pH and pOH

The relationship between hydrogen ion concentration [H⁺] and hydroxide concentration [OH^–] is a very straight forward inverse relationship. Mathematically, the equation is

\[K_{\rm w} = {\rm [H^+][OH^-]}\]

Where K_w is 1.0 × 10^–14 as is the auto-ionization constant for water. The inverse relationship means that what happens to one ion, the opposite happens to the other. So if [H⁺] doubles in concentration, [OH^–] will half in concentration. If [H⁺] decreases by a factor of 1000 (divide by 1000), [OH^–] will increase by a factor of 1000 (multiply by 1000). As a matter of fact, this changing concentrations by factors of 10 is what lead to the pH and pOH scales. We humans typically like our numeric scales to be nice human friendly whole numbers or at least in a friendly range of whole numbers. Concentrations in solution of H⁺ and OH^– vary from like 10 M all the way down to 10^–15 M. This is why a logarithmic scale makes a lot of sense.

The pH Scale is just a logarithm scale that we flip the annoying negative sign on so that most all the values are positive values. Simply put the formula for the scale is

\[{\rm pH} = -\log[{\rm H}^+]\]

Here is a diagram with several common substances and their associated pH.

The logarithm is base-10 by the way. It will match the "LOG" button on your scientific calculator. Calculating pOH is just as easy...

\[{\rm pOH} = -\log[{\rm OH}^–]\]

Cool. As a matter of fact, this scale is so useful for sets of increasingly small numbers that we use it for all sorts of little numbers (10 to some negative exponent). We have all kinds of equilibrium constants in chemistry which all are a K with some descriptive subscript attached. Like the one for water itself is K_w where the subscript "w" is for water. duh. So apply the "p" function of K_w and you get

\[{\rm p}K_{\rm w} = -\log[K_{\rm w}]\]

And, since we already know the value of K_w we can just calculate and hopefully memorize the value of pK_w

\[{\rm p}K_{\rm w} = -\log(1.0\times 10^{-14}) = 14.00\]

The pX Scale is backwards though as far as physical concentrations matching up with the pX value. The biggest values of X are at the low end of the pX scale... like the 0-1 end. Every unit change in pX increasing, like 1, 2, 3, 4... is really a factor of 10× LESS than the last (I guess that is really divided by 10 then). Think about it.

Back to good ol' pH. A pH of 7 has a [H⁺] concentration that is 100× LESS than the concentration in a pH 5 solution. The 2-unit pH difference matches a 10² factor difference. This even works with pH's that aren't perfect integers. For example, a solution with a pH of 3.75 is 10× more concentrated in H⁺ than a solution of pH 4.75.

Some More Math Coolness

So that original equation on this page, you know, the \(K_{\rm w} = {\rm [H^+][OH^-]}\) one. Let's take the log of both sides and see what happens...

\[-\log(K_{\rm w}) = -\log({\rm [H^+][OH^-]})\]

\[-\log(K_{\rm w}) = -\log{\rm [H^+]} -\log{\rm [OH^-]}\]

Those are all p-functions so we can now write

\[{\rm p}K_{\rm w} = {\rm pH + pOH}\]

\[14 = {\rm pH + pOH}\]

This is the logarithmic consequence of having an inverse relationship between to variables. The log version of an inverse relationship means that the two logarithms of the values will always sum to equal the same number. In our case, the pH and pOH will always sum to give 14. This means you'll always know one from the other by doing some very simple math.

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Neutral Water

We often talk about "neutral" water. What does neutral actually mean? It's really quite simple, it means there are equal amounts of hydrogen ions and hydroxide ions. Specifically:

neutral water definition: [H⁺] = [OH^–]

This also means that pH = pOH as well. At normal temperatures (specifically, at room temperature), this means that the concentrations of both [H⁺] and [OH^–] are 1.0 × 10^–7 M and the pH and pOH are both 7.00.

Definition of an ACIDIC Solution

An acidic solution is one where there is more acid than base. Specifically this means that

ACIDIC solutions: [H⁺] > [OH^–]

So this means that any pH less than 7 is an acidic solution.

Definition of an BASIC Solution

A basic solution is one where there is more base than acid. Specifically this means that

BASIC solutions: [OH^–] > [H⁺]

So this means that any pH greater than 7 is a basic solution. We also refer to basic solutions as being alkaline solutions.

A good reference page from the gchem site on pH and pOH

and a nice helpsheet on pH and pOH from the gchem site