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 p*K*_{w}

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

OK, LOOK above, we are now taking logarithms of numbers. Remember way back in chapter 1, the section on significant figures? I mentioned the LOG function and how to count sig figs there. Yeah here is the link: Chapter 1.4 - Significant Figures (Logarithms). So remember, to keep accuracy with pH and pOH (logarithms), your sig figs are all AFTER the decimal. OK back to our riveting narrative on the pX scale...

**The p X Scale** is backwards though as far as physical concentrations matching up with the p

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 10^{2} 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.

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.

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.

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.

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

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