Math Detours · Grade 11 · Connections · 3

What pH measures

Each step on the pH scale is ten times the acidity. A logarithm turns a range of a hundred trillion into the numbers 0 to 14.

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Tonight's idea

How acidic a liquid is depends on its concentration of hydrogen ions, written \([\text{H}^+]\) and measured in moles per litre. (A mole is a fixed, very large number of particles.) In everyday liquids that concentration runs from about 1 down to about \(10^{-14}\): a range of a hundred trillion. Nobody wants to compare 0.01 with 0.00000001 by eye. So chemists report the exponent instead.

\(\text{pH} = -\log_{10} [\text{H}^+]\), which is the same as \([\text{H}^+] = 10^{-\text{pH}}\).

A base-ten logarithm is a power of ten (where you met this). If \([\text{H}^+] = 10^{-3}\), the logarithm is \(-3\) and the pH is 3. The minus sign makes the usual values positive. It has a side effect: more acid means a lower pH.

0123456789101112131410⁰10⁻²10⁻⁴10⁻⁶10⁻⁸10⁻¹⁰10⁻¹²10⁻¹⁴ lemon juiceblack coffeepure waterseawaterhousehold ammonia [H⁺] in moles per litre acidicalkaline
Each step right divides \([\text{H}^+]\) by 10. The everyday values are approximate.

Because pH is an exponent, adding to it multiplies the concentration. One unit lower means ten times the hydrogen ions. Two units lower means a hundred times. Pure water sits at 7. Below 7 is acidic and above 7 is alkaline.

Watch one: both directions

A sample has \([\text{H}^+] = 2.5 \times 10^{-4}\). The log of a product is the sum of the logs:

\[\text{pH} = -\big(\log_{10} 2.5 + \log_{10} 10^{-4}\big) = -(0.40 - 4) = 3.60.\]

Sensible: the concentration is between \(10^{-4}\) and \(10^{-3}\), and the pH is between 4 and 3.

Now backwards. Seawater has a pH of about 8.1, so \([\text{H}^+] = 10^{-8.1} = 7.9 \times 10^{-9}\). Check: \(-\log_{10}(7.9 \times 10^{-9}) = 8.10\).

Your turn

  1. Find the pH when \([\text{H}^+]\) is \(10^{-3}\), when it is \(10^{-7}\), and when it is \(4 \times 10^{-5}\).
  2. Lemon juice has a pH of about 2 and black coffee about 5. How many times more concentrated are the hydrogen ions in the juice? Then compare two juices, one of pH 2.0 and one of pH 2.3.
  3. A rain sample has pH 5.2. Find \([\text{H}^+]\). How many times more concentrated are its hydrogen ions than those of pure water?
  4. Equal volumes of two strong acid solutions are mixed, one of pH 2 and one of pH 4. (In a strong acid the hydrogen ion concentrations simply average.) Explain why the mixture is not pH 3, and find its pH.

Stretch: a small number that is not small

The average pH of the ocean surface has fallen from about 8.2 to about 8.1 since the start of the industrial age. By what percentage has \([\text{H}^+]\) risen? What fall in pH would mean that \([\text{H}^+]\) had doubled?

Answers

  1. 3 and 7, straight from the exponents. For the third, \(-(\log_{10} 4 + \log_{10} 10^{-5}) = -(0.60 - 5) = 4.40\).
  2. Three units apart, so \(10^3 = 1000\) times. For the two juices the gap is 0.3, and \(10^{0.3} = 2.0\): the pH 2.0 juice has twice the hydrogen ions. A gap of 0.3 is a doubling because \(\log_{10} 2 = 0.30\).
  3. \([\text{H}^+] = 10^{-5.2} = 6.3 \times 10^{-6}\) moles per litre. Pure water has \(10^{-7}\). The ratio is \(10^{-5.2} \div 10^{-7} = 10^{1.8} = 63\).
  4. pH is a logarithm, and you cannot average logarithms and expect the average concentration. Average the concentrations themselves: \(\tfrac12(10^{-2} + 10^{-4}) = \tfrac12(0.0101) = 5.05 \times 10^{-3}\). Then \(\text{pH} = -\log_{10}(5.05 \times 10^{-3}) = 2.30\). The pH 2 solution has 100 times the hydrogen ions, so it dominates. Mixing has done little more than dilute it by half, which raises the pH by 0.3.

Stretch. The ratio of concentrations is \(10^{-8.1} \div 10^{-8.2} = 10^{0.1} = 1.26\), a rise of about 26%. Doubling needs \(10^{x} = 2\), so \(x = \log_{10} 2 = 0.30\): a fall of 0.3 in pH. On a logarithmic scale a small step is a large factor.

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