One simple number for acidity
In lesson 1.8.5 you measured acidity with pH paper: pH runs from about 0 to 14, below 7 is acidic, and above 7 is basic. But what is pH, exactly? In lesson 2.12.2, 0.10 M acetic acid had [H⁺] ≈ 1.3 × 10⁻³ M, and [H⁺] in water solutions ranges from around 1 M down to 10⁻¹⁴ M, more than a dozen powers of ten. Numbers like that are awkward to write and compare. In 1909 the Danish chemist Søren Sørensen suggested keeping just the exponent.
Defining pH
The pHpH of a solution is the negative base-10 logarithm of [H⁺]:
[H⁺] is in M (mol/L). For example, [H⁺] = 1 × 10⁻³ M gives pH = 3.
Each drop of 1 in pH means [H⁺] is 10 times higher. A solution at pH 2 has 10³ = 1000 times the [H⁺] of a solution at pH 5.
pOHpOH is defined the same way:
First: water ionizes on its own. In pure water, a tiny fraction of the water molecules ionize into H⁺ and OH⁻ (H₂O ⇌ H⁺ + OH⁻). So every water solution contains both H⁺ and OH⁻; only the amounts differ. Acidic solutions have more H⁺, and basic solutions have more OH⁻. At 25 °C, whatever is dissolved in the water, the product of the two concentrations is always 1.0 × 10⁻¹⁴. In pure water they’re equal, 1.0 × 10⁻⁷ M each. Lesson 2.12.4 explains why.
As an equation, at 25 °C every water solution has:
Taking the negative log of both sides gives, at 25 °C:
So at 25 °C a solution is acidic below pH 7, neutral at pH 7, and basic above pH 7.
Significant figures in pH
Lesson 2.1.1 promised that logarithms have their own sig-fig rule: the number of decimal places in pH equals the number of significant figures in [H⁺].
- [H⁺] = 2.5 × 10⁻³ M (2 sig figs) gives pH = 2.60 (2 decimal places), not 2.6 or 2.602.
- Going back, pH = 3.45 (2 decimal places) gives [H⁺] = 10−3.45 = 3.5 × 10⁻⁴ M (2 sig figs).
- Writing pH = 7.00 means [H⁺] = 1.0 × 10⁻⁷ M. Writing just pH = 7 strictly gives no significant figures at all.
The same goes for pOH, and converting with pH + pOH = 14.00 keeps the number of decimal places. The tool card below shows where the rule comes from.
🔧 Math tool card · LogarithmsSkip it if you already know this
A logarithm is a power of 10. If 10ˣ = N, then x is the base-10 logarithm of N, written x = log N. So log 100 = 2, log 1 = 0, and log 0.001 = −3.
Two rules are all you need:
So log(2.5 × 10⁻³) = log 2.5 + log 10⁻³ = 0.398 − 3 = −2.602, and pH = 2.602. Three values are handy to remember: log 2 ≈ 0.30, log 3 ≈ 0.48, log 5 ≈ 0.70.
Going back to concentration. Given pH, [H⁺] = 10−pH. On most calculators this is SHIFT (or 2nd) and then log, which gives the 10ˣ key.
Where the sig-fig rule comes from. In log(2.5 × 10⁻³) = −3 + 0.398, the whole-number part, −3, comes from the power of 10 and only places the decimal point. The decimal part, 0.398, comes from the coefficient 2.5. The coefficient has 2 significant figures, so only 2 digits of the decimal part are reliable. That is why pH = 2.60.
What is the pH when [H⁺] = 2 × 10⁻⁴ M? (log 2 ≈ 0.30)
pH = −log(2 × 10⁻⁴) = −(0.30 − 4) = 3.7. The concentration has 1 sig fig, so pH gets 1 decimal place. 4.3 has the sign of log 2 reversed; −3.7 forgets the minus sign in the definition.
Given any one of [H⁺], [OH⁻], pH, and pOH, you can find the other three. Enter a value below and look at the steps and the sig figs. Then press “[H⁺] × 10” and check that pH drops by exactly 1:
pH of strong acids and strong bases
Strong acids and bases ionize completely, so [H⁺] or [OH⁻] comes straight from the concentration:
- Monoprotic strong acid: [H⁺] = concentration of the acid. In 0.010 M HCl, [H⁺] = 0.010 M and pH = 2.00.
- Strong base: [OH⁻] = concentration of the base × the number of OH⁻ per formula unit. In 0.0010 M Ba(OH)₂, [OH⁻] = 2.0 × 10⁻³ M.
How Chinese textbooks differ
- Chinese textbooks write lg for the base-10 log and c(H⁺) for the concentration: pH = −lg c(H⁺), with c in mol/L.
- Chinese textbooks rarely use pOH. They usually find c(H⁺) from c(H⁺)·c(OH⁻) = 1.0 × 10⁻¹⁴ first and then take the pH.
- The decimal-places rule is a US convention. Chinese practice problems often give pH as a whole number (0.01 mol/L HCl has pH = 2); this site follows the rule throughout.
Example 1: pH of a strong base
At 25 °C, find [OH⁻], pOH, pH, and [H⁺] for 0.0010 M Ba(OH)₂.
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[OH⁻]. Ba(OH)₂ is a strong base and dissociates completely, giving 2 OH⁻ per formula unit:
so [OH⁻] = 2 × 0.0010 = 2.0 × 10⁻³ M (2 sig figs).
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pOH. log 2.0 = 0.30:
Two sig figs in the concentration means two decimal places in the pOH.
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pH. At 25 °C, pH + pOH = 14.00:
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[H⁺].
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Check. 2.0 × 10⁻³ × 5.0 × 10⁻¹² = 1.0 × 10⁻¹⁴, which is Kw.
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A common mistake. Forgetting the 2 and using [OH⁻] = 0.0010 M, which gives pH 11.00.
Example 2: from pH back to concentration
Stomach acid has a pH of about 1.20. What is its [H⁺]? How many times higher is that than in a solution at pH 3.20?
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[H⁺]. Undo pH = −log[H⁺]:
The pH has 2 decimal places, so the concentration gets 2 sig figs.
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Compare. The pH values differ by 3.20 − 1.20 = 2.00. Each pH unit is a factor of 10, so [H⁺] differs by 10² = 100 times.
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Check. At pH 3.20, [H⁺] = 6.3 × 10⁻⁴ M:
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A common mistake. Thinking pH 1.20 vs. 3.20 is “less than 3 times” different. pH is a logarithm: a difference of 2 means a factor of 100.
- [H⁺] = 1.0 × 10⁻² M
pH = −log(1.0 × 10⁻²) = 2.00. Two sig figs, two decimal places.
- [H⁺] = 0.10 M
0.10 = 1.0 × 10⁻¹, so pH = 1.00.
- [H⁺] = 2.5 × 10⁻³ M
pH = 3 − log 2.5 = 3 − 0.40 = 2.60.
- [H⁺] = 5.0 × 10⁻⁴ M
pH = 4 − log 5.0 = 4 − 0.70 = 3.30.
- [OH⁻] = 1.0 × 10⁻³ M
pOH = 3.00, so pH = 14.00 − 3.00 = 11.00.
- 0.010 M NaOH
[OH⁻] = 0.010 M, pOH = 2.00, pH = 12.00.
- A solution with pOH = 13.00
pH = 14.00 − 13.00 = 1.00.
- 0.0050 M Ba(OH)₂
[OH⁻] = 2 × 0.0050 = 0.010 M, pOH = 2.00, pH = 12.00.
Common mistakes
Mistake 1: “A solution at pH 3 is twice as acidic as one at pH 6.” pH is a log scale. A difference of 3 means [H⁺] differs by 10³ = 1000 times.
Mistake 2: “The calculator says pH = 2.6020599…, and more digits are more accurate.” [H⁺] = 2.5 × 10⁻³ M has only 2 significant figures, so the pH is 2.60. The extra digits come from the calculator, not from the measurement.
Mistake 3: “A basic solution has no H⁺, so it has no pH.” Every water solution contains both H⁺ and OH⁻. In a base [H⁺] is tiny but not zero, and the pH is still defined; 0.010 M NaOH has pH 12.00.
Quick check
At 25 °C a solution has [H⁺] = 3.0 × 10⁻⁵ M. How should its pH be written?
pH = −log(3.0 × 10⁻⁵) = 5 − 0.477 = 4.523. The concentration has 2 sig figs, so the pH gets 2 decimal places: 4.52.
At 25 °C a solution has pH 9.40. What is [OH⁻]?
pOH = 14.00 − 9.40 = 4.60, so [OH⁻] = 10−4.60 = 2.5 × 10⁻⁵ M. 4.0 × 10⁻¹⁰ M is [H⁺].
Summary
- pH = −log[H⁺] and pOH = −log[OH⁻]. A difference of 1 in pH is a factor of 10 in [H⁺].
- At 25 °C, [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ and pH + pOH = 14.00; [H⁺] = 10−pH.
- pH gets as many decimal places as [H⁺] has significant figures: [H⁺] = 2.5 × 10⁻³ M gives pH 2.60.
On to AP (for later)
AP topic 8.1 asks you to convert among [H₃O⁺], [OH⁻], pH, and pOH, and topic 8.2 asks for the pH of strong acid and base solutions. The “p” means “negative log of”; you’ll also meet pKa = −log Ka (lesson 2.12.5) and pKw. The decimal-places rule is standard in US textbooks, and following it is always safe.