Buffers and Hendersen-Hasselbalch

Let's say I have some weak acid.

I'll call it HA.

A is a place holder for

really a whole set of elements that I could put there.

It could be fluorine, it could be an ammonia molecule.

If you add H it becomes ammonium.

So this isn't any particular element I'm talking about.

This is just kind of a general way of writing an acid.

And let's say it's in equilibrium with, of course,

and you've seen this multiple times, a proton.

And all of this is in an aqueous solution.

Between this proton jumping off of this

and its conjugate base.

And we also could have written a base equilibrium,

where we say the conjugate base could disassociate,

or it could essentially grab a hydrogen from the water

and create OH.

And we've done that multiple times.

But that's not the point of this video.

So let's just think a little bit about

what would happen to this equilibrium

if we were to stress it in some way.

And you can already imagine

that I'm about to touch on Le Chatelier's Principal,

which essentially just says, look,

if you stress an equilibrium in any way,

the equilibrium moves in such a way

to relieve that stress.

So let's say that the stress that I apply to the system

--Let me do a different color.

I'm going to add some strong base.

That's too dark.

I'm going to add some NaOH.

And we know this is a strong base

when you put it in a aqueous solultion,

the sodium part just kind of disassociates,

but the more important thing,

you have all this OH in the solution,

which wants to grab hydrogens away.

So when you add this OH to the solution,

what's going to happen for every mole that you add,

not even just mole,

for every molecule you add of this into the solution,

it's going to eat up a molecule of hydrogen.

Right?

So for example, if you had

1 mole oh hydrogen molecules in your solution

right when you do that,

all this is going to react with all of that.

And the OHs are going to react with the Hs

and form water,

and they'll both just kind of

disappear into the solution.

They didn't disappear, they all turned into water.

And so all of this hydrogen will go away.

Or at least the hydrogen that was initially there.

That 1 mole of hydrogens will disappear.

So what should happen to this reaction?

Well, know this is an equilibrium reaction.

So as these hydrogen disappear,

because this is an equilibrium reaction

or because this is a weak base,

more of this is going to be converted

into these two products

to kind of make up for that loss of hydrogen.

And you can even play with it on the math.

So this hydrogen goes down initially,

and then it starts getting to equilibrium very fast.

But this is going to go down.

This is going to go up.

And then this is going to go down less.

Because sure, when you put the sodium hydroxide there,

it just ate up all of the hydrogens.

But then you have this

-- you can kind of view as

the spare hydrogen capacity here to produce hydrogens.

And when these disappear,

this weak base will disassociate more.

The equilibrium we'll move more in this direction.

So immediately, this will eat all of that.

But then when the equilibrium moves in that direction,

a lot of the hydrogen will be replaced.

So if you think about what's happening,

if I just threw this sodium hydroxide in water.

So if I just did NaOH in an aqueous solution

so that's just throwing it in water

-- that disassociates completely

into the sodium cation and hydroxide anion.

So you all of a sudden immediately

increase the quantity of OHs

by essentially the number of moles of

sodium hydroxide you're adding,

and you'd immediately increase the pH, right?

Remember. When you increase the amount of OH,

you would decrease the pOH, right?

And that's just because it's the negative log.

So if you increase OH, you're decreasing pOH,

and you're increasing pH.

And just think OH-- you're making it more basic.

And a high pH is also very basic.

If you have a mole of this, you end up with a pH of 14.

And if you had a strong acid, not a strong base,

you would end up with a pH of 0.

Hopefully you're getting a little bit familiar

with that concept right now,

but if it confuses you,

just play around with the logs a little bit

and you'll eventually get it.

But just to get back to the point,

if you just did this in water,

you immediately get a super high pH

because the OH concentration goes through the roof.

But if you do it here

--if you apply the sodium hydroxide to this solution,

the solution that contains

a weak acid and it's conjugate base,

the weak acid and its conjugate base,

what happens?

Sure, it immediately reacts with all of this hydrogen

and eats it all up.

And then you have this extras supply here

that just keeps providing more and more hydrogens.

And it'll make up a lot of the loss.

So essentially, the stress won't be as bad.

And over here, you dramatically increase the pH

when you just throw it on water.

Here, you're going to increase the pH by a lot less.

And in future videos, we'll actually do the math of

how much less it's increasing the pH.

But the way you could think about it is,

this is kind of a shock absorber for pH.

Even though you threw this strong based

into this solution,

it didn't increase the pH

as much as you would have expected.

And you can make it the other way.

If I just wrote this exact same reaction

as a basic reaction

--and remember, this is the same thing.

So if I just wrote this as, A minus

--so I just wrote its conjugate base--

is in equilibrium with the conjugate base

grabbing some water from the surrounding aqueous solution.

Everything we're dealing with right now

is an aqueous solution.

And of course that water that it grabbed from

is not going to be an OH.

Remember, are just equivalent reactions.

Here, I'm writing it as an acidic reaction.

Here, I'm writing it is a basic reaction.

But they're equivalent.

Now.

If you were to add a strong acid to the solution,

what would happen?

So if I were to throw hydrogen chloride into this.

Well hydrogen chloride, if you just throw it

into straight up water without the solution,

it would completely disassociate into

a bunch of hydrogens and a bunch of chlorine anions.

And it would immediately make it very acidic.

You would get to a very low pH.

If you had a mole of this

--if your concentration was 1 molar,

then this will go to a pH of 0.

But what happens if you add hydrochloric acid to

this solution right here?

This one that has this weak base

and its conjugate weak acid?

Well, all of these hydrogen protons that

disassociate from the hydrochloric acid

are all going to react with these OHs you have here.

And they're just going to cancel each other out.

They're just going to merge with these and turn into water

and become part of the aqueous solution.

So this, the OHs are going to go down initially,

but then you have this reserve of weak base here.

And Le Chatelier's Principal tells us.

Look, if we have a stressor

that is decreasing our overall concentration of OH,

then the reaction is going to

move in the direction that relieves that stress.

So the reaction is going to go in that direction.

So you're going to have more of our weak base

turning into a weak acid and producing more OH.

So the pH won't go down as much as you would expect

if you just threw this in water.

This is going to lower the pH,

but then you have more OH that could be produced as

this guy grabs more and more hydrogens from the water.

So the way to think about it is it's kind of like

a cushion or a spring in terms of what

a strong acid or base could do to the solution.

And that's why it's called a buffer.

Because it provides a cushion on acidity.

If you add a strong base to water,

you immediately increase its pH.

Or you decrease its acidity dramatically.

But if you add a strong base to a buffer,

because of Le Chatelier's Principal,

essentially, you're not going to affect the pH as much.

Same thing.

If you add and acid to that same buffer,

it's not going to affect the pH

as much as you would have expected

if you had thrown that acid in water

because the equilibrium reaction can always

kind of refill the amount of OH that you lost

if you're adding acid,

or it can refill the amount of hydrogen you lost

if you're adding a base.

And that's why it's called buffer. It provides a cushion.

So it give some stability to the solution's pH.

The definition of a buffer is

just a solution of a weak acid in equilibrium

with its conjugate weak base.

That's what a buffer is, and it's called a buffer

because it provides you this kind of cushion of pH.

It's kind of a stress absorber,

or a shock absorber for the acidity of a solution.

Now, with that said,

let's explore a little bit the math of a buffer,

which is really just the math of a weak acid.

So if we rewrite the equation again,

so HA is in equilibrium.

Everything's in an aqueous solution.

With hydrogen and its conjugate base.

We know that there's an equilibrium constant for this.

We've done many videos on that.

The equilibrium constant here is equal to

the concentration of our hydrogen proton

times the concentration of our conjugate base.

When I say concentration, I'm talking molarity.

Moles per liter divided by

the concentration of our weak acid.

Now.

Let's solve for hydrogen concentration.

Because what I want to do is

I want to figure out a formula,

and we'll call it the Hendersen-Hasselbalch Formula,

which a lot of books want you to memorize,

which I don't think you should.

I think you should always just be able to

go from this kind of basic assumption and get to it.

But let's solve for the hydrogen

so we can figure out a relationship

between pH and all the other stuff that's in this formula.

So, if we want to solve for hydrogen,

we can multiply both sides

by the reciprocal of this right here.

And you get hydrogen concentration.

Ka times

--I'm multiplying both sides times a reciprocal of that.

So times the concentration of our weak acid

divided by the concentration of our weak base

is equal to our concentration of our hydrogen.

Fair enough.

Now.

Let's take the negative log of both sides.

So the negative log of all of that stuff,

of your acidic equilibrium constant,

times HA, our weak acid divided by our weak base,

is equal to the negative log of

our hydrogen concentration.

Which is just our pH, right?

Negative log of hydrogen concentration is

--that's the definition of pH.

I'll write the p and the H in different colors.

You know a p just means negative log.

Minus log. That's all. Base 10.

Let's see if we can simplify this any more.

So our logarithmic properties.

We know that when you take the log of something

and you multiply it,

that's the same thing as taking the log of this

plus the log of that.

So this can to be simplified to minus log of our Ka minus

the log of our weak acid concentration divided by

its conjugate base concentration.

Is equal to the pH.

Now, this is just the pKa of our weak acid,

which is just the negative log of

its equilibrium constant.

So this is just the pKa.

And the minus log of HA over A.

What we can do is we could make this a plus,

and just take this to the minus 1 power. Right?

That's just another logarithm property,

and you can review the logarithm videos

if that confused you.

And this to the minus 1 power just means invert this.

So we could say,

plus the logarithm of our conjugate base concentration

divided by the weak acid concentration

is equal to the pH.

And this right here,

this is called the Hendersen-Hasselbalch Equation.

And I really encourage you not to memorize it.

Because if you do attempt to memorize it,

within a few hours, you're going to forget

whether this was a plus over here.

You're going to forget this,

and you're going to forget whether you

put the A minus or the HA

on the numerator or the demoninator,

and if you forget that, it's fatal.

The better thing is to just start

from your base assumptions. And trust me.

It took me a couple minutes to do it,

but if you just do it really fast on paper,

you don't have to talk it through the way I did

--it'll take in no time at all to come to this equation.

It's much better than memorizing it,

and you won't forget it when you're 30 years old.

But what's useful about this?

Well, it immediately relates pH to our pKa,

and this is a constant, right, for an equilibrium?

Plus the log of the ratios between the acid

and the conjugate base.

So the more conjugate base I have,

and the less acid I have,

the more my pH is going to increase. Right?

If this goes up and this is going down,

my pH is going to increase. Which makes sense

because I have more base in the solution.

And if I have the inverse of that,

might be just going...