Question

In the lime soda process once used in large scale municipal water softening, calcium hydroxide prepared from lime and sodium carbonate are added to precipitate \(\mathrm{Ca}^{2+}\) as \(\mathrm{CaCO}_{3}(s)\) and \(\mathrm{Mg}^{2+}\) as \(\mathrm{Mg}(\mathrm{OH})_{2}(s) :\) $$\begin{array}{c}{\mathrm{Ca}^{2+}(a q)+\mathrm{CO}_{3}^{2-}(a q) \longrightarrow \mathrm{CaCO}_{3}(s)} \\ {\mathrm{Mg}^{2+}(a q)+2 \mathrm{OH}^{-}(a q) \longrightarrow \mathrm{MgOH}_{2}(a q)}\end{array}$$ How many moles of \(\mathrm{Ca}(\mathrm{OH})_{2}\) and \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) should be added to soften (remove the \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+} ) 1200 \mathrm{L}\) of water in which $$\begin{array}{l}{\left[\mathrm{Ca}^{2+}\right]=5.0 \times 10^{-4} M \text { and }} \\ {\left[\mathrm{Mg}^{2+}\right]=7.0 \times 10^{-4} \mathrm{M?}}\end{array}$$

Short answer

To soften 1200 L of water with given concentrations of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\), 0.6 moles of \(\mathrm{Ca(OH)_2}\) and 0.84 moles of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) should be added.

Step-by-step solution

Calculate the moles of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) in the water

First, we need to determine the number of moles of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) ions in the 1200 L of water. We can use the formula: moles \(= \text{concentration} \times \text{volume}\), and convert the volume to liters. For \(\mathrm{Ca}^{2+}\) ions: Moles = \(\left(5.0 \times 10^{-4}\text{ M}\right) \times (1200 \text{ L})\) For \(\mathrm{Mg}^{2+}\) ions: Moles = \(\left(7.0 \times 10^{-4}\text{ M}\right) \times (1200 \text{ L})\)

Calculate the moles of \(\mathrm{Ca(OH)_2}\) needed

According to the first reaction, one mole of \(\mathrm{Ca}^{2+}\) requires one mole of \(\mathrm{CO}_{3}^{2-}\) to form one mole of \(\mathrm{CaCO}_{3}(s)\). Since \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) provides one mole of \(\mathrm{CO}_{3}^{2-}\) per mole, we can say that one mole of \(\mathrm{Ca}^{2+}\) requires one mole of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\). To find the moles of \(\mathrm{Ca(OH)_2}\) required, we can simply use the moles of \(\mathrm{Ca}^{2+}\) ions: Moles of \(\mathrm{Ca(OH)_2}\) = Moles of \(\mathrm{Ca}^{2+}\)

Calculate the moles of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) needed

According to the second reaction, one mole of \(\mathrm{Mg}^{2+}\) requires two moles of \(\mathrm{OH}^{-}\) to form one mole of \(\mathrm{Mg(OH)}_{2}(s)\). Since \(\mathrm{Ca(OH)_2}\) provides two moles of \(\mathrm{OH}^{-}\) per mole, we can say that one mole of \(\mathrm{Mg}^{2+}\) requires one mole of \(\mathrm{Ca(OH)_2}\). To find the moles of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) required, we can simply use the moles of \(\mathrm{Mg}^{2+}\) ions: Moles of \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) = Moles of \(\mathrm{Mg}^{2+}\)

Calculate the number of moles required for \(\mathrm{Ca(OH)_2}\) and \(\mathrm{Na}_{2}\mathrm{CO}_{3}\)

Now that we have the moles of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) ions, we can calculate the moles required for \(\mathrm{Ca(OH)_2}\) and \(\mathrm{Na}_{2}\mathrm{CO}_{3}\): Moles of \(\mathrm{Ca(OH)_2} = \left(5.0 \times 10^{-4}\text{ M}\right) \times (1200 \text{ L})\) Moles of \(\mathrm{Na}_{2}\mathrm{CO}_{3} = \left(7.0 \times 10^{-4}\text{ M}\right) \times (1200 \text{ L})\) So, the number of moles required for each compound to soften 1200 L of water is: \(\mathrm{Ca(OH)_2}: 0.6\,\text{moles}\) \(\mathrm{Na}_{2}\mathrm{CO}_{3}: 0.84\,\text{moles}\)

Key concepts

Lime Soda Process

The lime soda process is a classical method used for water softening, in which the hardness of water is reduced by removing calcium and magnesium ions. This is achieved by adding chemical compounds, specifically calcium hydroxide, also known as slaked lime, and sodium carbonate, often referred to as soda ash. These two substances react with the calcium and magnesium ions to form insoluble precipitates that can be filtered out from the water.

In larger scale applications, such as municipal water treatment plants, this process has been widely adopted due to its effectiveness and simplicity. The solid precipitate is then removed through sedimentation or filtration, resulting in softened water that is more suitable for household and industrial use.

Calcium Hydroxide

Calcium hydroxide, represented by the chemical formula \(\mathrm{Ca(OH)_2}\), plays a critical role in the water softening process. When introduced into water containing calcium and magnesium ions, it facilitates a chemical reaction that eventually leads to precipitation. Each molecule of calcium hydroxide provides two hydroxide ions \(\mathrm{OH}^-\), which are key for removing magnesium ions by forming \(\mathrm{Mg(OH)_2}\), an insoluble compound that precipitates out of the solution.

More than just a reactant, calcium hydroxide also helps to maintain a basic pH in the water, which is necessary for the optimal precipitation of magnesium hydroxide. It can easily be produced by adding water to calcium oxide, also known as lime.

Sodium Carbonate

Sodium carbonate (\(\mathrm{Na_2CO_3}\)) is another essential component used in the lime soda softening process. It is sometimes simply called soda ash. In this context, sodium carbonate provides carbonate ions \(\mathrm{CO_3^{2-}}\) that react with calcium ions in the water to form calcium carbonate \(\mathrm{CaCO_3}\), a white precipitate which is also insoluble in water.

This conversion step is pivotal for the removal of calcium ions and contributes to the reduction of water hardness. The end product, calcium carbonate, is a common mineral widely found in rocks such as limestone and is also the primary component of chalk and marble.

Chemical Precipitation

Chemical precipitation is a process where solutes are transformed into insoluble solids and separated from a solution. In the context of water softening, chemical precipitation involves the addition of substances like calcium hydroxide and sodium carbonate to induce the formation of insoluble calcium carbonate \(\mathrm{CaCO_3}\) and magnesium hydroxide \(\mathrm{Mg(OH)_2}\).

This removal method is based on the solubility limits of these compounds. The key to successful precipitation is to create conditions in which the solubility of targeted ions is exceeded, making it energetically favorable for them to come out of solution as a separate phase. Thus, selecting the right amounts and concentration of precipitating agents is crucial for the effectiveness of the process.

Molar Concentration

Molar concentration, represented with the unit \(M\), refers to the number of moles of a solute present in one liter of solution. It's a fundamental measure in chemistry, especially relevant in reaction stoichiometry and quantitative analysis. For example, the molar concentration of calcium ions \(\mathrm{Ca^{2+}}\) and magnesium ions \(\mathrm{Mg^{2+}}\) in the water determines the amount of reactants needed to carry out water softening.

To accurately calculate the necessary chemicals for the lime soda process, a precise understanding of the water's molar concentration in terms of its calcium and magnesium ion content is imperative. This concentration provides the starting point for computing the stoichiometric amounts of calcium hydroxide and sodium carbonate required to fully soften the water.

Stoichiometry

Stoichiometry is the field of chemistry that involves the calculation of reactants and products in chemical reactions. For the lime soda process, understanding stoichiometry is crucial because it dictates the exact proportions of calcium hydroxide and sodium carbonate necessary to react with a given amount of calcium and magnesium ions.

In the given exercise, stoichiometry helps in determining the moles of calcium hydroxide and sodium carbonate based on the molar concentrations of calcium and magnesium ions. Through stoichiometric calculations, one can ensure the right amounts of chemicals are added to achieve a complete reaction, preventing the usage of excess chemicals and optimizing the softening process for effective and economic outcomes.