Grade 12 Chemistry · The pH scale
p is a machine, not a letter
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p is a machine, not a letter

Grade 11 taught you to read a pH. Grade 12 asks what the symbol MEANS, because the same move is about to be played on four other quantities. Lowercase p is an operator: it takes the base-ten logarithm of whatever follows and flips the sign. So pH=log[H+]\mathrm{pH} = -\log[\mathrm{H^+}] — read aloud: pH equals negative log of H-plus. Naming the parts: [H+][\mathrm{H^+}] is the hydrogen-ion concentration in mol/L (the square brackets are read the concentration of), and pH is a pure number — the logarithm ate the units on the way through. The hydroxide side gets the identical treatment: pOH=log[OH]\mathrm{pOH} = -\log[\mathrm{OH^-}], where [OH][\mathrm{OH^-}] is the hydroxide-ion concentration, also in mol/L.

Why two scales chained together? Because water quietly ionizes itself, and the product of the two ion concentrations is fixed: Kw=[H+][OH]=1.0×1014K_w = [\mathrm{H^+}][\mathrm{OH^-}] = 1.0 \times 10^{-14} at 25 °C, where KwK_w is the ion product of water, a pure number. Run p over both sides — the log of a product is a sum — and the whole of grade 11's see-saw drops out as a theorem: pH+pOH=14\mathrm{pH} + \mathrm{pOH} = 14. It was never a rule to memorise. It is KwK_w wearing a p.