TL;DR:
pH is defined as
pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in mol/L.Each drop of one pH unit means ten times more hydrogen ions: pH 3 has
10⁻³M, pH 4 has10⁻⁴M.At 25 °C pure water is neutral at pH 7, since
[H⁺] = 10⁻⁷ M.
Hydrogen ion concentrations in water solutions span an enormous range, from about 1 mol/L in strong acid to a tiny 10⁻¹⁴ mol/L in strong base. Writing those numbers every time would be clumsy. The pH scale squeezes them into a handy range, usually 0 to 14, using a logarithm.
A quick look at logarithms
The base-10 logarithm of a number answers the question: what power of 10 gives this number? So log 1000 = 3 because 10³ = 1000, and log 0.01 = −2 because 10⁻² = 0.01. A step of 1 in the logarithm means a factor of 10 in the number.
The definition is pH = −log[H⁺]. The minus sign is there so that small concentrations give positive pH numbers. If [H⁺] = 10⁻³ M, then log = −3 and pH = 3.
Worked example: pH 3 compared with pH 4
At pH 3:
[H⁺] = 10⁻³ = 0.001 M.At pH 4:
[H⁺] = 10⁻⁴ = 0.0001 M.Compare:
10⁻³ ÷ 10⁻⁴ = 10¹ = 10.So the pH 3 solution has ten times the hydrogen ion concentration of the pH 4 solution.
The steps add up. Going from pH 3 to pH 6 is three steps, so the ratio is 10³ = 1000. The pH 3 solution has a thousand times more hydrogen ions than pH 6. A difference of two pH units is a factor of 100, and so on.
This is why pH numbers that look close can hide big differences. A solution at pH 2 is not a bit more acidic than one at pH 3: it has ten times the hydrogen ion concentration.
When the number is not a neat power of ten
Say [H⁺] = 2.0 × 10⁻⁴ M. A calculator gives log(2.0 × 10⁻⁴) = −3.70, so pH = 3.70. Going backward, [H⁺] = 10⁻³·⁷⁰ ≈ 2.0 × 10⁻⁴ M. A pH of 3.70 sits between 3 and 4, and the concentration lies between 10⁻³ and 10⁻⁴ too, closer to the smaller one.
For a strong acid that fully splits, the hydrogen concentration equals the acid concentration. So 0.0010 M hydrochloric acid has [H⁺] = 1.0 × 10⁻³ M and pH 3.00. This is the situation in the starting point of a titration curve.
Neutral water and the basic side
Water splits a little into H⁺ and OH⁻. At 25 °C the product of their concentrations is fixed: [H⁺][OH⁻] = 1.0 × 10⁻¹⁴. In pure water the two are equal, each 1.0 × 10⁻⁷ M, so pH = 7. That is neutral.
Taking logs of the fixed product gives pH + pOH = 14, where pOH = −log[OH⁻]. A solution with [OH⁻] = 10⁻³ M has pOH 3, so pH = 14 − 3 = 11. Below 7 is acidic, above 7 is basic. The neutral value is 7 only near 25 °C. At other temperatures the product of the ion concentrations changes, so neutral shifts slightly.
Do not taste or mix to find out: Never test a household liquid by tasting or smelling it. Never mix cleaning products, especially bleach with an acid or with ammonia, because that can release toxic gas. In a lab, use pH paper or a meter, with goggles and gloves.
Here is how the scale looks in everyday terms, without quoting exact measurements. Stomach acid is strongly acidic, black coffee is mildly acidic, pure water is neutral, and household bleach and drain cleaners are strongly basic. The exact pH of any real liquid varies, so treat these as ordering, not as figures to memorise.
Another worked case: what is the pH of a solution with [H⁺] = 0.316 M? log 0.316 = −0.50, so pH = 0.50. And the reverse: for pH 0.50, [H⁺] = 10⁻⁰·⁵⁰ = 0.316 M. Each half step on the scale is a factor of about 3.16, because 10^0.5 ≈ 3.16, and two half steps make a full factor of 3.16 × 3.16 ≈ 10.
How to check your answer
Sign. For
[H⁺]below 1 M, pH must be positive. If your answer is negative, check the minus sign.Direction. More hydrogen ions means a lower pH.
Round trip. Convert pH to concentration with
10⁻ᵖᴴand back.Range. Typical solutions fall between 0 and 14, but stronger solutions can be outside.
Mistakes that cost marks
Treating pH as a linear scale, so that pH 2 seems a little more acidic than pH 3.
Dropping the minus sign in the definition.
Taking the log of moles instead of concentration.
Writing the concentration after a dilution without recomputing it. See molarity and dilution.
Assuming a weak acid has
[H⁺]equal to its concentration. Only strong acids fully split.
Where you will use it next
pH feeds into buffers, equilibrium and solubility. The next idea to learn is how a system at equilibrium responds when disturbed, in Le Chatelier's principle with examples.
Frequently asked questions
Why do we use a logarithm?
Concentrations span many powers of ten. A logarithm turns each power of ten into one step, giving numbers that are easy to say and compare.
Can pH be negative or above 14?
Yes, for very concentrated strong acids or bases. For example, 2.0 M strong acid would have a pH near −0.3, since −log 2.0 = −0.30, though real behaviour in concentrated solutions deviates from the simple formula.
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