Water is never just water. A tiny fraction of it is always split into H⁺ and OH⁻, and the product of those two is fixed — push one up and the other must come down. pH is the logarithm of the first, which is why a single step down the scale is a tenfold change and why stomach acid and drain cleaner are a hundred trillion times apart in something you can write with two digits.
Every bar below is a factor of ten. The blue and orange numbers are the real concentrations of H⁺ and OH⁻ — multiply them together at any point on the scale and you get the same 10⁻¹⁴, which is the one fact the whole subject rests on. Nothing is ever purely one or the other.
A strong acid hands over every proton it has. A weak one hands over a small fraction and keeps the rest, and how small a fraction is its Ka. Put the same amount of hydrochloric and acetic acid in the same water and the pH differs by nearly two units — not because there is less acid, but because most of the acetic acid never let go. Vinegar and battery acid differ in kind, not in dose.
| acid | Ka | pKa | pH at 0.1 M | fraction dissociated |
|---|
Add base a drop at a time and the pH barely moves — then moves enormously — then barely moves again. The vertical cliff is the equivalence point, where the last of the acid has just been neutralised. Notice the flat stretch before it, and where its middle sits: at exactly half an equivalent, pH equals pKa. That is not a coincidence, and it is how pKa gets measured.
A buffer is a weak acid sitting next to its own conjugate base, so it has something to mop up whatever you add in either direction. Below, the same dose of strong acid is dripped into plain water and into a buffer. The water plunges; the buffer barely notices — until its capacity is spent, at which point it gives way all at once.
Every pH on this page comes from solving the full charge-balance equation [H⁺] = [OH⁻] + [A⁻] together with Ka and Kw, by bisection, to fourteen digits. That matters because the shortcuts taught alongside it break in ways worth seeing. The usual pH = −log C for a strong acid says that 10⁻⁸ molar hydrochloric acid has a pH of 8 — that adding acid to water makes it alkaline. Set the concentration that low and this page gives 6.98, because it has not forgotten the water's own H⁺. Likewise [H⁺] = √(Ka·C) for a weak acid is fine at 0.1 M and drifts once the acid is dilute enough to be mostly dissociated.
What is left out: this is 25 °C, dilute, and assumes ions do not notice each other. Real solutions need activity coefficients rather than concentrations once they get crowded, Kw itself changes with temperature — pure water at 60 °C is neutral at pH 6.5, not 7 — and acids with more than one proton to give have a curve with a step for each. None of that changes the shape of what is here; it moves the numbers by a few tenths.