Keq Intuition (mathy and not necessary to progress) A probabilistic look at how molecules react to develop the intuition behind the equilibrium constant formula.
Keq Intuition (mathy and not necessary to progress)
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- Let's see if we can develop some intuition as to why the
- equilibrium constant equation looks the way it does.
- Just as review, this is it: equilibrium constant.
- It would be the concentration of our molecule Y raised to
- its coefficient power or, if we're thinking in moles,
- raised to the number of moles.
- If we think of these as kind of the mole ratios, or the
- molar ratios-- or we could just view them as the
- molecular ratios, either way-- times the concentration of our
- molecule Z.
- Now, we're not doing some calculus here.
- d is just the number of moles we need of Z for every c moles
- of Y, b moles of X, and a moles of V.
- So it's Z to the d power divided by the concentration
- of V to the a power and X to the b power.
- So it's a nice, little, clean equation, but why does
- it look this way?
- And I actually made a video earlier today where I started
- exploring this with natural logs.
- And I think I got someplace, but that one
- started to break down.
- And I think I've come up with a much simpler reason why this
- looks this way.
- So I've deleted that video, and I think I've come up with
- a much more intuitive one that explains more of why this
- works and actually some of the other things we're going to
- learn about the equilibrium constants in future videos.
- So what makes a reaction happen?
- Or what does equilibrium mean?
- It means the rate at which the forward reaction is happening.
- So that means that the rate of this happening, of V plus X
- turning into Y plus Z-- I can't forget the
- coefficients-- is going to be equal to the reverse reaction,
- is equal to the rate of the reverse reaction.
- So our c moles of Y plus d moles of Z turning and going
- the other way, turning into the V and the X
- with certain ratios.
- It doesn't mean necessarily that the concentrations are
- the same, because we could have one where we end up
- heavily favoring the forward reaction.
- Where we end up with much higher concentrations of Y and
- Z, or we might heavily favor the backwards reactions where
- we have more V and X.
- But when we're in equilibrium, we're saying that our
- concentrations have reached a stability point, which implies
- that the rate of going in this direction is equal to the rate
- going into that direction.
- So let's just think a little bit about what drives these
- rates, what drives these rates of reactions.
- In order for this forward reaction to happen that I drew
- in purple, what needs to happen?
- We have to have a molecules of V roughly.
- And let's say in any volume of space, we have to have some V
- molecules, and preferably a V molecules, being in the
- vicinity of b X molecules.
- So there's got to be b of these X molecules, and they
- have to be in the right configuration and in the right
- place and kind of close enough in order for
- the reaction to happen.
- So the reaction is really going to be driven by, if you
- think about it, the probability of finding a V
- molecules and b molecules all within close enough confines
- that they can actually react.
- So you could say that the rate is going to be driven by--
- maybe it's going to be proportional.
- Let's say it's just equal to-- let's say some constant that
- takes into account things like temperature and how the
- molecules are actually configured.
- Because it's not dependent just on them being there.
- You have to have worry about their kinetic energies.
- You have to worry about their shape, because some shapes are
- going to be more conducive to reaction than others.
- So let's just let that be taken into account with a K.
- And we're talking about the forward reaction, right?
- So in order for the forward reaction to happen, let's call
- that K plus for the forward reaction.
- We have to have a molecules of V there and b molecules of X.
- So what's the probability of having a molecules of X?
- Or what's a rough approximation of the
- Well, the concentration.
- Let's think about this a second.
- When we write the concentration of the molecule
- V, which I think when I did this was the blue one right
- here, what is that given in?
- That is given in moles per liter.
- Moles is just a number, so this tells us, look, in any
- given volume, roughly how many of the molecules do
- you expect to find?
- That's what concentration is.
- So if I wanted to figure out the probability of finding a
- of these molecules, because that's how many I need, I need
- to multiply this by itself a times,
- because I need a of them.
- The probability of having just one molecule in just some
- small fraction, you would just use the concentration once.
- But you're going to use it a times, because you want a of
- those molecules there, right?
- You could look at it like what's the probability of
- having five heads?
- Well you would multiply the probability of
- one head five times.
- So the forward reaction probability is going to be the
- concentration of V to the a power, and, of course, that's
- not enough to have the reaction happen.
- You also need to have b of the X molecules there.
- So you have the concentration of X to the b power.
- And I want to make sure you understand this.
- My claim is that this is approximation-- or actually
- it's a pretty good way of calculating-- the probability.
- So let me write it this way.
- The rate is equal to some constant that takes into
- account the temperature and the molecular configurations
- times the probability of having a V molecules and b X
- molecules in a sufficiently small area
- all at the same time.
- And the best way to approximate that is with their
- Obviously, the higher the concentration, the higher the
- moles per liter, the more likely you're going to find
- that many of molecules in kind of that little small space
- that you care about, and the temperature and the
- configuration are going to matter more.
- But if you use the concentration as the
- probability of a-- let me switch colors.
- If the probability of having a V molecule in some volume-- if
- we assume that the solution is homogeneous, that the V
- molecules are roughly evenly distributed, it's going to
- be-- this isn't even an approximation.
- It's going to be the concentration of the V
- molecules times the volume under which we care about.
- If we want the probability of a, where a is a number, it
- could be five V molecules, a V's in some volume, it's the
- probability of finding this a times.
- So it's going to be equal to-- and this is just from the
- probability concepts that we learned in the whole
- probability playlist.
- So if you want to have five heads in a row, it's 1/2 to
- the fifth power.
- If you want to have V molecules there, five of them
- at the same time in some volume, or a of them, it's
- going to be V to the a power times the volume.
- If you also care about the probability so you want all of
- that, so a V's and b X's in some volume, then you're going
- to have to multiply all of them together.
- So it's going to be equal to the concentration of V to the
- a power times the concentration of X to the b
- power times the volume.
- So the probability of finding the right number of V
- particles and X particles in the right place in some volume
- is going to be proportional to exactly this.
- And we're saying that the reaction rate, the forward
- reaction rate, is also proportional to this thing.
- So that's where we get the forward reaction rate.
- So the rate forward is equal to the concentration of our V
- molecules to the a power times the concentration of our X
- molecules to the b power.
- Now, if we want to find the reverse rate, so this is the
- rate forward.
- If we want to find the rate of the reverse reaction, let's
- say that that's equal to some other constant-- let's call
- that K-minus-- the same exact logic holds.
- We're just going in this direction now.
- If we look at our original one, we're
- going in that direction.
- So for this reaction, we do the same thing.
- We literally just do different letters, so the reverse
- reaction is just going to be the concentration of the Y
- molecule to the c power, because we need c of them
- there roughly at the same time, times the concentration
- of the Z molecule to the d power.
- Now, just at the beginning of the video, we said that
- equilibrium is when these rates equal each other.
- I wrote it down right here.
- So if the reverse rate is equal to some constant times
- this, and the forward rate is equal to some constant times
- that, then we reach equilibrium when these two are
- equal to each other.
- Let me clear up some space here.
- Let me clear this up, too.
- So when are they going to be equal to each other?
- When the forward rate-- the forward rate is this.
- That's our forward constant, which took into account a
- whole bunch of temperature and molecular structure and all of
- that-- times the concentration of our V
- molecule to the a power.
- You can kind of view that as what's the probability of
- finding in a certain volume-- and that certain volume can be
- factored into that K factor as well-- but what's the
- probability of finding V things, a V
- molecules in some volume.
- And it's the concentration of V to the a power times
- concentration of X to the V power-- that's the forward
- reaction-- and that has to equal the reverse reactions.
- So K-minus times the concentration of Y to the c
- power times the concentration of Z to the d power.
- Now, if we divide both sides by-- let me erase more space.
- Nope, not with that.
- All right.
- So let's divide both sides by K-minus and both sides by
- this, so you get K-plus over K-minus is equal to that, is
- equal to Y to the c times Z to the d.
- All of that over that-- V to the a times the concentration
- of X to the V.
- Let me put this in magenta just so you know that this was
- this K-minus right here.
- And then, these are just two arbitrary constants, so we
- could just replace them and call them
- the equilibrium constant.
- And we're there where we need to be.
- We're at the formula for the equilibrium constant.
- Now, I know this was really hand wavy, but I want you to
- at least get the sense that this doesn't come from out of
- the blue, and there is-- at least I think there is--
- there's an intuition here.
- These are really calculating the probabilities of finding--
- this is the forward reaction rate probabilities
- proportional to this.
- Because the more V concentration you have, the
- more likely you're able to find it.
- Although if you need more of those particles around, you're
- going to have to multiply that concentration by each other,
- because the probability's going to get lower.
- Because you need more of them together in order for the
- reaction to happen.
- Same thing for everything there.
- But all this is derived from is that the forward reaction
- should be equal to some constant
- times the reverse reaction.
- Or actually, their rates should be equal, but then when
- you actually calculate the probability, you'll have a
- constant in there.
- Anyway, hopefully, I didn't confuse you, but I just wanted
- to give you that this isn't just some random equation.
- It really does, I think, come from the reality that the
- higher the concentration you have, the more the probability
- you have of the actual molecules
- bumping into each other.
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