Question

The solubility of \(\mathrm{AgCl}\) in water is \(0.001435 \mathrm{~g}\) per litre at \(15^{\circ} \mathrm{C}\). The solubility product of \(\mathrm{AgCl}\) is \((\mathrm{Ag}=108, \mathrm{Cl}=35.3)\) (a) \(10^{-5}\) (b) \(10^{-10}\) (c) \(2 \times 10^{-10}\) (d) \(10^{-9}\)

Short answer

The solubility product of AgCl at 15°C is \(1 \times 10^{-10}\) which matches choice (b).

Step-by-step solution

Understanding the Concept of Solubility Product (Ksp)

The solubility product, Ksp, is an equilibrium constant for a solid substance dissolving in an aqueous solution. It represents the level at which a solute dissolves to form a saturated solution. For the dissolution of AgCl in water, the equation is as follows: AgCl (s) ↔ Ag+ (aq) + Cl- (aq). The Ksp expression for this equation is: Ksp = [Ag+] [Cl-].

Calculating Molar Solubility

Using the molar mass of AgCl (the sum of the atomic masses of Ag and Cl), convert the given solubility in grams per liter to moles per liter. Molar mass of AgCl = 108 + 35.5 = 143.5 g/mol. Now calculate the molar solubility: 0.001435 g/L / 143.5 g/mol = 0.00001 mol/L.

Writing the Expression for Ksp

Ksp = [Ag+] [Cl-]. Since the dissolution of AgCl yields one mole of Ag+ ions and one mole of Cl- ions for each mole of AgCl dissolved, the concentrations of Ag+ and Cl- are equal to the molar solubility of AgCl.

Calculating the Ksp

Substitute the molar solubility into the Ksp expression. Ksp = (0.00001) (0.00001) = 1 x 10^-10

Key concepts

Understanding the Equilibrium Constant (Ksp)

The solubility product constant, often referred to as the equilibrium constant (Ksp), is a special type of equilibrium constant that measures the solubility of sparingly soluble ionic compounds. It's important to recognize that Ksp is applicable only to compounds that partially dissolve in solution, reaching a point of dynamic equilibrium.

At this equilibrium, the rate of the solid dissolving into ions is equal to the rate at which the ions recombine to form the solid. The value of Ksp gives an idea of the extent to which a compound can dissolve in water. This is crucial for predicting the solubility behavior of substances in various chemical processes, including biological systems and industrial applications.

Calculating Molar Solubility from Ksp

Molar solubility is the number of moles of a solute that can be dissolved per liter of solution before the solution becomes saturated. Calculating molar solubility from Ksp involves a bit of reverse engineering; given the Ksp, we can determine the maximum amount of substance that can dissolve in a solution.

To do this, we use stoichiometry of the dissolving process and the Ksp value to establish a relationship between the concentrations of the ions in a saturated solution. In our exercise example with AgCl, for each mole of AgCl that dissolves, one mole of Ag+ and one mole of Cl- ions are produced. The equilibrium concentrations of these ions, which are equal since they dissociate in a 1:1 ratio, define the molar solubility of AgCl.

The Role of Ionic Product in Solubility

Ionic product is similar to Ksp but refers to the product of concentrations of the ionic species at any given moment, not just at equilibrium. When this ionic product equals the Ksp value, the solution is saturated, and no more solute can dissolve. If the ionic product is less than Ksp, more solute can still dissolve, while if it's greater, precipitation occurs as the solution is supersaturated.

In our AgCl example, when the molar solubility matches the point where the concentration product of Ag+ and Cl- ions equals the Ksp, the solution is at the brink of saturation. Recognizing the connection between ionic product and Ksp helps understand precipitation reactions, bioavailability of minerals, and other solution dynamics.