Question

\(50 \mathrm{~mL}\) of a solution containing \(10^{-3}\) mole of \(\mathrm{Ag}^{+}\) is mixed with \(50 \mathrm{~mL}\) of a \(0.1 \mathrm{M}\) HCl solution. How much \(\mathrm{Ag}^{+}\) remains in solution? \(\left(K_{s p}\right.\) of \(\left.\mathrm{AgCl}=1.0 \times 10^{-10}\right)\) (a) \(2.5 \times 10^{-9}\) (b) \(2.5 \times 10^{-7}\) (c) \(2.5 \times 10^{-8}\) (d) \(2.5 \times 10^{-10}\)

Short answer

The concentration of Ag+ remaining in solution is \(2.5 \times 10^{-8}\) M, which corresponds to option (c).

Step-by-step solution

Determine the Final Concentrations of the Ions

Combine the volumes of the two solutions to find the new volume, which is 100 mL. Since volume is additive but moles are conserved, recalculate the concentrations of Ag+ and Cl-. The initial moles of Ag+ are given (10^-3 mol). For Cl-, use the concentration of the HCl solution (0.1 M) and its volume (50 mL) to calculate the moles of Cl-. The new concentration of each ion is the moles of each ion divided by the new volume (0.1 L).

Write the Solubility Product Expression

Use the solubility product constant (Ksp) for AgCl (1.0 x 10^-10) to create the equation Ksp = [Ag+][Cl-], where [Ag+] and [Cl-] are the molar concentrations of the silver and chloride ions respectively.

Calculate the Concentration of Ag+ Remaining in Solution

Since the concentration of Cl- is in excess, assume it stays constant at 0.05 M (from Step 1). Use the equation from Step 2 and solve for the concentration of Ag+ ([Ag+]) by dividing Ksp by the concentration of Cl-. This will give you the concentration of Ag+ that can remain in solution at equilibrium.

Key concepts

Solubility Equilibrium

Solubility equilibrium is a type of dynamic equilibrium that exists when a chemical compound in the solid state is in chemical equilibrium with a solution of that compound. It is a balance between the process of the solid dissolving and the dissolved ions recombining to form the solid. Solubility product constant (\(K_{sp}\)) is a measure of this equilibrium and is defined for a saturated solution at a given temperature.

In our exercise, when the AgCl starts to dissolve in water, Ag⁺ and Cl^- ions begin to form. If more salt is added, it will continue to dissolve until the maximum solubility limit is reached. If the product of the concentration of the ions exceeds the solubility product constant, the excess ions precipitate out as solid AgCl. This is the essence of reaching the solubility equilibrium in the context of the exercise.

Chemical Concentration

Chemical concentration refers to the amount of a substance per defined space; in other words, it is the ratio of solute in a solution to either solvent or total solution. The concentration is usually expressed in moles per liter (Mc), which is molarity. For our exercise, understanding the concept of concentration is essential since the moles of Ag⁺ and Cl^- must be recalculated after their solutions are mixed together.

In the context of our exercise, the concentration of Cl^- can be assumed as constant because it is present in a large excess, making it the limiting reactant in this equilibrium scenario. It's important to calculate the new concentrations accurately because those values are used to determine whether a precipitate will form and to calculate the K_aspt when mixed together.

Precipitation Reactions

A precipitation reaction occurs when two aqueous solutions combine and form a precipitate, or insoluble solid. The formation of a precipitate is indicative of a chemical reaction taking place. In the context of our exercise, when Ag⁺ ions from the silver solution and Cl^- ions from the HCl solution are mixed, a precipitation reaction can occur if the product of their molar concentrations exceeds the solubility product constant (K_aspt) for AgCl.

To determine if AgCl will precipitate, we compare the actual product of the concentrations of Ag⁺ and Cl^- with the K_aspt for AgCl. If the product is greater than the K_aspt, precipitation occurs until equilibrium is re-established. This outcome is based on Le Chatelier’s principle, which states that the system will adjust to minimize the change, resulting in the formation of a solid precipitate until the product of the concentrations is equal to the K_aspt.