Question

The total concentration of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) in a sample of hard water was determined by titrating a 0.100-L sample of the water with a solution of EDTA \(^{4-}\). The EDTA \(^{4-}\) chelates the two cations: $$ \begin{array}{r} \mathrm{Mg}^{2+}+[\mathrm{EDTA}]^{4-}--\rightarrow[\mathrm{Mg}(\mathrm{EDTA})]^{2-} \\\ \mathrm{Ca}^{2+}+[\mathrm{EDTA}]^{4-}--\rightarrow[\mathrm{Ca}(\mathrm{EDTA})]^{2-} \end{array} $$ It requires \(31.5 \mathrm{~mL}\) of \(0.0104 M[\mathrm{EDTA}]^{4-}\) solution to reach the end point in the titration. A second \(0.100-\mathrm{L}\) sample was then treated with sulfate ion to precipitate \(\mathrm{Ca}^{2+}\) as calcium sulfate. The \(\mathrm{Mg}^{2+}\) was then titrated with \(18.7 \mathrm{~mL}\) of \(0.0104 M[\mathrm{EDTA}]^{4-} .\) Calculate the concentrations of \(\mathrm{Mg}^{2+}\) and \(\mathrm{Ca}^{2+}\) in the hard water in \(\mathrm{mg} / \mathrm{L}\).

Short answer

The concentration of \(\mathrm{Mg}^{2+}\) in the hard water is approximately \(4.72\,\mathrm{mg/L}\), and the concentration of \(\mathrm{Ca}^{2+}\) in the hard water is approximately \(5.33\,\mathrm{mg/L}\).

Step-by-step solution

Determine moles of EDTA in the first titration.

We have \(31.5 \mathrm{~mL}\) of \(0.0104 M[\mathrm{EDTA}]^{4-}\) used in the first titration. To calculate the number of moles of EDTA \(^{4-}\), we use the formula: Moles = Molarity × Volume (in L) Moles of EDTA \(^{4-}\) = \(0.0104 \,M \times 0.0315\, L\) Moles of EDTA \(^{4-}\) = \(0.0003276\) mol Since each mole of EDTA \(^{4-}\) binds with one mole of \(\mathrm{Mg}^{2+}\) or \(\mathrm{Ca}^{2+}\), the total moles of \(\mathrm{Mg}^{2+}\) and \(\mathrm{Ca}^{2+}\) together in the initial water sample are \(0.0003276\) moles.

Determine moles of EDTA in the second titration.

We have \(18.7 \mathrm{~mL}\) of \(0.0104 M[\mathrm{EDTA}]^{4-}\) used in the second titration. To calculate the number of moles of EDTA \(^{4-}\), we use the formula: Moles of EDTA \(^{4-}\) = \(0.0104 \,M \times 0.0187\, L\) Moles of EDTA \(^{4-}\) = \(0.00019448\) mol Since each mole of EDTA \(^{4-}\) binds with one mole of \(\mathrm{Mg}^{2+}\) in the second titration, the moles of \(\mathrm{Mg}^{2+}\) in the initial water sample are \(0.00019448\) moles.

Calculate moles of \(\mathrm{Ca}^{2+}\) in the initial water sample.

To find the moles of \(\mathrm{Ca}^{2+}\) in the initial water sample, subtract the moles of \(\mathrm{Mg}^{2+}\) that we found in Step 2 from the total moles of \(\mathrm{Mg}^{2+}\) and \(\mathrm{Ca}^{2+}\) together that we found in Step 1: Moles of \(\mathrm{Ca}^{2+}\) = Total moles of \(\mathrm{Mg}^{2+}\) and \(\mathrm{Ca}^{2+}\) - Moles of \(\mathrm{Mg}^{2+}\) Moles of \(\mathrm{Ca}^{2+}\) = \(0.0003276 - 0.00019448\) Moles of \(\mathrm{Ca}^{2+}\) = \(0.00013312\) mol

Calculate concentrations of \(\mathrm{Mg}^{2+}\) and \(\mathrm{Ca}^{2+}\) in mg/L.

Now we can convert moles of \(\mathrm{Mg}^{2+}\) and \(\mathrm{Ca}^{2+}\) to concentrations in mg/L using their molar masses. The molar mass of \(\mathrm{Mg}^{2+}\) = \(24.305\,\mathrm{g/mol}\) and the molar mass of \(\mathrm{Ca}^{2+}\) = \(40.078\,\mathrm{g/mol}\). Concentration of \(\mathrm{Mg}^{2+}\) = \(\frac{0.00019448 \,\mathrm{mol} \times 24.305\,\mathrm{g/mol}}{0.1 \,\mathrm{L}} = 4.72\,\mathrm{mg/L}\) Concentration of \(\mathrm{Ca}^{2+}\) = \(\frac{0.00013312 \,\mathrm{mol} \times 40.078\,\mathrm{g/mol}}{0.1 \,\mathrm{L}} = 5.33\,\mathrm{mg/L}\) So, the concentration of \(\mathrm{Mg}^{2+}\) in the hard water is approximately \(4.72\,\mathrm{mg/L}\), and the concentration of \(\mathrm{Ca}^{2+}\) in the hard water is approximately \(5.33\,\mathrm{mg/L}\).

Key concepts

Hard Water Analysis

Hard water contains dissolved minerals, primarily calcium (\(\text{Ca}^{2+}\)) and magnesium (\(\text{Mg}^{2+}\)) ions. These ions can cause a variety of problems, including scaling in pipes and reduced effectiveness of soaps and detergents. Analyzing hard water involves measuring the concentration of these metallic ions. EDTA titration is a technique commonly used for this purpose. EDTA (\(\text{Ethylene Diamine Tetra-acetic Acid}\)) is a chelating agent that binds with metal ions, forming stable complexes. This method allows us to determine the concentration of calcium and magnesium separately, providing a complete picture of water hardness. During the titration process, EDTA effectively captures all the \(\text{Ca}^{2+}\) and \(\text{Mg}^{2+}\) ions in the sample. The endpoint of the titration indicates that all metal ions have reacted with the EDTA. By calculating the amount of EDTA used, we can break down the specific levels of calcium and magnesium in the sample.

• This provides a crucial insight into the overall quality of water.
• It helps with the formulation of solutions to mitigate water hardness.

Calcium Concentration

Calculating calcium concentration in hard water requires understanding its interaction with EDTA. When performing an EDTA titration, the amount of EDTA used in the first titration accounts for both \(\text{Ca}^{2+}\) and \(\text{Mg}^{2+}\) ions. After accounting for the magnesium through a second titration, the remaining amount of EDTA corresponds to calcium.The calculation involves subtracting the moles of magnesium from the total moles initially determined. Then, the moles of \(\text{Ca}^{2+}\) can be transformed into a concentration using the conversion factor of its molar mass:- Molar Mass of \(\text{Ca}^{2+}\) = 40.078 g/mol.- Use this conversion to get the concentration in mg/L.
This step ensures that not only is the total sum of ions considered, but also each ion is correctly represented in terms of concentration.This information is critical:- It quantifies the calcium content, essential for evaluating water treatment methods.- Provides data for potential adjustments needed in industrial or residential water systems.

Magnesium Concentration

The concentration of magnesium in hard water is identified through titration that specifically accounts for its reaction with EDTA. During the processing of hard water samples, magnesium is often left in solution after calcium is precipitated out.By introducing a sulfate agent, the calcium ions form calcium sulfate and are effectively removed from the solution. This leaves magnesium ions to be titrated as a single species, facilitating a more accurate assessment of their concentration. The procedure uses this distinct titration to directly calculate the magnesium content based on the amount of EDTA required to reach the endpoint:- Molar Mass of \(\text{Mg}^{2+}\) = 24.305 g/mol.- Convert the moles of \(\text{Mg}^{2+}\) into mg/L using its molar mass. This gives the precise concentration of magnesium in the original sample. Understanding magnesium levels is essential for:- Diagnosing the potential sources of water hardness.- Informing the subsequent steps for water softening.