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What is a Buffer?  And why are they used?

Lets say you live by a lake - There are fish in your lake, lucky you!  But then one day, there's a lot of rain.  Crap, all the pollution in the air has made the rain into acid-rain.  Now all that acid-rain makes your lake into a big green acid-lake (now I'm making up words Colbert-style).  All your fish dissolve into goo and you can't go fishing anymore.  Shit.

A bit exaggerated, but the point remains the same - without buffers this kind of thing would happen - though not to the extremes described!  Luckily, many of our lakes are actually buffered systems.  All that means is that you can add in an acid or a base and the pH (measure of how acidic or basic something is) won't change very much.  Your fish are alive again, yay!

A buffer is just a weak acid and its conjugate base in a solution.  If you were to add another source of acid to the solution, it would react with the conjugate base, making more of the weak acid and barely affecting the pH of the solution.  Ok, before we get too much farther, I better give you some background on this weak-acid stuff and go over some terminology.  If you just want to know how to make a buffer, click on down to the bottom.


What is an acid?

For our purposes, acids are like Santa, they give and give and give, never taking anything.  Except instead of presents, acids give their protons (H+) to other molecules (children).  Strong acids like hydrochloric acid (HCl) are generous, and give away all of their protons, even if there are no molecules (kids) to take them!  Weak acids are cheap - they only give away some of their protons (H+) and keep some for themselves. (did I just call Santa cheap?)

What is a base?

Bases are all those pesky little kids out there asking for presents (protons - H+).  Now as we all know, some kids are greedier than others.  Weak bases are those normal, unspoiled kids who will take a present (H+) if Santa gives it to them.  Strong bases like sodium hydroxide (NaOH) are those pesky little spoiled brat kids who take every single present (H+) they can, and are even willing to steal from Santa (weak acid) if he's being cheap!

*I am describing here the Bronsted-Lowry acid base theory.  There are other acid-base theories, the most common of which being Lewis Acids and Lewis Bases.


Examples always help - and since I like the smell of vinegar, we're going to look at acetic acid (CH3COO-H) as an example of a weak acid.  It's conjugate base is the acetate anion (CH3COO-).  A conjugate base is just what the acid turns into when it gives away its proton (H+).  The partial splitting reaction looks like this

(CH3COO-H) ↔ (CH3COO-) (H+)

Now this is probably the most important bit of terminology you will need to understand this stuff.  That reaction can be written generically as

(HA) ↔ (H+) + (A-)

It's easy to see that the (H) stays the same through both reactions.  What hangs people up sometimes is realizing that you can write (CH3COO-) as simply (A-).  This is true for ANY conjugate base.  That's important so I'll say it again. 

You can write ANY conjugate base as (A-).


That brings us to the underlying theory used to make buffers - equilibrium.  This section talks about equilibrium as it pertains to buffers - a more complete treatment can be found here.  Every weak acid has a value called its Ka.  The Ka value tells us how much of the acid (HA) will split into (H+) and (A-).  Thinking about Santa and the kids - the Ka tells us how cheap Santa is (how much he wants to give away H+). We define Ka as

ka equation

You can see that Ka is just a ratio - Acetic acids Ka = 1.74x10-5 .  What does that mean though?  Just think about how you divide.  If you want the answer to be a small number, you have to divide by a BIG number.  That BIG number is [HA] and the bigger it is, the weaker our acid is (its not giving away its protons).  To sum up

Smaller Ka = weaker acid              Bigger Ka = stronger acid.


Buffer Calculations

Now we're getting to the useful part - follow along with the math if you'd like, or just skip to the final result.  First we need to define a couple things. 

pH = -log[H+]                pKa = -log(Ka)

First we are going to take the -log of our Ka equation.

and separate the proton

then substitute pH for -log(H) and pKa for -log(Ka)

you can then just rearrange this equation

henderson-haselbalch equation

We're going to stop here because this equation is useful - it is known as the Henderson-Hasselbalch equation.  This is the equation most often used when performing buffer calculations, but it is exactly the same as the definition of Ka, we just manipulated it. 

One of the most interesting things we can see when we look at the equation is that when [A-] = [HA], then the pKa will equal the pH.  Why is this important?  It tells us that the middle of our buffer range is where our pKa is. 

A weak acid with a pKa of 7.2 will not be a good buffer for a solution we want to be at a pH of 3.6 - all of our weak acid will be in the [HA] form so if we add more acid, there won't be anything for it to react with and it will lower the pH!  The general rule is that a buffer works within 1pH of its pKa.  So if we have a pKa of 7.2, it will work between a pH of 6.2 and 8.2.

The last thing to mention about buffers, is something called their capacity.  The buffer capacity is directly related to the concentration of your weak acid.  The higher the concentration, the higher your buffer capacity!  Remember that buffers work by reacting with other acids or bases so that they don't change the pH.  The more stuff these new acids/bases have to react with, the less they'll change the pH.


How to Make a Buffer

There are a few ways to actually make a buffer.  The easiest by far (after the calculations) is to use both the weak acid and its conjugate base.  To do this, we start with a target pH, a weak acid with a pKa no more than 1 more or less than the pH, and decide what we want our [HA] concentration to be (buffer capacity).  Plug these numbers into the Henderson-Haselbalch equation and solve for [A]

Now we know the [HA] and the [A] that we want.  Now we just convert that to grams of a salt, or liters of a solution we need using stoichiometry.  Mix everything together and presto! you have a buffer solution at your target pH.  You can use my buffer calculator to do all of this for you.

But what if you only have the weak acid form???

Not a problem - you just add the weak acid to your solution and then slowly add a strong acid or a strong base until you get to the pH you want.  To make things easier for yourself, you can calculate how much acid or base you need to add using Henderson-Haselbalch.  This time however, you also need to use the Ka definition and an ICE table to first determine your initial [A] and [HA] concentrations.  ICE stands for Initial, Change, Equilibrium. 

Lets say we start with 0.5M acetic acid (Ka = 1.74x10-5).  This will partially split up into [H] and [A] which start at a concentration of 0.  These are our I - initial conditions.  Now the amounts are going to change by some number we're going to call x.  The [HA] will split up and lose x, while both [H] and [A] will gain x.  This is our C - change.  Our E (equilibrium) step is just putting it all together.

  [HA] [H] [A]
I 0.5 0 0
C -x +x +x
E 0.5-x 0+x 0+x

This E - equilibrium row - is what we put into our Ka equation

Break out those algebra skills and solve for x.  In this case, x=0.00295.  So now we know our [A] = x = 0.00295 and we know our [HA] = 0.5-x = 0.49705. 

At this point, you could calculate your pH and see if you needed more acid or more base.  Too many steps for me - I find it easier to think of it this way. Remembering that for every mole of a strong acid we add, it will completely react with [A] to form [HA] we can write the Henderson-Hasselbalch equation to look like this.

Lets say you have 1 Liter of buffer solution, and you want to get to a pH of 4.5, plug in your numbers

More algebra, I know it sucks - but you solve for x and you find that x = -0.174.  Well how is that possible?  I can't add negative acid!  Sure you can, its called base.  Now you know that you need to add 0.174 moles of a strong base like NaOH to get your desired pH.  If you have 6M NaOH, you can use stoichiometry to find that you will need to add 29mL to get to your desired pH.  I don't have a calculator to do this for you yet, would you like to see one?


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