The pH scale.

Step 1 of 6

Minus the logarithm

The hydronium concentrations of everyday solutions run from about $1$⁠ mole per litre down to $10^{-14}$⁠: fourteen orders of magnitude. Rather than write powers of ten each time, chemists use the exponent itself. In 1909 Søren Sørensen, a chemist at the Carlsberg laboratories, proposed the pH:

$\mathrm{pH} = -\log[\mathrm{H_3O^+}]$$\mathrm{pH} =$$-\log[\mathrm{H_3O^+}]$⁠

The logarithm of a concentration below one is negative, and the minus makes it positive. The first chart is the pH scale, with the beaker’s pH marked on it.

Step 2 of 6

Powers of ten

Because pH is a logarithm, each step of $1$⁠ on the scale is $10$⁠ times in concentration. Here pH is fixed at 4. Predict, then look at the second chart.

Step 3 of 6

How many times more acidic

So comparing two pH values is subtracting exponents: a gap of $n$⁠ units is $10^n$⁠ times in hydronium concentration.

Step 4 of 6

From a concentration to pH

When the concentration is not a whole power of ten, use a calculator: $\mathrm{pH} = -\log[\mathrm{H_3O^+}]$$\mathrm{pH} =$$-\log[\mathrm{H_3O^+}]$⁠. The other way: $[\mathrm{H_3O^+}] = 10^{-\mathrm{pH}}$$[\mathrm{H_3O^+}] =$$10^{-\mathrm{pH}}$⁠.

Step 5 of 6

Dilution

When you add water to an acid, the same ions spread through a larger volume. In a strong acid all the ions are already out, so their concentration falls by exactly the factor the volume grows.

Step 6 of 6

Bases on the same scale

pH describes basic solutions too, through the ion product: $[\mathrm{H_3O^+}] = \dfrac{10^{-14}}{[\mathrm{OH^-}]}$$[\mathrm{H_3O^+}] =$$\dfrac{10^{-14}}{[\mathrm{OH^-}]}$⁠. A shorter way: define $\mathrm{pOH} = -\log[\mathrm{OH^-}]$$\mathrm{pOH} =$$-\log[\mathrm{OH^-}]$⁠, and then $\mathrm{pH} + \mathrm{pOH} = 14$$\mathrm{pH} + \mathrm{pOH} =$$14$⁠.