Question

The total concentration of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Mg}^{2+}\) in a sample of hard water was determined by titrating a \(0.100-\mathrm{L}\) sample of the water with a solution of EDTA \(^{4-}\). The EDTA \(^{4-}\) chelates the two cations: $$ \begin{aligned} \mathrm{Mg}^{2+}+[\mathrm{EDTA}]^{4-} \longrightarrow &[\mathrm{Mg}(\mathrm{EDTA})]^{2-} \\ \mathrm{Ca}^{2+}+[\mathrm{EDTA}]^{4-} \longrightarrow &[\mathrm{Ca}(\mathrm{EDTA})]^{2-} \end{aligned} $$ It requires \(31.5 \mathrm{~mL}\) of \(0.0104 \mathrm{M}[\mathrm{EDTA}]^{4-}\) 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 \(0.0104 \mathrm{M}\) \([\mathrm{EDTA}]^{4-},\) Calculate the concentrations of \(\mathrm{Mg}^{2+}\) and \(\mathrm{Ca}^{2+}\) in the hard water in \(\mathrm{mg} / \mathrm{L}\).

Short answer

The concentrations of Ca²⁺ and Mg²⁺ in the hard water sample are 52.9 mg/L and 47.1 mg/L, respectively.

Step-by-step solution

Determine the total amount of moles reacted with EDTA in the first titration

In the first titration, 0.100 L of the sample was titrated with 31.5 mL of 0.0104 M EDTA⁴⁻ solution. We will convert volume and concentration to moles by using the formula: Moles of EDTA⁴⁻ = Volume of EDTA⁴⁻ solution (L) × Molarity of EDTA⁴⁻ solution (mol/L) Moles of EDTA⁴⁻ = 0.0315 L × 0.0104 mol/L = \(3.26 \times 10^{-4}\) mol Both Ca²⁺ and Mg²⁺ react with EDTA⁴⁻ in a 1:1 molar ratio, so the total moles reacted with EDTA⁴⁻ in the first titration will be \(3.26 \times 10^{-4}\) mol.

Determine the amount of moles of Mg²⁺ left in the second titration

In the second titration, 0.100 L of the sample was titrated with 18.7 mL of 0.0104 M EDTA⁴⁻ solution after precipitating Ca²⁺ as calcium sulfate. We will similarly convert the volume and concentration to moles. Moles of EDTA⁴⁻ = Volume of EDTA⁴⁻ solution (L) × Molarity of EDTA⁴⁻ solution (mol/L) Moles of EDTA⁴⁻ = 0.0187 L × 0.0104 mol/L = \(1.94 \times 10^{-4}\) mol Since Mg²⁺ reacts with EDTA⁴⁻ in a 1:1 molar ratio, the moles of Mg²⁺ in the second titration will be \(1.94 \times 10^{-4}\) mol.

Calculate the moles and concentration of Ca²⁺ and Mg²⁺

From the information obtained in the previous steps, we can now calculate the moles of Ca²⁺ and Mg²⁺ separately. Moles of Ca²⁺ = Total moles in the first titration - Moles of Mg²⁺ in the second titration Moles of Ca²⁺ = \(3.26 \times 10^{-4}\) mol - \(1.94 \times 10^{-4}\) mol = \(1.32 \times 10^{-4}\) mol Now we will calculate the molar concentrations of both ions in the sample: Concentration of Ca²⁺ = Moles of Ca²⁺ / Volume of sample (L) = \(\frac{1.32 \times 10^{-4}\, \text{mol}}{0.100 \, \text{L}}\) = \(1.32 \times 10^{-3}\) M Concentration of Mg²⁺ = Moles of Mg²⁺ / Volume of sample (L) = \(\frac{1.94 \times 10^{-4}\, \text{mol}}{0.100 \, \text{L}}\) = \(1.94 \times 10^{-3}\) M

Convert concentrations to mg/L

We will use molar mass to convert the concentrations of Ca²⁺ and Mg²⁺ to mg/L. Molar mass of Ca²⁺ = 40.08 g/mol Molar mass of Mg²⁺ = 24.30 g/mol Concentration of Ca²⁺ (mg/L) = Concentration of Ca²⁺ (M) × Molar mass of Ca²⁺ (g/mol) × 1000 (to convert g to mg) Concentration of Ca²⁺ (mg/L) = \(1.32 \times 10^{-3} \frac{\text{mol}}{\text{L}}\) × 40.08 g/mol × 1000 = 52.9 mg/L Concentration of Mg²⁺ (mg/L) = Concentration of Mg²⁺ (M) × Molar mass of Mg²⁺ (g/mol) × 1000 (to convert g to mg) Concentration of Mg²⁺ (mg/L) = \(1.94 \times 10^{-3} \frac{\text{mol}}{\text{L}}\) × 24.30 g/mol × 1000 = 47.1 mg/L The concentrations of Ca²⁺ and Mg²⁺ in the hard water sample are 52.9 mg/L and 47.1 mg/L, respectively.

Key concepts

Hard Water Analysis

Analysis of hard water involves determining the presence and concentration of cations, primarily calcium (Ca²⁺) and magnesium (Mg²⁺), responsible for water hardness. Water is considered 'hard' when it has a high mineral content, which can lead to scaling in pipes and inefficiency in soaps and detergents. In our exercise, the total concentration of these cations was assessed using a titration method with a chelating agent, EDTA^4-, which forms complexes with the cations.

To enhance understanding of water hardness, it's meaningful to discuss the source of these minerals; they generally enter the water supply by dissolving from minerals like limestone or dolomite as water percolates through the ground. The hardness can be temporary, caused by bicarbonate minerals that precipitate when heated, or permanent, from sulfate or chloride minerals that do not. The exercise focuses on the total hardness, which is a combined measure of both.

When evaluating the quality of water for domestic or industrial use, knowledge of water hardness is crucial. Softening of hard water is common, and understanding the precise levels of calcium and magnesium is essential for determining the appropriate treatment process.

Complexometric Titration

Complexometric titration, a key concept in our exercise, is a form of volumetric analysis where a solution (titrant) with a known concentration is used to determine the unknown concentration of a metal ion in a sample. This titration employs a complexing agent, such as EDTA^4-, that forms a stable, water-soluble complex with metal ions.

In complexometric titration, the point at which the titrant has completely reacted with the ion of interest is known as the endpoint. This is often detected with the help of an indicator that changes color when the reaction is complete, or by instruments measuring changes in the solution's properties. In the context of our exercise, the color change might not be visible, hence students could understand that techniques such as a pH meter or conductivity meter might have been used to detect the endpoint.

Complexometric titrations are highly accurate and can selectively determine one type of ion in the presence of others. The technique is widely used in areas such as water hardness analysis, pharmaceutical and food industries for quality control, and in clinical laboratories for blood serum analyses.

Cation Concentration Calculation

The calculation of cation concentration following a titration process is critical for quantifying specific ions in a solution, as in our hard water example. The exercise demonstrates the determination of calcium and magnesium ion concentrations in a water sample using EDTA titration.

The calculation begins with the titration data to determine the moles of EDTA required to react with the cations in the water sample. With the assumption that EDTA reacts in a 1:1 molar ratio with both Ca²⁺ and Mg²⁺, we can ascertain the total moles of ions present. Following the titration steps, the concentration (in moles per liter) is calculated by dividing the moles of each ion by the volume of the water sample.

However, expressing these concentrations in mg/L is more practical for comparison to water quality standards. Thus, we convert molarity to mg/L using the molar mass of each ion, providing a clear view of the hardness level of the water. From a practical standpoint, students may benefit from knowing that these concentrations help in formulating water softening treatments and assessing compliance with environmental regulations.