Problem 97 - Chemistry: An Atoms-Focused Approach

Question

What are the equilibrium concentrations of \(\mathrm{Cu}^{+}\) and \(\mathrm{Cl}^{-}\) in a saturated solution of copper(I) chloride at \(25^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

Answer: The equilibrium concentrations of Cu⁺ and Cl⁻ in a saturated solution of copper(I) chloride at 25°C are equal to √(Ksp), where Ksp is the solubility product constant for copper(I) chloride.

Step 1: Write the balanced dissolution equation

First, we should write the balanced dissolution equation for copper(I) chloride. It will dissociate in the following manner: \[ \mathrm{CuCl} \rightleftharpoons \mathrm{Cu}^{+} + \mathrm{Cl}^- \]

Step 2: Determine the solubility product constant (\(K_{sp}\))

To find the equilibrium concentrations, we need the solubility product constant (\(K_{sp}\)) for copper(I) chloride. For this exercise, we assume the \(K_{sp}\) of copper(I) chloride at \(25^{\circ}\mathrm{C}\) is given or can be looked up in a table. Let's indicate its value as \(K_{sp}\).

Step 3: Set up the equilibrium expression using \(K_{sp}\)

Now we set up the equilibrium equation using the \(K_{sp}\) value. For the dissociation of copper(I) chloride, we have: \[ K_{sp} = [\mathrm{Cu}^{+}] [\mathrm{Cl}^{-}] \] Let the equilibrium concentrations of \(\mathrm{Cu}^{+}\) and \(\mathrm{Cl}^{-}\) be equal to \(x\). Then we have: \[ K_{sp} = [x] [x] = x^2 \]

Step 4: Solve for the equilibrium concentrations of \(\mathrm{Cu}^{+}\) and \(\mathrm{Cl}^{-}\)

To find the equilibrium concentrations of \(\mathrm{Cu}^{+}\) and \(\mathrm{Cl}^{-}\), we need to solve for \(x\) using the established equilibrium expression: \[ x^2 = K_{sp} \] Take the square root of both sides of the equation to solve for \(x\): \[ x = \sqrt{K_{sp}} \] Therefore, the equilibrium concentrations of \(\mathrm{Cu}^{+}\) and \(\mathrm{Cl}^{-}\) in a saturated solution of copper(I) chloride at \(25^{\circ}\mathrm{C}\) are equal to \(\sqrt{K_{sp}}\).

Key Concepts

copper(I) chloride

Copper(I) chloride, composed of copper and chloride ions, is symbolized as CuCl. It is important in understanding the basic principles of solubility equilibrium. CuCl, a solid at room temperature, dissolves in water to release copper(I) ions \(\mathrm{Cu}^{+}\) and chloride ions \(\mathrm{Cl}^{-}\).

When a compound like CuCl dissolves in water, it breaks down into its ionic components. This process is known as dissociation. For CuCl, the dissociation equation is given by:

• \(\mathrm{CuCl} \rightleftharpoons \mathrm{Cu}^{+} + \mathrm{Cl}^{-}\)

Copper(I) chloride is a classic example for studying equilibrium in chemical reactions, as it demonstrates the principles of how ions distribute in a solution at equilibrium. The simple 1:1 stoichiometry in the dissociation formula helps in easily understanding the concentration effects in solution.

equilibrium concentration

Equilibrium concentration refers to the concentration of each ion or molecule in a solution when the rate of dissolution equals the rate of precipitation. In other words, the system reaches a state where concentrations remain constant over time.

For the copper(I) chloride example, when the system reaches equilibrium in a saturated solution, there will be certain concentrations for \(\mathrm{Cu}^{+}\) and \(\mathrm{Cl}^{-}\). By assigning these concentrations the variable \(x\), the equilibrium expressions can be simplified:

• For a given solubility product constant, the concentrations of both \(\mathrm{Cu}^{+}\) and \(\mathrm{Cl}^{-}\) are each \(x\).

This concept is crucial, as it allows chemists to predict how much of a substance can dissolve in a solvent at equilibrium. Using known values, one can compute specific concentrations of ions in a mixture, assisting in various scientific and industrial applications.

solubility product constant (Ksp)

The solubility product constant, known as \(K_{sp}\), is a special type of equilibrium constant that applies to the dissolving of sparingly soluble salts. It provides a quantitative measure of a substance's solubility.

For copper(I) chloride, \(K_{sp}\) helps determine how much of the compound can dissolve in a given solvent at a particular temperature. The equilibrium expression for the dissolution of CuCl is \([\mathrm{Cu}^{+}][\mathrm{Cl}^{-}] = K_{sp}\).

In solving for the equilibrium concentrations of \(\mathrm{Cu}^{+}\) and \(\mathrm{Cl}^{-}\), we set:

• \(K_{sp} = x^2\)

Where \(x\) represents the concentration of ions in the solution.
The computation of the square root of \(K_{sp}\) is used to find the equilibrium concentration of ions. A higher \(K_{sp}\) value typically indicates greater solubility, while a lower value signifies a less soluble compound.Understanding \(K_{sp}\) is essential in fields such as chemistry, environmental science, and pharmaceuticals where controlling and predicting solubility is critical.