Why pH is logarithmic in the first place

Hydrogen-ion concentrations in real solutions span fourteen orders of magnitude — from 1 M in concentrated HCl down to 10⁻¹⁴ M in concentrated NaOH. Plotting that range linearly is hopeless: the difference between stomach acid (0.1 M H⁺) and seawater (10⁻⁸ M) would be invisible on any reasonable chart. So Søren Sørensen took the negative log in 1909, compressed the whole range into 0–14, and gave us a number you can fit on a meter dial.

That logarithmic compression is also why a one-unit pH change feels innocuous but isn’t. Acid rain at pH 4 is ten times more acidic than normal rain at pH 5, and a hundred times more acidic than pure water at pH 7. Aquariums, blood chemistry, soil management — every system that cares about acidity cares about pH because small numbers map to enormous concentration shifts.

The four-quantity web

At 25 °C, every aqueous solution sits at a unique point defined by four interlocking values:

pH = −log₁₀[H⁺]
pOH = −log₁₀[OH⁻]
pH + pOH = 14.00
[H⁺] × [OH⁻] = K_w = 1.0 × 10⁻¹⁴

Tell me any one and I can give you the other three. That’s the whole job.

The cheat sheet:

pH from [H⁺]: pH = −log[H⁺]
[H⁺] from pH: [H⁺] = 10⁻ᵖᴴ
pOH from pH: pOH = 14.00 − pH
[OH⁻] from pOH: [OH⁻] = 10⁻ᵖᴼᴴ

Worked examples — start with the easy direction

[H⁺] → pH

A solution has [H⁺] = 3.5 × 10⁻⁴ M.

pH = −log(3.5 × 10⁻⁴) = −(log 3.5 + log 10⁻⁴) = −(0.544 − 4) = 3.46

Acidic, as expected.

pH → [H⁺]

pH = 9.25.

[H⁺] = 10⁻⁹·²⁵ = 5.6 × 10⁻¹⁰ M

Basic — only half a nanomolar of free protons.

Full conversion from pH = 4.72

pOH = 14.00 − 4.72 = 9.28
[H⁺] = 10⁻⁴·⁷² = 1.91 × 10⁻⁵ M
[OH⁻] = 10⁻⁹·²⁸ = 5.25 × 10⁻¹⁰ M

Confirm with K_w: (1.91 × 10⁻⁵)(5.25 × 10⁻¹⁰) ≈ 1.0 × 10⁻¹⁴. Checks out.

Strong acid — 0.0020 M HCl

HCl dissociates 100%, so [H⁺] = 0.0020 M directly.

pH = −log(0.0020) = 2.70

The shortcut only works because HCl is a strong acid. Don’t try this with acetic acid.

Strong base — 0.015 M NaOH

[OH⁻] = 0.015 M.

pOH = −log(0.015) = 1.82 pH = 14.00 − 1.82 = 12.18

Strong vs. weak — the move that matters

Strong acids (HCl, HBr, HI, HNO₃, HClO₄, H₂SO₄ for the first proton) dissociate completely. [H⁺] equals the formal acid concentration. For diprotic H₂SO₄ in intro courses you can approximate [H⁺] ≈ 2[H₂SO₄], though the second dissociation is actually only partial.

Strong bases (NaOH, KOH) likewise give [OH⁻] = formal concentration. For Ca(OH)₂ or Ba(OH)₂, double it: [OH⁻] = 2[base] because each formula unit liberates two hydroxides.

Weak acids and bases break the shortcut. Acetic acid at 0.10 M does not give pH = 1.00. Only ~1.3% of CH₃COOH actually dissociates, so [H⁺] ≈ 1.3 × 10⁻³ M and pH ≈ 2.87. To get there you need K_a (1.8 × 10⁻⁵ for acetic acid) and an ICE table:

CH₃COOH ⇌ H⁺ + CH₃COO⁻ K_a = [H⁺][A⁻] / [HA] = x² / (0.10 − x) ≈ x² / 0.10

x² = (1.8 × 10⁻⁵)(0.10) → x = 1.34 × 10⁻³ M → pH = 2.87.

Treat a weak acid as strong and you’ll be off by two pH units. That’s a factor of 100 in [H⁺].

Sig figs in pH — the rule that surprises everyone

For pH, the digits to the right of the decimal point count as significant figures. The whole-number part just locates the order of magnitude.

If [H⁺] = 2.5 × 10⁻³ M (2 sig figs), then pH = 2.60 — two decimals, not three. Writing pH = 2.602 implies you knew [H⁺] to four significant figures, which you didn’t.

This is unusual: in most calculations sig figs count from the first non-zero digit, but logs flip the convention. Lab grading rubrics enforce it.

Common mistakes

Dropping the negative sign. pH = −log[H⁺]. Forget the minus and you get pH = −3.46 for 3.5 × 10⁻⁴ M HCl, which is nonsense.

Mixing pH and pOH. Low pH means acidic. Low pOH means basic. They are mirror images that sum to 14 at 25 °C.

Treating weak as strong. The single biggest pH error in general chemistry. If the acid isn’t on the strong-acid list, you need K_a and an equilibrium calculation.

Wrong sig figs. Decimal places in pH = sig figs in [H⁺], not the total digit count.

Assuming 14.00 always. K_w changes with temperature. At 50 °C, K_w ≈ 5.5 × 10⁻¹⁴ and pH + pOH ≈ 13.26, not 14. Pure water is still neutral, but its pH at 50 °C is 6.63, not 7.00.

The pH Calculator takes any of the four inputs and returns the other three in one pass — handy for buffer prep or for sanity-checking equilibrium calculations.

How to Calculate pH from Concentration