Question

Calcium hydroxide is a strong base. Compute \(\left[\mathrm{Ca}^{2+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\) for a solution that is prepared by dissolving \(0.60 \mathrm{~g}\) of \(\mathrm{Ca}(\mathrm{OH})_{2}\) in enough water to make a \(1500 \mathrm{~mL}\) of solution. [Atomic weights : \(\mathrm{Ca}=40, \mathrm{O}=16, \mathrm{H}=1\) ] (a) \(5.4 \times 10^{-3}, 9.1 \times 10^{-13}\) (b) \(5.4 \times 10^{-3}, 1.08 \times 10^{-2}\) (c) \(5.4 \times 10^{-3}, 5.4 \times 10^{-3}\) (d) \(8.1 \times 10^{-3}, 8.1 \times 10^{-3}\)

Short answer

(b) \(\left[\mathrm{Ca}^{2+}\right]\) = 5.4 \times 10^{-3} \text{ mol/L}, \(\left[\mathrm{OH}^{-}\right]\) = 1.08 \times 10^{-2} \text{ mol/L}

Step-by-step solution

Calculate Molar Mass of Calcium Hydroxide

To find the molar mass of calcium hydroxide, \text{Ca(OH)_2}, sum the atomic weights of all atoms: molar mass = atomic weight of Ca + 2 * (atomic weight of O + atomic weight of H). That amounts to \(40 + 2 * (16 + 1) = 40 + 2 * 17 = 40 + 34 = 74 \text{ g/mol}\).

Convert Mass of \text{Ca(OH)_2} to Moles

The amount of substance in moles can be calculated by the mass of the calcium hydroxide divided by its molar mass. For this case, \(0.60 \text{ g} \times \frac{1 \text{ mol}}{74 \text{ g/mol}} \text{gives} \frac{0.60}{74} \text{mol}\).

Calculate the Concentration of \text{Ca}^{2+}

The concentration of calcium ions, \text{Ca}^{2+}, is equal to the moles of \text{Ca(OH)_2} since it dissolves to release one mole of Ca^2+ per mole of \text{Ca(OH)_2}. For a 1500 mL solution, \([\text{Ca}^{2+}] = \frac{0.60}{74 \times 1.5} \text{ mol/L}\).

Calculate the Concentration of \text{OH}^{-}

Each mole of \text{Ca(OH)_2} releases two moles of hydroxide ions, \text{OH}^{-}. So, \([\text{OH}^{-}] = 2 \times [\text{Ca}^{2+}] = 2 \times \frac{0.60}{74 \times 1.5} \text{ mol/L}\).

Simplify and Calculate Final Concentrations

Dividing \(0.60 \text{ g}\) by \(74 \text{ g/mol}\), we get \(0.0081081 \text{ mol}\) of \text{Ca(OH)_2}. Dividing these moles by the 1.5 L of solution yields a \text{Ca}^{2+} concentration of \(0.0054054 \text{ mol/L}\) or \(5.4 \times 10^{-3} \text{ mol/L}\). The \text {[OH}^{-}] concentration will be twice this value, \(1.08 \times 10^{-2} \text{ mol/L}\).

Key concepts

Molar Mass Calculation

Understanding the molar mass of substances is fundamental in chemistry, especially when preparing solutions. Molar mass represents the mass of one mole of a substance, usually expressed in grams per mole (g/mol). To calculate the molar mass, you sum the atomic masses of each element in the compound, taking into account the number of atoms of each element present. For example, calcium hydroxide, Ca(OH)₂, has a molar mass calculated as follows:
Molar mass = Atomic weight of Ca + 2 × (Atomic weight of O + Atomic weight of H).
Using the atomic weights given: Ca = 40, O = 16, H = 1, the molar mass of Ca(OH)₂ is:
40 + 2 × (16 + 1) = 40 + 2 × 17 = 40 + 34 = 74 g/mol.
This calculation is essential to convert between the mass of a substance and the amount of substance in moles.

Moles to Mass Conversion

The conversion of moles to mass or vice versa is heavily reliant on the molar mass of the substance in question. Once you have the molar mass, you can easily convert the mass of a substance to moles by dividing the mass by the molar mass of the substance. For instance, to find out how many moles of calcium hydroxide are present in 0.60 g of Ca(OH)₂, you would use the following formula:
Number of moles = Mass (g) / Molar mass (g/mol).
In our case: Number of moles = 0.60 g / 74 g/mol ≈ 0.00811 mol.
This step is vital for stoichiometric calculations to determine the amount of reagents or products in a chemical reaction.

Calculation of Ion Concentrations in Solution

To calculate the concentration of ions in a solution, you start by understanding the dissociation of the compound in water. For calcium hydroxide, which dissociates into calcium ions, Ca²⁺, and hydroxide ions, OH^-, the stoichiometry is 1:2 as one mole of Ca(OH)₂ yields one mole of Ca²⁺ and two moles of OH^-.
Concentration is typically measured in moles per liter (mol/L). When you dissolve a known mass of a compound in a certain volume of solution, you can find the concentration of each ion by dividing the moles of compound by the volume of the solution, considering the stoichiometry of the dissociation. If you dissolve 0.60 g of Ca(OH)₂ in 1500 mL (or 1.5 L), the concentration of Ca²⁺ would be the number of moles of Ca(OH)₂ divided by the volume in liters. The concentration of OH^- would be twice the concentration of Ca²⁺ due to the stoichiometry of the dissociation.

Stoichiometry in Chemistry

Stoichiometry involves the calculation of the reactants and products in chemical reactions. It is grounded in the conservation of mass and the law of constant proportions. The key to solving stoichiometric problems is balancing the chemical equation and using mole ratios to convert between the amounts of different substances involved. In the context of a solution, stoichiometry allows you to predict the amount of solute needed to achieve a desired concentration in a given volume of solvent.
For the calcium hydroxide solution mentioned, stoichiometry is used to determine the concentrations of the resulting ions. Since each formula unit of Ca(OH)₂ yields one Ca²⁺ ion and two OH^- ions upon dissociation, the stoichiometry directly translates to the ion concentrations in the solution, proving crucial for accurate calculations in both laboratory and industrial chemical applications.