Hydrogen ion concentrations in ordinary solutions range across fifteen powers of ten. Nobody wants to write those numbers out, so we take a logarithm instead — and that one move is the whole of pH.
📚 What you need to know
pH = –log10[H+], where [H+] is in mol dm–3.
Rearranged: [H+] = 10–pH.
The scale is logarithmic to base 10: one pH unit is a factor of ten in [H+].
A lower pH means a higher [H+] and a more acidic solution.
At 298 K, pH < 7 is acidic, pH = 7 is neutral and pH > 7 is alkaline.
Values are quoted to two decimal places, and are not restricted to the range 0–14.
A pH meter gives an accurate value; universal indicator gives a rough one.
The definition
The acidity of a solution comes down to how many H+ ions — strictly H3O+ ions — are floating about in it. pH is just a compact way of stating that concentration.
Definition of pH
pH = –log10[H+]
The colours are those of universal indicator. They are a convenient guide, not a measurement — you cannot read two decimal places off a colour chart.
What the logarithm is doing
A logarithm answers the question “ten to the what?”. If [H+] is 10–3 mol dm–3, then log10 of it is –3, and the minus sign in the definition flips that to a friendly pH of 3.
Read the table downwards and the pH rises while the concentration falls. The relationship is inverse, which is exactly what the minus sign in the definition is for.
Why bother with logs at all? Because the numbers are otherwise unmanageable. A strong acid might have [H+] of 1 mol dm–3 and an oven cleaner 0.000 000 000 000 01 mol dm–3. On a log scale those become 0 and 14 — two numbers you can plot on the same axis.
Going backwards
If a question hands you a pH and asks for the concentration, you undo the logarithm.
From pH to concentration
[H+] = 10–pH
Find the 10x button on your calculator before the exam, not during it. On some models it is labelled ALOG or reached with INV then LOG. And check the sign: a pH of 3.40 needs 10–3.40, not 103.40.
WORKED EXAMPLE
A solution of hydrochloric acid has [H+] = 2.5 × 10–3 mol dm–3. Calculate its pH.
Step 1 — put it straight into the definitionpH = −log₁₀(2.5 × 10⁻³)Step 2 — evaluate= −(−2.602) = 2.602pH = 2.60A sensible answer: the concentration is between 10⁻³ and 10⁻², so the pH must land between 2 and 3.
WORKED EXAMPLE
A solution has a pH of 3.40. Calculate the concentration of hydrogen ions.
Step 1 — rearrange[H⁺] = 10 to the power (−pH)Step 2 — substitute[H⁺] = 10⁻³·⁴⁰ = 3.98 × 10⁻⁴3.98 × 10⁻⁴ mol dm⁻³Check it sits between 10⁻⁴ and 10⁻³, as a pH between 3 and 4 demands.
WORKED EXAMPLE
10.0 cm3 of a nitric acid solution of pH 1.00 is added to distilled water to make 1000.0 cm3 of solution. Calculate the pH of the diluted solution.
Step 1 — starting concentration[H⁺] = 10⁻¹·⁰⁰ = 0.100 mol dm⁻³Step 2 — the dilution factor1000.0 ÷ 10.0 = 100Step 3 — new concentration0.100 ÷ 100 = 1.00 × 10⁻³ mol dm⁻³pH = −log₁₀(1.00 × 10⁻³)pH = 3.00Diluting by a factor of 100 raises the pH by exactly 2. Every tenfold dilution is worth one pH unit — a shortcut worth having.
Measuring it
Method
How it works
Strengths and limits
pH meter
An electrode dipped in the solution produces a voltage that depends on [H+]; the meter converts it to a reading
Accurate to two decimal places, but needs calibrating with buffer solutions first
Universal indicator
A mixture of dyes that changes colour over the whole range; the colour is matched to a chart
Quick and cheap, but gives a whole number at best and is useless with coloured solutions
Careful with the phrase “pH 7 is neutral”. That is only true at 298 K. Warm the water up and neutral water has a pH below 7, which is the subject of the next page.
💡 Exam tip
Quote pH to two decimal places unless told otherwise.
Do not lose the minus sign. If your pH comes out negative for a dilute acid, you have dropped it.
Estimate before you calculate: a concentration of 4 × 10–5 must give a pH between 4 and 5.
The formula uses [H+]. If a question gives you [OH–], you have more work to do first.
A tenfold dilution raises pH by one unit. Use it to check dilution answers instantly.
Never describe a change of one pH unit as “slightly” more acidic.
⚠️ Common mix-up
Treating the scale as linear. pH 2 is ten times more acidic than pH 3, not half as much again.
Forgetting the negative sign in either direction, giving 10pH instead of 10–pH.
Putting [OH–] into the pH formula and quoting the answer as a pH.
Believing the scale stops at 0 and 14. Both ends can be exceeded.
Confusing concentration with strength. A dilute strong acid can have a higher pH than a concentrated weak one.
Up next: The Ionic Product of Water — pH only tracks the hydrogen ions. To handle alkalis, and to see why “neutral” is not quite the same as “pH 7”, you need the other half of the picture.
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