If a solution has a [OH-] 1.30 x 10-5 M, what is the pOH and the pH?

Question

If a solution has a [OH-] 1.30 x 10-5 M, what is the pOH and the pH?

pOH = 13 pH = 1

pOH = 4.89 pH = 9.11

pOH = 9.11 pH = 4.89

pOH = 1 pH = 13

Answer 1

The answer is pOH = 4.89 and pH = 9.11, the second choice.

Two steps, and the second one is only subtraction:

pOH = -log[OH-] = -log(1.30 x 10^-5)
    = 5 - log(1.30)
    = 5 - 0.114
    = 4.89

pH = 14.00 - pOH
   = 14.00 - 4.89
   = 9.11

Before you touch a calculator you can throw out two of the four options. A hydroxide concentration of 1.30 x 10^-5 M is much bigger than the 1.0 x 10^-7 M of pure water, so this solution is basic, so its pH has to sit above 7. That kills "pOH = 9.11, pH = 4.89" and "pOH = 13, pH = 1" on sight. Asking "acidic or basic?" before doing any arithmetic turns a four-way guess into a two-way one, and on a timed exam that habit is worth more than the algebra.

Where the wrong options come from:

pOH = 9.11, pH = 4.89  -> the right two numbers, swapped
pOH = 1,    pH = 13    -> treating 1.30 x 10^-5 M as if it were 0.1 M
pOH = 13,   pH = 1     -> that same slip, and then swapped too

One shortcut worth keeping: the log of any number between 1 and 10 falls between 0 and 1, and log(1.30) is only 0.11. So the moment you read 1.30 x 10^-5 you already know the pOH is a hair under 5 and the pH a hair over 9. If what your calculator hands back is not near those, the mistake happened before you pressed a key.

A footnote for later: pOH + pH = 14.00 is Kw at 25 C, not a law of nature. Kw rises with temperature, so at 50 C the two add to about 13.26. Neutral does not mean pH 7 either, it means [H3O+] = [OH-], which at 50 C is a pH near 6.63. At room temperature, though, 14.00 is exactly right.