Question

Approximately \(1.5 \times 10^{-3} \mathrm{~g}\) of iron(II) hydroxide, \(\mathrm{Fe}(\mathrm{OH})_{2}(s),\) dissolves per liter of water at 18 ". Calculate \(K_{\text {sp }}\) for \(\mathrm{Fe}(\mathrm{OH})_{2}(s)\) at this temperature.

Short answer

The solubility product constant (K_sp) for iron(II) hydroxide, Fe(OH)_2, at 18 °C is approximately \(1.84 \times 10^{-14}\).

Step-by-step solution

Write the balanced chemical equation

: The chemical equation for the dissolution of iron(II) hydroxide, Fe(OH)_2, in water is as follows: Fe(OH)_2(s) ⇌ Fe^(2+)(aq) + 2OH^(-)(aq)

Write the expression for K_sp

: From the balanced chemical equation, we know that the solubility product constant (K_sp) is the product of the concentrations of ions produced in the equilibrium: K_sp = [Fe^(2+)] [OH^(-)]^2

Calculate the concentration of each ion

: We are given that 1.5 x 10^(-3) g of Fe(OH)_2 dissolves per liter of water. The molar mass of Fe(OH)_2 is approximately 89.86 g/mol (55.84 g/mol for Fe + 34.02 g/mol for 2 OH groups). So, to convert grams to moles, we use the following formula: Moles = (Mass in grams) / (Molar mass) Moles = (1.5 x 10^(-3) g) / (89.86 g/mol) ≈ 1.67 x 10^(-5) mol Since we have 1 L of solution, the moles are also equal to the concentration in mol/L. So, the concentration of Fe^(2+) is 1.67 x 10^(-5) M, and the concentration of OH^(-) ions is double that of Fe^(2+) ions, so it is 2 x (1.67 x 10^(-5) M) = 3.33 x 10^(-5) M.

Calculate K_sp

: Now that we have the concentration of each ion, we can plug these values into the K_sp expression: K_sp = [Fe^(2+)][OH^(-)]^2 K_sp = (1.67 x 10^(-5)) * (3.33 x 10^(-5))^2 K_sp ≈ 1.84 x 10^(-14) Hence, the solubility product constant (K_sp) for Fe(OH)_2 at 18 °C is approximately 1.84 x 10^(-14).

Key concepts

Chemical Equilibrium

In a solution, chemical equilibrium occurs when a chemical reaction and its reverse happen at the same rate, resulting in the amounts of reactants and products remaining constant over time. For the dissolution of iron(II) hydroxide, Fe(OH)₂, a state of equilibrium is reached when the rate at which it dissolves equals the rate at which its ions recombine to form solid Fe(OH)₂. This can be indicated by the equilibrium expression of the balanced chemical equation: Fe(OH)₂(s) ⇌ Fe²⁺(aq) + 2OH⁻(aq).
At equilibrium, although the concentrations of the solid Fe(OH)₂ don't appear to change because it's a solid, the concentrations of its ions in solution do impact the system's overall behavior. In this context, the Solubility Product Constant (K_sp) signifies these ion concentrations at equilibrium. It quantifies the saturated solution's ability to hold the dissolved ions, indicating how much of the ionic compound can dissolve in water before settling into equilibrium.

Molar Concentration

Molar concentration or molarity measures how much of a substance is present in a given volume of solution. It's represented in moles per liter (mol/L). For a substance like iron(II) hydroxide dissolving in water, understanding molarity helps us determine how the solubility impacts the surrounding solution. For example, if 1.5 x 10⁻³ g of Fe(OH)₂ dissolves in one liter of water, we convert this mass into moles using its molar mass (roughly 89.86 g/mol): Moles = \((1.5 \times 10^{-3} \text{ g}) / (89.86 \text{ g/mol}) \approx 1.67 \times 10^{-5} \text{ mol} \).
Given that these moles dissolve in 1 L of solution, this is also the molarity of Fe²⁺ ions. Since each Fe(OH)₂ releases one Fe²⁺ ion and two OH⁻ ions, the concentration of OH⁻ ions is double that of Fe²⁺ ions, resulting in \(3.33 \times 10^{-5} \text{ M} \) for OH⁻ ions.

Balanced Chemical Equation

A balanced chemical equation is vital in chemistry, ensuring atoms are conserved and provides insight into the relationship between reactants and products. The balanced equation for iron(II) hydroxide's dissolution is: Fe(OH)₂(s) ⇌ Fe²⁺(aq) + 2OH⁻(aq).
This equation tells us that when one unit of Fe(OH)₂ dissolves, it produces one Fe²⁺ ion and two OH⁻ ions. Balancing involves equalizing the number of each type of atom on both sides of the equation. Here, we see:

• 1 iron (Fe) atom on both sides.
• 2 hydroxide (OH⁻) ions on both sides.

This balancing acts as a map to understand the stoichiometric proportions in a chemical reaction. In the context of a solubility problem, it helps derive the stoichiometric coefficients needed to calculate the molar concentrations of ions in a solution, ultimately allowing us to compute the K_sp value for the compound.