# pH Scale

Skills to be Tested

- Discuss the pH scale.
- Point out the neat things about the pH scale.
- Tell the origin and the logic of using the pH scale.
- Apply the same strategy for representing other types of quantities such as p
*K*_{a}, p*K*_{b}, p*K*_{w}.

From the simple definition of pH being the negative log of the \(\ce{H+}\) ion concentration, \(\ce{[H+]}\),

\[\mathrm{pH = - \log [H^+]}\]

you already know the following:

- This scale is convenient to use, because it converts some odd expressions such as 1.23x10
^{-4}into a single number of 3.91. - This scale covers a very large range of \(\ce{[H+]}\), from 0.1 to 10
^{-14}. When \(\ce{[H+]}\) is high, we usually do not use the pH value, but simply the \(\ce{[H+]}\). For example, when \(\mathrm{[H^+] = 1.0}\), pH = 0. We seldom say the pH is 0, and that is why you consider pH = 0 such an odd expression. A pH = -0.30 is equivalent to a \(\ce{[H+]}\) of 2.0 M. Negative pH values are only for academic exercises. Using the concentrations directly conveys a better sense than the pH scales. - The pH scale expands the division between zero and 1 in a linear scale or a compact scale into a large scale for comparison purposes. In mathematics, you learned that there are infinite values between 0 and 1, or between 0 and 0.1, or between 0 and 0.01 or between 0 and any small value. Using a log scale certainly converts infinite small quantities into infinite large quantities.
- The non-linearity of the pH scale in terms of \(\ce{[H+]}\) is easily illustrated by looking at the corresponding values for pH between 0.1 and 0.9 as follows:

pH = |
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|

[H^{+}] = |
0.79 | 0.63 | 0.50 | 0.40 | 0.32 | 0.25 | 0.20 | 0.16 | 0.13 |

- Because the negative log of \(\ce{[H+]}\) is used in the pH scale, the pH scale usually has positive values. Furthermore, the larger the pH, the smaller the \(\ce{[H+]}\).

Other Interesting Facts about the pH Scale

The pH scale was originally introduced by the Danish biochemist S.P.L. Sorensen in 1909 using the symbol p_{H}. Other symbols such as *p*_{H} have been used in the past. The letter *p* is derived from the German word *potenz* meaning power or exponent of, in this case, 10. You may argue, what have we learned by looking into the historical origin of the term? Well, the origin of concept is interesting in that we sometimes need to develop concepts ourselves. A concept or tool becomes important if many people find it convenient and elegant.

The reality is that many chemists have used the pH scale and the p scales for many other quantities and often take it for granted, without realizing the logic behind their usages. For example, we have used the p*K*_{a}, p*K*_{b}, p*K*_{w}, notations by analogy to the pH notations without asking a question. Now that you know the pH is an exponent, the following relationship is obvious:

\[\mathrm{[H^+] = 10^{-pH}}\]

### Questions

- What is the \(\ce{[H+]}\) in a solution whose pH is 3.21?

### Solutions

- Answer 6.17e-4

Consider...

\(\mathrm{[H^+] = 10^{-pH}}\)

### Contributors

Chung (Peter) Chieh (Chemistry, University of Waterloo)