# How can the pH value be unitless? [duplicate]

How can the definition of the pH value $\mathrm{pH}=-\log\left[\ce{H+}\right]$ yield a unitless result? Since $\left[\ce{H+}\right]$ is a concentration, wouldn't it have units $\mathrm{mol/dm^3}$ and therefore following the product rule you would get: $-\log\left(x\ \mathrm{mol/dm^3}\right)=-\log\left(x\right)-\log\left(\mathrm{mol/dm^3}\right)$.

How can these units in a mathematical sense suddenly disappear?

• Good question. I didn't know the answer so here is a very enlightening IUPAC link from wikipedia. I guess chemists we are not very good at rigour. The pH definition could make a mathematician or a physicist cry. From the paper: "(pH) involving as it does a single ion activity, it is immeasurable". How you can introduce a definition of something that measures a common and useful quantity (acidity) that is itself "immeasurable" (what do they even mean by immeasurable?) is an interesting discussion topic. – K_P Oct 27 '14 at 19:54
• I always thought that it was the number of H+ ions divided by the total number, so a relative ratio. Note that the acidity should not depend on the density of the solution as it does in your viewpoint. – Jon Custer Oct 27 '14 at 20:07
• The reason there is a difference between the notional definition and the operational definition is that the activity of a single ion cannot actually be measured independently because it is not independent of its counter-ion. Even though it's not directly measurable, it's still useful conceptually, just as many quantum mechanical systems do not have exact solutions, numerical approximations are still of value. – Michael DM Dryden Oct 27 '14 at 20:37
• As an aside, the log of a value never has a unit, even if the original value had a unit. I can't explain why, though. – Popher Sep 20 '15 at 19:57