pH scale

The pH scale is a scale that expresses the hydronium ion concentration, [H₃O⁺], in an aqueous solution using log base 10.

When you are dealing with very large or very small numbers it is convenient to use logarithms. Note: We will be using the logarithm base 10, log₁₀, which is denoted as "log" (this is not the log "base 2", log₂; or the log base "e", log_e, that is the natural logarithm noted as ln).

Pure neutral water at room temperature has a hydronium ion concentration, [H₃O⁺] = 1x10^-7M.

The pH is defined as:

\[pH = -log([H_3O^+])\]

This means for pure water at room temperature

\[pH = -log([H_3O^+])= -log(1x10^{-7}) = -(-7) = 7\]

Thus the pH for neutral water at room temperature is 7. (Again it is important to realize there is nothing "magic" about 7. It just so happens that at room temperature, K_w is nearly exactly a perfect factor of 10 so the math works out in nice round numbers.). Why take the logarithm? Convenience. Rather than having to say "[H₃O⁺] = 7.2x10^-5M" you can simply say "pH = 4.14" (-log(7.2x10^-5)=-(-4.14)=4.14). Also, in the days before calculators the use of logarithms simplified many calculations. The irony being that now the use of logarithms makes the mathematics more difficult for many students.

You need to remember how to perform basic logarithm math. In particular, you need to be able to take the inverse log. Thus to convert from pH to [H₃O⁺]

\[\rm{[H_3O^+]} = 10^{-pH}\]

pH can vary widely. However, there are some natural limits to the value since [H₃O⁺] has some limits. For example, it is not possible in any aqueous solution for [H₃O⁺] = 200 M. This is simply absurd since one liter of water has only 55 moles of water. It is not possible to have 200 moles of H₃O⁺ in solution that has only 55 moles of water. However, the hydronium ion concentration can easily be larger than 1 M. The pH of these solutions is negative (as the log of numbers larger than 1 is a positive number and pH = -log([H₃O⁺]). On the dilute end of the spectrum, it is not possible to obtain hydronium ion concentrations much less than 1x10^-14 M as such extremely low values imply extremely large values of [OH^-] as the two are linked by K_w.

Practically speaking [H₃O⁺] range from 1x10^-14 M up to 1 M ( but they can be slightly smaller or larger than this). That means that typical pH values range from 14 to 0. Below is a pH scale that shows both the pH and the [H₃O⁺] concentration relative to pH = 7. It is important to note that since pH is a log scale, changes by one unit are really changes in concentration of 10x.