Ion exchange suits a building. A city softens by precipitation: add a cheap alkali, turn the dissolved hardness into a solid, and settle it out. Two chemicals do the work, and each has its own sum. Both sums run entirely in mg/L as CaCO₃, the common currency from the first lesson of this chapter, and that is the only reason their terms can be added at all. Metric throughout.

Lime. $L = \mathrm{CO_2} + \mathrm{Alk} + \mathrm{Mg} + \mathrm{Ex}$, read aloud L equals C-O-two plus Alk plus M-g plus E-x. $L$ is the lime dose, the quantity you are solving for. $\mathrm{CO_2}$ is the free carbon dioxide in the raw water, $\mathrm{Alk}$ the bicarbonate alkalinity, $\mathrm{Mg}$ the magnesium hardness that is to be removed, and $\mathrm{Ex}$ the excess lime added on top. Every one of the five is in mg/L as CaCO₃.

Each term is a separate demand on the lime. The carbon dioxide buys nothing: it consumes lime and precipitates no hardness, which is why aerating a groundwater ahead of the softener is often the cheapest chemical saving in the plant. The alkalinity term is the carbonate hardness coming out as calcium carbonate. Magnesium only precipitates above about pH 10.6, so removing it costs the lime that lifts the pH that high; a plant that leaves magnesium in enters zero. And the excess is what makes the reactions finish inside a real basin. 62.5 mg/L is the textbook figure, and 30 to 70 covers ordinary practice.

Soda ash. $S = \mathrm{TH} - \mathrm{Alk}$, read aloud S equals T-H minus Alk. $S$ is the soda ash dose, $\mathrm{TH}$ the total hardness and $\mathrm{Alk}$ the total alkalinity, all in mg/L as CaCO₃. The difference has a name: the noncarbonate hardness, calcium and magnesium paired with sulphate or chloride instead of bicarbonate. Lime works by turning bicarbonate into carbonate, so where no bicarbonate is paired with the hardness, more lime raises the pH and removes nothing. Soda ash brings its own carbonate. It also costs several times what lime does, so it is dosed only against what lime cannot reach. When the alkalinity is the larger figure there is no noncarbonate hardness, and the plant needs no soda ash at all.

Then the step that catches people: as CaCO₃ is not what you buy. The dose converts to product equivalent for equivalent, with the same 50.04 from lesson one now underneath: multiply by $28.04/50.04$ for quicklime, CaO, and by $52.99/50.04$ for soda ash, Na₂CO₃. Quicklime is the lighter equivalent, so the lime figure shrinks. Soda ash is slightly the heavier, so its figure grows. Commercial purity comes off after that.

Treat both answers as the starting point for a jar test and not as a setpoint. Stoichiometry says what the water demands, and says nothing about how readily a cold basin will give it up. Softened water also leaves at pH 10 or above, so recarbonation is part of the process and not a refinement of it.