What ratio of lime must be added to effectively remove magnesium carbonate hardness during water softening?

To effectively remove magnesium carbonate hardness during water softening, the process involves the addition of lime, which is calcium hydroxide, to precipitate magnesium ions. The stoichiometric relationship between lime and magnesium carbonate is significant in determining the appropriate amount of lime needed.

When lime is added to water containing magnesium carbonate hardness, the magnesium ions are precipitated as magnesium hydroxide. The reaction can be represented as follows:

[ \text{MgCO}_3 + \text{Ca(OH)}_2 \rightarrow \text{Mg(OH)}_2 \downarrow + \text{CaCO}_3 ]

For each mole of magnesium carbonate present, two moles of lime (calcium hydroxide) are required to completely precipitate the magnesium ions. This is because one mole of lime not only reacts with one mole of magnesium carbonate but also produces calcium carbonate as a byproduct, which does not remove any additional hardness itself.

Consequently, the effective removal of magnesium carbonate hardness necessitates that two times as much lime, in terms of molar ratio, be added to ensure complete precipitation of magnesium as magnesium hydroxide. This is in alignment with standard water treatment practices for softening.

Therefore, the required ratio of lime to effectively remove