Extended reading: why the pH scale is a logarithm

A scale that is not even

On a ruler, a step of $1\ \text{cm}$ is always the same length. The pH scale is not like that. A step of $1$ on the pH scale means ten times more or less of something.

What pH measures

An acid gives hydrogen ions, $\text{H}^+$, in water. The amount in each litre is the concentration $[\text{H}^+]$, in mol/L. In pure water it is a very small number, $0.0000001\ \text{mol/L}$, which is $10^{-7}\ \text{mol/L}$. In $0.1\ \text{M}$ hydrochloric acid it is $0.1\ \text{mol/L}$, which is $10^{-1}$.

Numbers like $10^{-7}$ are awkward. The chemist Søren Sørensen suggested a plain number instead: the power of ten, with the minus sign removed.

If you do not know logarithms yet, read it this way. For a concentration written as $10^{-n}$, the pH is just $n$.

One step is a factor of ten

Between pH $3$ and pH $4$ the concentration of hydrogen ions falls from $10^{-3}$ to $10^{-4}$. That is ten times less. Between pH $2$ and pH $4$ the change is $10 \times 10 = 100$ times.

• A solution of pH $3$ has $100$ times as many hydrogen ions as one of pH $5$.
• Stomach acid, pH $2$, has $100\,000$ times as many as pure water, pH $7$.

This is the common mistake: pH $2$ is not twice as acidic as pH $4$. It is a hundred times as acidic.

The same for bases

A base gives hydroxide ions, $\text{OH}^-$. In water, at $25\ ^{\circ}\text{C}$, the two concentrations multiply to a fixed number:

So the two powers of ten add up to $14$: $\text{pH} + \text{pOH} = 14$. For $0.00001\ \text{M}$ sodium hydroxide, $[\text{OH}^-] = 10^{-5}$, so $\text{pOH} = 5$ and $\text{pH} = 14 - 5 = 9$. This is why the scale in the textbook runs from $0$ to $14$.

You should find that the blue dot is always one step higher for each power of ten, however strong the acid was.

Diluting an acid

A strong acid is diluted $10$ times. The concentration of hydrogen ions falls to a tenth, so the pH goes up by $1$. It is diluted $100\,000$ times: the pH goes up by $5$.

An acid never becomes a base by dilution. If you dilute an acid a million times, its pH moves towards $7$ (the water takes over), and it does not pass $7$.

Practise with new numbers

Try it in Python

This program changes a concentration to a pH, and compares two pH values.

Remember these

• $\text{pH} = -\log_{10}[\text{H}^+]$: for $10^{-n}$ the pH is $n$.
• One step of pH is a factor of $10$ in the hydrogen ions. pH $2$ against pH $4$ is $100$ times.
• $\text{pH} + \text{pOH} = 14$ at $25\ ^{\circ}\text{C}$.
• Diluting a strong acid $10$ times raises the pH by $1$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What is the pH of a solution with $[\text{H}^+] = 10^{-8}\ \text{M}$?

2. How many times as many hydrogen ions does stomach acid, pH $2$, have as pure water, pH $7$?

3. Can an acid become a base if it is diluted enough?

4. Why does the pH scale in school run from $0$ to $14$?