Question

The solubility product for Zn (OH) \(_{2}\) is \(3.0 \times 10^{-16} .\) The formation constant for the hydroxo complex, Zn (OH) \(_{4}^{2-}\) , is 4.6 \(\times 10^{17} .\) What concentration of \(\mathrm{OH}^{-}\) is required to dissolve 0.015 mol of \(\mathrm{Zn}(\mathrm{OH})_{2}\) in a liter of solution?

Short answer

The concentration of OH⁻ required to dissolve 0.015 mol of Zn(OH)₂ in a liter of solution can be determined by solving the solubility equilibrium expressions for Zn(OH)₂ and Zn(OH)₄²⁻ using given solubility product and formation constant. Letting the concentration of OH⁻ be denoted as x, the concentration of Zn²⁺ and Zn(OH)₄²⁻ can be expressed in terms of x. Substituting these expressions into the solubility equilibrium expressions and solving for x, we obtain the concentration of OH⁻ by employing a numerical method like Newton's method.

Step-by-step solution

Write down the solubility equilibria expressions for Zn(OH)₂ and Zn(OH)₄²⁻

For the dissolution of Zn(OH)₂, the solubility equilibrium expression is: \[K_{sp}=[Zn^{2+}][OH^-]^2\] For the formation of the hydroxo complex Zn(OH)₄²⁻, the equilibrium expression is: \[K_f=[Zn(OH)_4^{2-}]/[Zn^{2+}][OH^-]^4\]

Express [Zn(OH)₄²⁻] and [Zn²⁺] in terms of [OH⁻]

Let the concentration of OH⁻ be denoted as x. Then, the concentration of Zn²⁺ produced by the dissolution of Zn(OH)₂ will also be x, and the concentration of Zn(OH)₄²⁻ will be 0.015-x because 0.015 moles of Zn(OH)₂ are dissolving. Therefore, \[Zn^{2+} = x\] \[Zn(OH)_4^{2-} = 0.015 - x\]

Substitute the concentrations into the solubility equilibrium expressions and solve for [OH⁻]

Substitute the expressions for [Zn²⁺] and [Zn(OH)₄²⁻] into the solubility equilibrium expressions: \[K_{sp} = (x)(x)^2\] \[K_f = (0.015 - x)/(x)(x)^4\] Now, solve for x (the concentration of OH⁻): \[3.0 \times 10^{-16} = x^3\] \[x^3 = 3.0 \times 10^{-16}\] \[x = \sqrt[3]{3.0 \times 10^{-16}}\] Next, substitute the value of x into the \[K_f\] expression: \[4.6 \times 10^{17} = (0.015 - x)/(x^5)\] \[x^5 = (0.015 - x)/(4.6 \times 10^{17})\] From here, a numerical method like Newton's method can be used to find the roots of the equation and obtain the value for x. After solving for x, the concentration of OH⁻ required to dissolve 0.015 mol of Zn(OH)₂ in a liter of solution can be determined.

Key concepts

Understanding Solution Equilibrium

When considering dissolving substances in a solvent, solution equilibrium is a fundamental concept to grasp. It refers to the state where the rate of dissolution and the rate of precipitation of a solute are equal, resulting in a constant concentration of the solute in the solution.

In our exercise, the solubility product (\(K_{sp}\)) for zinc hydroxide (\text{Zn(OH)₂}) is crucial. The equilibrium for its dissolution can be represented by the equation \[Zn(OH)₂(s) \rightleftharpoons Zn^{2+}(aq) + 2OH^-(aq)\] and the expression for the solubility product becomes \[K_{sp}=[Zn^{2+}][OH^-]^2\] which indicates that when the product of the ions' concentration reaches a particular value, the solution is saturated, and excess solute will not dissolve. Understanding this equilibrium allows us to predict how changes in conditions, such as the addition of other substances that affect the ion concentration, will shift the equilibrium to either favor more dissolution or precipitation.

Complex Ion Formation in Solution

Complex ion formation is another key concept in understanding solubility and dissolution. Complex ions are species formed from a central metal ion and surrounding ligands, molecules or ions that 'donate' a pair of electrons to the metal. These complex ions can greatly affect the solubility of some compounds.

In the exercise, a complex ion Zn(OH)₄²⁻ forms when Zn²⁺ ions bind with hydroxide ions. This is represented by the formation constant (\text{Kf}), which is quite high (\text{4.6 x 10¹⁷}). It means the complex ion is very stable and will preferentially form in solution, and thus, it can 'pull' more Zn(OH)₂ into solution by reducing the concentration of Zn²⁺ ions.

The chemical equation for the complex ion formation is: \[ Zn^{2+}(aq) + 4OH^-(aq) \rightleftharpoons Zn(OH)_4^{2-} (aq)\] The related equilibrium expression is \[K_f = \frac{[Zn(OH)_4^{2-}]}{[Zn^{2+}][OH^-]^4}\] Complex ion formation, especially those with large formation constants, is an essential consideration when predicting the behavior of ions in solution, as it significantly enhances the solubility of certain substances.

Concentration Calculation in Chemical Solutions

Accurately calculating concentrations in chemical solutions is vital for various applications, including scientific research, medical diagnosis, and industrial processes. In our exercise, concentration calculations are essential for determining the concentration of hydroxide ions (\text{OH⁻}) needed to dissolve a specific amount of zinc hydroxide (\text{Zn(OH)₂}) in solution.

The set up of concentration expressions and substitution into equilibrium expressions guides us to a set of mathematical equations. We then often rely on algebraic or numerical methods to solve for the unknown concentration values.

For example, we define the concentration of hydroxide ions as x, then set up the equations: \[K_{sp} = (x)(x)^2\] \[K_f = \frac{(0.015 - x)}{(x)(x)^4}\] By solving for x, we find the concentration. Even if it can't be solved analytically, numerical methods can be employed, demonstrating the importance of not just understanding the chemistry but also the mathematics involved in concentration calculations. Knowing the concept of concentration allows one to manipulate conditions to achieve desired dissolution or precipitation results, crucial in controlling reactions in industrial and laboratory settings.