The hardness of water is reported as milligrams of calcium...

The hardness of water is reported as milligrams of calcium carbonate per liter of water. Calculate the quantity of sodium carbonate in grams needed to soften 10.0 L of water with a hardness of $255 \mathrm{mg} / \mathrm{L}$. (Hint: How much $\mathrm{Ca}^{2+}$ is present in the water?)

Key Concepts

Water Hardness

Water hardness refers to the concentration of dissolved divalent cations, primarily calcium (Ca2+) and magnesium (Mg2+), in water. It is typically expressed in terms of milligrams per liter (mg/L) of calcium carbonate (CaCO3) equivalent, which provides a standardized measure for the potential of the water to form scale and interfere with soap performance.

Calcium Carbonate Equivalence

Although hardness is measured as mg CaCO3 per liter, the value represents the combined effect of various divalent ions in water. In many calculations, this measurement is used as a proxy to estimate the concentration of calcium ions (Ca2+) since CaCO3 is closely related to the precipitation reaction that removes hardness. This equivalence is essential for translating the measured hardness into a stoichiometric requirement for treatment reagents.

Precipitation Reaction in Water Softening

Water softening often involves a precipitation reaction where a chemical such as sodium carbonate is added to react with calcium ions. The reaction forms an insoluble compound, calcium carbonate, which can then be removed from the water. Understanding this reaction is key since it determines the stoichiometric ratio between the added reagent (sodium carbonate) and the hardness-causing calcium ions.

Stoichiometry and Mole Ratios

Stoichiometry is used to relate the amount of substance measured (hardness in terms of CaCO3) to the required amount of sodium carbonate needed. It involves converting the mass of calcium carbonate equivalent into moles using its molar mass and then using the reaction mole ratios (typically 1:1 between Ca2+ and CO3^2-) to determine how many moles of sodium carbonate are needed for complete precipitation.

Unit Conversions and Molar Mass Calculations

Performing calculations for water softening requires converting the concentration of ions from mg/L to grams, and further to moles, using the molar masses of the compounds involved. These unit conversions are essential for ensuring that the stoichiometric calculations are accurate, and they facilitate the determination of the exact mass of reagents (like sodium carbonate) needed to achieve the desired water treatment.

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