pH and Hydrogen Ion Concentration

Introduction

Lemon juice has a pH of about 2 and black coffee a pH of about 5. It is tempting to think the lemon juice is a little more than twice as acidic. In fact it contains roughly a thousand times more hydrogen ions. The pH scale is not an ordinary scale like a ruler; each step represents a jump by a factor of ten. Understanding this is the key to using pH values properly.

Core explanation

A tenfold scale. When the pH decreases by 1, the concentration of hydrogen ions, [H⁺], increases by a factor of 10. When the pH increases by 1, [H⁺] falls to one-tenth of its value. So:

- pH 3 has 10 times the [H⁺] of pH 4; - pH 3 has 100 times the [H⁺] of pH 5; - pH 3 has 1000 times the [H⁺] of pH 6.

In general, a difference of n pH units corresponds to a factor of 10ⁿ in hydrogen ion concentration.

Linking pH to concentration. For whole-number values, the pH is the power of ten with the sign removed:

pH [H⁺] / mol/dm³ --- --- 0 1 (= 10⁰) 1 0.1 (= 10⁻¹) 2 0.01 (= 10⁻²) 3 0.001 (= 10⁻³) 7 0.000 000 1 (= 10⁻⁷) 12 10⁻¹² 14 10⁻¹⁴

So hydrochloric acid of concentration 0.01 mol/dm³, which is fully ionised, has [H⁺] = 10⁻² mol/dm³ and pH 2.

Why use such a scale? Hydrogen ion concentrations in everyday solutions range from about 1 mol/dm³ down to about 0.000 000 000 000 01 mol/dm³ — a range of a hundred million million. Writing these directly would be clumsy. The pH scale compresses this enormous range into manageable numbers from 0 to 14. Scales that work like this are called logarithmic ; the decibel scale for sound and the Richter magnitude scale for earthquakes are other examples.

Non-whole values. A pH of 2.5 lies between 2 and 3; its [H⁺] is about 0.003 mol/dm³, roughly three times that at pH 3. A change of 0.3 pH units corresponds to about a doubling of [H⁺]. These small changes are significant in biology: blood with a pH of 7.1 instead of 7.4 contains about twice as many hydrogen ions.

What about alkalis? As pH rises above 7, [H⁺] keeps falling by a factor of ten per unit, and [OH⁻] rises by a factor of ten per unit. A solution at pH 12 has 10 times the hydroxide ion concentration of one at pH 11.

Formulae

pH = −log₁₀[H⁺], equivalently [H⁺] = 10^(−pH) mol/dm³. Factor change in [H⁺] between two solutions = 10^(difference in pH).

Step-by-step reasoning

To compare the acidity of two solutions:

1. Find the difference between their pH values. 2. Raise 10 to the power of that difference. 3. The solution with the lower pH has that many times more H⁺ ions. 4. State the answer clearly, for example "pH 2 has 1000 times the H⁺ concentration of pH 5".

Visual explanation

Draw a staircase where each step down in pH makes the column of H⁺ ions ten times taller. At pH 6 there is one small block; at pH 5, ten blocks; at pH 4, a hundred blocks; at pH 3, a thousand blocks. The columns grow so fast that pH 1 would tower off the page.

Real-world analogy

Think of money in units: pennies, ten-pence coins, pound coins, ten-pound notes. Moving up one type multiplies the value by ten. Moving three steps, from a penny to a ten-pound note, multiplies by a thousand. pH steps work the same way, but in reverse: each step down multiplies [H⁺] by ten.

Real-world example

Healthy rain water is slightly acidic, around pH 5.6, because it dissolves carbon dioxide. In areas badly affected by acid rain, rain with a pH of about 4.2 to 4.6 was often recorded. A drop from 5.6 to 4.6 means ten times as many hydrogen ions, enough to damage lakes, forests and limestone buildings.

Why?

Why does one pH unit equal a factor of ten rather than a fixed amount? Because pH is defined as the negative logarithm (base ten) of [H⁺]. A logarithm counts powers of ten, so adding 1 to the pH means dividing [H⁺] by ten. This definition was chosen to make huge ranges of concentration easy to handle.

Common misconception

"pH 2 is twice as acidic as pH 4 because 4 is double 2." pH differences represent powers of ten, not simple ratios. A solution at pH 2 has 10² = 100 times the hydrogen ion concentration of one at pH 4.

Worked example

Question: A sample of lake water changes from pH 6.5 to pH 4.5 over several years. By what factor has the hydrogen ion concentration changed?

Reasoning: Difference in pH = 6.5 − 4.5 = 2. Factor = 10² = 100. The pH has fallen, so [H⁺] has increased.

Answer: The hydrogen ion concentration has increased by a factor of 100.

Quick check

1. How many times greater is [H⁺] at pH 3 than at pH 6? Answer: 10³ = 1000 times greater.

Exam focus

Examiners test the tenfold rule with "how many times more concentrated" questions. Always state which solution has more H⁺ and by what factor. Link decreasing pH to increasing [H⁺]. At this level you may be asked for [H⁺] at whole-number pH values; use powers of ten.

Advanced insight

Because [H⁺] × [OH⁻] = 1 × 10⁻¹⁴ (mol/dm³)² in water at 25 °C, you can find the OH⁻ concentration of any solution from its pH. At pH 11, [H⁺] = 10⁻¹¹, so [OH⁻] = 10⁻¹⁴ ÷ 10⁻¹¹ = 10⁻³ mol/dm³. This relationship, the ionic product of water, underlies the whole 0–14 scale.

Summary

pH is a logarithmic scale: each decrease of one unit means a tenfold increase in hydrogen ion concentration, and a difference of n units means a factor of 10ⁿ. For whole-number values, [H⁺] = 10^(−pH) mol/dm³. The scale compresses an enormous range of concentrations into convenient numbers, and small pH changes can represent large changes in acidity.

Practice questions

1. What is the hydrogen ion concentration of a solution with pH 4? Answer: 10⁻⁴ mol/dm³, which is 0.0001 mol/dm³. 2. Solution X has pH 1 and solution Y has pH 5. Compare their hydrogen ion concentrations. Answer: X has 10⁴ = 10 000 times the hydrogen ion concentration of Y. 3. A fully ionised acid has [H⁺] = 0.001 mol/dm³. What is its pH? Answer: 0.001 = 10⁻³, so the pH is 3. 4. If the pH of a solution rises from 8 to 10, what happens to its hydroxide ion concentration? Answer: It increases by a factor of 10² = 100.