pH - The Mathematical Definition

pH, as explained here, is a way of understanding how acidic or basic a solution is. This, of course, is actually a function of the concentration of hydronium (H₃O⁺¹) and hydroxide (OH^-1) so it is important that you understand how they relate to each other. This idea is explored here.

What led chemists to create the pH scale was the huge range of possible concentrations for hydronium in aqueous solutions. A concentrated solution of a strong acid can have [H₃O⁺¹] over 10.0 M. A solution of strong base could have [H₃O⁺¹] less than 10^-14 M. This is a huge range, so in order to make it seem less extreme, chemists decided to focus on the power of ten in the concentration. In this way, the range described above runs from 1 to -15, rather than from 10 to 0.000000000000001.

Of course, JUST looking at the power of 10 isn’t quite right because 0.01 and 0.0999 are very different, but both have the same power of 10 (1x10^-2 vs 9.000x10^-2). The mathematical way to look at the power of 10, but still keep track of the rest of the number is to take the logarithm (or log) of the number. The log of .01 = -2, while the log of 0.0999 = -1.004. In this way we can see the very real difference between those two concentrations.

Just taking the log of [H₃O⁺¹] has one other issue attached to it. Except for extremely concentrated solutions of strong acids, log[H₃O⁺¹] will be negative. So, for convenience sake, we decided to multiple the answer by -1. In that way, the answer will be positive except in those rare extreme cases. This leaves us with the final definition:

Other Uses of “p” 

Having formed a useful way of looking at concentrations of hydronium, chemists figured that the same idea could be used elsewhere, and they used the “p” in the same way with other values. So, for instance:

and

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