration.

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Obviously, the higher the concentration, the higher the

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moles per liter, the more likely you're going to find

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that many of molecules in kind of that little small space

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that you care about, and the temperature and the

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configuration are going to matter more.

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But if you use the concentration as the

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probability of a-- let me switch colors.

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If the probability of having a V molecule in some volume-- if

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we assume that the solution is homogeneous, that the V

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molecules are roughly evenly distributed, it's going to

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be-- this isn't even an approximation.

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It's going to be the concentration of the V

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molecules times the volume under which we care about.

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If we want the probability of a, where a is a number, it

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could be five V molecules, a V's in some volume, it's the

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probability of finding this a times.

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So it's going to be equal to-- and this is just from the

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probability concepts that we learned in the whole

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probability playlist.

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So if you want to have five heads in a row, it's 1/2 to

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the fifth power.

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If you want to have V molecules there, five of them

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at the same time in some volume, or a of them, it's

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going to be V to the a power times the volume.

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If you also care about the probability so you want all of

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that, so a V's and b X's in some volume, then you're going

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to have to multiply all of them together.

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So it's going to be equal to the concentration of V to the

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a power times the concentration of X to the b

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power times the volume.

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So the probability of finding the right number of V

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particles and X particles in the right place in some volume

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is going to be proportional to exactly this.

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And we're saying that the reaction rate, the forward

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reaction rate, is also proportional to this thing.

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So that's where we get the forward reaction rate.

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So the rate forward is equal to the concentration of our V

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molecules to the a power times the concentration of our X

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molecules to the b power.

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Now, if we want to find the reverse rate, so this is the

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rate forward.

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If we want to find the rate of the reverse reaction, let's

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say that that's equal to some other constant-- let's call

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that K-minus-- the same exact logic holds.

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We're just going in this direction now.

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If we look at our original one, we're

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going in that direction.

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So for this reaction, we do the same thing.

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We literally just do different letters, so the reverse

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reaction is just going to be the concentration of the Y

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molecule to the c power, because we need c of them

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there roughly at the same time, times the concentration

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of the Z molecule to the d power.

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Now, just at the beginning of the video, we said that

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equilibrium is when these rates equal each other.

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I wrote it down right here.

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So if the reverse rate is equal to some constant times

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this, and the forward rate is equal to some constant times

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that, then we reach equilibrium when these two are

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equal to each other.

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Let me clear up some space here.

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Let me clear this up, too.

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So when are they going to be equal to each other?

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When the forward rate-- the forward rate is this.

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That's our forward constant, which took into account a

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whole bunch of temperature and molecular structure and all of

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that-- times the concentration of our V

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molecule to the a power.

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You can kind of view that as what's the probability of

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finding in a certain volume-- and that certain volume can be

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factored into that K factor as well-- but what's the

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probability of finding V things, a V

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molecules in some volume.

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And it's the concentration of V to the a power times

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concentration of X to the V power-- that's the forward

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reaction-- and that has to equal the reverse reactions.

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So K-minus times the concentration of Y to the c

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power times the concentration of Z to the d power.

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Now, if we divide both sides by-- let me erase more space.

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Nope, not with that.

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All right.

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So let's divide both sides by K-minus and both sides by

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this, so you get K-plus over K-minus is equal to that, is

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equal to Y to the c times Z to the d.

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All of that over that-- V to the a times the concentration

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of X to the V.

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Let me put this in magenta just so you know that this was

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this K-minus right here.

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And then, these are just two arbitrary constants, so we

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could just replace them and call them

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the equilibrium constant.

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And we're there where we need to be.

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We're at the formula for the equilibrium constant.

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Now, I know this was really hand wavy, but I want you to

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at least get the sense that this doesn't come from out of

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the blue, and there is-- at least I think there is--

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there's an intuition here.

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These are really calculating the probabilities of finding--

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this is the forward reaction rate probabilities

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proportional to this.

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Because the more V concentration you have, the

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more likely you're able to find it.

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Although if you need more of those particles around, you're

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going to have to multiply that concentration by each other,

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because the probability's going to get lower.

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Because you need more of them together in order for the

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reaction to happen.

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Same thing for everything there.

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But all this is derived from is that the forward reaction

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should be equal to some constant

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times the reverse reaction.

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Or actually, their rates should be equal, but then when

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you actually calculate the probability, you'll have a

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constant in there.

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Anyway, hopefully, I didn't confuse you, but I just wanted

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to give you that this isn't just some random equation.

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It really does, I think, come from the reality that the

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higher the concentration you have, the more the probability

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you have of the actual molecules

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bumping into each other.