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### Mixing Coffee and Milk: Ph and Buffers

```Date: 07/10/2003 at 19:09:51
From: Libby
Subject: Ph level

Can you tell me if I did this right?

The question is:

If I mix coffee (pH 5) and milk (pH 6.9) (50%/50%), what will be the pH
of the coffee/milk mixture?

I used the formula pH = -log [h+]

ph = -log [.5(10^-5) + .5(10^-6.9)]
-pH =  log [.5(10^-5) + 10^-6.9)]
-pH =  log 5.0629^-6
-pH = -5.295
pH = approx. 5.3
```

```
Date: 07/10/2003 at 20:10:46
From: Doctor Achilles
Subject: Re: Ph level

Hi Libby,

Thanks for writing to Dr. Math.

Given the information you have available, that is the best
approximation. However, this method makes two assumptions:

1) That you are mixing equal parts coffee and milk. If, for example,
you were to mix 3 parts coffee to 1 part milk, then you would have to

2) That neither the coffee nor the milk has any buffers in it. A
buffer is something that resists changes in pH. Here are the basics of
how a buffer works:

pH is a measure of the concentration of free H+ ions in solution.  An
acid is a chemical that releases H+ ions and a base is a chemical
that traps H+ ions.

The simplest type of acid is a "strong acid."  HCl (hydrochloric acid)
is a typical example of a strong acid. In a strong acid, every
hydrogen is immediately released as an H+ ion when you put it in
"aqueous" (mostly water) solution. So if you put HCl in aqueous
solution, every HCl molecule will dissociate (separate) into an H+ and
a Cl-. So if you add 100 trillion HCl molecules, you will effectively
have added 100 trillion H+ ions.

There are also "strong bases." NaOH (sodium hydroxide) is a typical
strong base. For a strong base, every molecule of base you add will
take a proton out of the solution. What happens in the case of NaOH
is actually a 2-step process. First, the molecule dissociates into
Na+ and OH-. Then the OH- finds an H+ and they get together to make
HOH, which is more commonly known as H2O, or water. This effectively
removes one H+ from the solution.

Now, there are also "weak acids." In a weak acid, only some fraction
of the molecules will add an H+ to the solution. A good example of a
weak acid is phosphate (H3O4P). The difficult part about weak acids is
that what fraction of the molecules actually *do* add an H+ to the
solution depends on the pH of that solution. If the pH is low
(therefore a high concentration of H+ ions), then a small fraction of
weak acid molecules will release an H+ ion. If the pH is high (a low
concentration of H+ ions), then a large fraction of weak acid
molecules will release an H+ ion. What that means is that weak acids
are very weak when you put them in something that is already acidic,
and they are pretty strong when you put them in something that is
basic.

If you turn that logic around, if you have a weak acid that is at some
pH, and you try to add a base to increase the pH, the weak acid will
just respond by releasing more H+ ions, and thus keep the pH
relatively constant. Similarly, if you try to add acid to decrease the
pH, the weak acid will just take back some of the extra H+ ions and
thus keep the pH relatively constant. So, weak acids can act as bases.
In fact, there isn't really a distinction between a weak acid and a
weak base; they are both called "buffers."  If you add H+ ions to a
buffer, then it will respond by taking some of them up, and if you
remove H+ ions, it will respond by releasing them.

Buffers aren't perfect, of course, and you can "saturate" a buffer by
forcing it to give up all its hydrogens or take up all the H+ ions it
can hold. And even a not-quite-saturated buffer can exist over a
fairly wide range of pH values.

Buffers are essential for survival. Your body has to maintain all its
fluids at a constant pH; even one or two tenths of a pH unit of change
can be *extremely* damaging. So all of your fluids have a mixture of
several different types of buffers.

So what does this have to do with milk and coffee? The analysis you
used works only if all the acids and bases in the milk and coffee
are "strong." However, both milk and coffee have buffers. Milk has
more. My guess is that this will cause the pH of the mixture to tend
toward the pH of the milk. How much depends on what buffers are in
each and what the dynamic ranges of those buffers are. That I can't
tell you off hand. The way I would answer the question is to measure
the pH of the mixture. Your other option is to analyze the milk and
coffee to find the concentrations of all the buffers and then look up
their dynamic ranges and relative buffering capabilities, which is a

Hope this helps.  If you have other questions or you'd like to talk

- Doctor Achilles, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Physics
High School Physics/Chemistry

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