The pH Scale

Square brackets, like $[{\text{H}}^{+}]$, mean 'concentration of' in mol/L. Because pH uses a negative logarithm, a LOWER pH means a HIGHER $[{\text{H}}^{+}]$ — it can feel backwards at first, but that inverse relationship is exactly what compresses a huge concentration range into a small, manageable number range. For concentrations written as $1\times 10^{-n}$ the log is easy: $\log (10^{-n})=-n$, so the pH is simply $n$. For example $[{\text{H}}^{+}]=1\times 10^{-3}$ mol/L gives pH 3. For any other coefficient, use the log key on your calculator; the pH then lands between two whole numbers. A useful estimate: $[{\text{H}}^{+}]=2.0\times 10^{-4}$ must have a pH between 3 and 4, and closer to 4 because 2.0 is small. In a strong acid, $[{\text{H}}^{+}]$ equals the acid concentration (because the acid ionizes completely), so 0.010 mol/L HCl has pH 2.00.

At 25°C, water itself is very slightly ionized, and the product of $[{\text{H}}^{+}]$ and $[{\text{OH}}^{-}]$ is always constant: $K_w=[{\text{H}}^{+}][{\text{OH}}^{-}]=1.0\times 10^{-14}\text{.}$ Why a constant product? Water molecules are always, very occasionally, splitting into ${\text{H}}^{+}$ and ${\text{OH}}^{-}$ and recombining. Adding an acid floods the water with ${\text{H}}^{+}$, so more of them meet and recombine with ${\text{OH}}^{-}$ — $[{\text{OH}}^{-}]$ falls as $[{\text{H}}^{+}]$ rises. That is why even a strongly acidic solution still contains a tiny amount of hydroxide, and a basic solution still contains a tiny amount of hydrogen ion: the two concentrations act like a seesaw. In pure water they are equal, $1.0\times 10^{-7}$ mol/L each, which is why neutral is pH 7 at 25 °C. Taking $-\log$ of both sides of this relationship gives a simple, extremely useful rule. It lets you go from any one of the four quantities — $[{\text{H}}^{+}]$, $[{\text{OH}}^{-}]$, pH, pOH — to all the others: a base problem usually starts with $[{\text{OH}}^{-}]$, so find pOH first and subtract from 14.

• pH < 7: acidic ($[{\text{H}}^{+}]>[{\text{OH}}^{-}]$)
• pH = 7: neutral ($[{\text{H}}^{+}]=[{\text{OH}}^{-}]=1.0\times 10^{-7}$ mol/L, at 25°C)
• pH > 7: basic / alkaline ($[{\text{OH}}^{-}]>[{\text{H}}^{+}]$)
• Each whole-number step on the pH scale represents a 10-fold change in $[{\text{H}}^{+}]$ — pH 3 has 10× the $[{\text{H}}^{+}]$ of pH 4, and 100× the $[{\text{H}}^{+}]$ of pH 5.

Since $\text{pH}=-\log [{\text{H}}^{+}]$, you can undo the logarithm to recover the concentration whenever you're given a pH value. On most calculators this is the $10^x$ key (often SHIFT + log): enter the NEGATIVE of the pH. A pH of 3.70 gives $10^{-3.70}=2.0\times 10^{-4}$ mol/L. Significant figures work differently for logs: only the digits AFTER the decimal point in a pH count, and they match the significant figures in the concentration. So $[{\text{H}}^{+}]=2.0\times 10^{-4}$ (two significant figures) gives pH 3.70 (two decimal places), and pH 9.00 converts back to $1.0\times 10^{-9}$ mol/L.

Ontario's operational guideline for drinking water is a pH between 6.5 and 8.5 — a treatment target for good-quality water rather than a health-based limit. Water that is too acidic slowly dissolves metal from pipes and plumbing fittings, which can release lead and copper into tap water; water that is too basic tastes bitter, leaves scale and makes chlorine disinfection less effective. Treatment plants therefore measure pH continuously and adjust it by adding a base such as lime or sodium hydroxide, or an acid such as carbon dioxide. Rain is naturally slightly acidic (about pH 5.6) because dissolved carbon dioxide forms carbonic acid; sulfur and nitrogen oxides from burning fuels make acid rain, which lowered the pH of many lakes on the Canadian Shield in Ontario. Lakes on limestone bedrock resist the change because carbonate rock neutralizes the acid.