Problem 22 - Introductory Chemistry Atoms First

Question

Sparingly soluble aluminum hydroxide dissolves in water to yield an equilibrium hydroxide ion concentration of \(8.58 \times 10^{-9} \mathrm{M}\). (a) Write the balanced equilibrium equation for aluminum hydroxide dissolving in water. (b) Write the \(K_{\mathrm{sp}}\) expression for aluminum hydroxide. (c) What is the equilibrium concentration of aluminum ion? (d) Calculate the value of \(K_{\mathrm{sp}}\) for aluminum hydroxide (show your calculation).

Step-by-Step Solution

Answer

(a) The balanced equilibrium equation for aluminum hydroxide dissolving in water is: Al(OH)3(s) \<--> Al3+ (aq) + 3OH- (aq) (b) The Ksp expression for aluminum hydroxide is: Ksp = [Al3+][OH-]^3 (c) The equilibrium concentration of aluminum ion (Al3+) is \(2.86 \times 10^{-9} M\). (d) The value of Ksp for aluminum hydroxide is approximately \(3.00 \times 10^{-27}\).

Step 1: Write the balanced equilibrium equation

First, we need to write the balanced equilibrium equation for aluminum hydroxide dissolving in water. Aluminum hydroxide has the chemical formula Al(OH)3. When it dissolves in water, it dissociates into aluminum ions (Al3+) and hydroxide ions (OH-): Al(OH)3(s) \<--> Al3+ (aq) + 3OH- (aq)

Step 2: Write the Ksp expression

Next, we need to write the Ksp (Solubility Product Constant) expression for aluminum hydroxide. The general formula for Ksp is the product of the concentrations of the products raised to the power of their coefficients (in the balanced equation), divided by the concentration of the reactants: Ksp = [Al3+][OH-]^3

Step 3: Calculate the equilibrium concentration of aluminum ions

We are given that the equilibrium hydroxide ion concentration is \(8.58 \times 10^{-9} M\). Since the ratio of aluminum ions to hydroxide ions in the balanced equation is 1:3, the concentration of aluminum ions in equilibrium will be a third of the concentration of hydroxide ions. Equilibrium concentration of Al3+ = \(\frac{1}{3}\) x (equilibrium concentration of OH-) Equilibrium concentration of Al3+ = \(\frac{1}{3}\) x \(8.58 \times 10^{-9} M\) Equilibrium concentration of Al3+ = \(2.86 \times 10^{-9} M\)

Step 4: Calculate the value of Ksp

Now that we have the equilibrium concentrations of aluminum ions and hydroxide ions, we can calculate the value of Ksp for aluminum hydroxide using the expression we found in step 2: Ksp = [Al3+][OH-]^3 Ksp = \((2.86 \times 10^{-9})(8.58 \times 10^{-9})^3\) Ksp = \(3.00 \times 10^{-27}\) The value of Ksp for aluminum hydroxide is approximately \(3.00 \times 10^{-27}\).

Key Concepts

Understanding the Solubility Product Constant

The solubility product constant, or \(K_{sp}\), is a crucial concept in chemistry, especially when dealing with sparingly soluble compounds like aluminum hydroxide. It represents the product of the molar concentrations of dissolved ions, each raised to the power of its respective coefficient from the balanced dissolution equation. For aluminum hydroxide, when it dissolves, it dissociates into aluminum ions \([\text{Al}^{3+}]\) and hydroxide ions \([\text{OH}^-]\). Thus, the \(K_{sp}\) expression is \([\text{Al}^{3+}][\text{OH}^-]^3\). The \(K_{sp}\) value helps predict whether a precipitate will form when solutions are mixed and is crucial in calculating the extent to which a compound will dissolve in the solvent.
Understanding \(K_{sp}\) assists in determining the solubility limits of compounds, which is pivotal in fields like pharmacology and environmental science.

Equilibrium Concentration Calculations

To calculate the equilibrium concentrations in a solution, especially for a compound like aluminum hydroxide, knowing the dissolution equation is key. The ratio of ions in the equation directly guides these calculations. When aluminum hydroxide dissolves, a ratio of 1:3 exists between the aluminum ions and hydroxide ions.
Given the equilibrium hydroxide ion concentration is \(8.58 \times 10^{-9} \, \text{M}\), the concentration of aluminum ions is calculated as one-third of this value, due to the 1:3 stoichiometry. Hence, the concentration of \([\text{Al}^{3+}]\) becomes \(2.86 \times 10^{-9} \, \text{M}\). This simple division helps determine the exact amount of each ion present at equilibrium, aiding significantly in further calculations, like determining \(K_{sp}\) or assessing the solution's properties.

The Importance of a Balanced Chemical Equation

A balanced chemical equation is foundational in understanding how reactions occur and how products form. In the case of aluminum hydroxide dissolving, the balanced equation is \(\text{Al(OH)}_3(s) \leftrightarrow \text{Al}^{3+}(aq) + 3\text{OH}^-(aq)\). This equation tells us that for every molecule of aluminum hydroxide that dissolves, one aluminum ion and three hydroxide ions are produced.
Balancing chemical equations ensures that the law of conservation of mass is obeyed, allowing accurate predictions of concentrations and the behavior of substances in reactions. The coefficients from the balanced equations are integral in writing the \(K_{sp}\) expression accurately, making them indispensable to solving equilibrium problems.

Dissolution of Sparingly Soluble Compounds

Sparingly soluble compounds, like aluminum hydroxide, dissolve minimally in water, reaching a point of dynamic equilibrium where no more solid dissolves and some dissolved ions start to re-form the solid. This equilibrium is characterized by a particular concentration of ions in the solution, often much lower than those of more soluble compounds.
Understanding how these compounds dissolve and establish equilibrium concentrations is key to many practical applications, such as analyzing water hardness or determining the behavior of drugs in the body. The low solubility also means that changes in the surrounding conditions, such as pH or temperature, can significantly affect their solubility and the position of equilibrium, which can be easily monitored by changes in the calculated \(K_{sp}\).