Question

Approximately $9.0\times {10}^{-4}$ g of silver chloride, $\mathrm{AgCl}(s),$ dissolves per liter of water at ${10}^{\circ }\mathrm{C}.$ Calculate $K_{\mathrm{sp}}$ for $\mathrm{AgCl}(s)$ at this temperature.

Short answer

The solubility product constant, $K_{sp}$, for AgCl at 10°C is approximately $3.95\times {10}^{-11}$.

Step-by-step solution

Write the dissociation equation

We need to write the chemical equation for the dissociation of AgCl in water to know the ions involved in the process. The dissociation reaction is as follows: $$\mathrm{AgCl}(s)\rightleftharpoons {\mathrm{Ag}}^{+}(aq)+{\mathrm{Cl}}^{-}(aq)$$

Write the expression for Ksp

The solubility product constant, Ksp, is a measure of the equilibrium between a solid and its dissociation ions in a solution. The expression for Ksp of AgCl based on the dissociation equation is the product of the equilibrium concentrations of the ions: $$K_{sp}=[{\mathrm{Ag}}^{+}][{\mathrm{Cl}}^{-}]$$

Convert given solubility to molar concentration

We are given that approximately $9.0\times {10}^{-4}$ g of AgCl dissolves in 1 liter (L) of water. We need to convert this into molar concentration to use in our Ksp calculation. First, we need to know the molar mass of AgCl: Molar mass of silver (Ag): 107.87 g/mol Molar mass of chloride (Cl): 35.45 g/mol Molar mass of AgCl = 107.87 g/mol + 35.45 g/mol = 143.32 g/mol Now, we can convert the given solubility to molar concentration: Molar concentration = $\frac{9.0\times {10}^{-4}\text{g}}{143.32\text{g/mol}}=6.28\times {10}^{-6}\text{mol/L}$

Determine the equilibrium concentrations of the ions

Since AgCl dissociates into a 1:1 ratio of Ag⁺ and Cl⁻, the concentrations of these ions at equilibrium will both be the molar concentration of AgCl that dissolves in water: $$[{\mathrm{Ag}}^{+}{]}_{eq}=[{\mathrm{Cl}}^{-}{]}_{eq}=6.28\times {10}^{-6}\text{mol/L}$$

Calculate Ksp

Using the Ksp expression from Step 2 and the equilibrium concentrations from Step 4, we can now calculate Ksp for AgCl: $$K_{sp}=[{\mathrm{Ag}}^{+}][{\mathrm{Cl}}^{-}]$$ $$K_{sp}=(6.28\times {10}^{-6})(6.28\times {10}^{-6})=3.95\times {10}^{-11}$$ Thus, the solubility product constant, Ksp, for AgCl at 10°C is approximately $3.95\times {10}^{-11}$.

Key concepts

Dissociation of Ionic Compounds

The concept of dissociation in chemistry refers to the process by which an ionic compound separates into its individual ions when it is dissolved in a solvent, such as water. This separation occurs because the water molecules interact with the ions in a way that overcomes the ionic bond holding them together in the solid.

Consider the dissociation of silver chloride (AgCl), which is represented by the reaction equation:
\[ \mathrm{AgCl}(s) \rightleftharpoons \mathrm{Ag}^{+}(aq) + \mathrm{Cl}^{-}(aq) \]
This equation shows that solid AgCl, when in contact with water, establishes an equilibrium where it breaks apart into Ag⁺ (silver ions) and Cl⁻ (chloride ions). The double arrow signifies a reversible reaction, illustrating that the dissociation is an equilibrium process where both reactants and products coexist.

The degree to which this dissociation happens is determined by factors such as the temperature and the nature of the ionic compound itself. Silver chloride is considered to be poorly soluble in water, which means only a small fraction will dissociate at any given time.

Chemical Equilibrium

Chemical equilibrium occurs when a chemical reaction and its reverse are proceeding at the same rate, resulting in no overall change in the concentrations of the reactants and products. It is a dynamic state where the forward and reverse reactions continue to happen but offset one another exactly.

In the case of silver chloride's dissociation in water:
\[ \mathrm{AgCl}(s) \rightleftharpoons \mathrm{Ag}^{+}(aq) + \mathrm{Cl}^{-}(aq) \]
The point at which the rate at which AgCl precipitates out of the solution to form a solid equals the rate at which it dissolves and ionizes into Ag⁺ and Cl⁻ ions. At equilibrium, the molar concentrations of the ions remain constant over time, despite the ongoing processes.

The concept of the solubility product constant (Ksp) is closely tied to chemical equilibrium. It represents the extent to which a compound will dissolve in a solution and helps to predict the solubility of the compound under specific conditions.

Molar Concentration

Molar concentration, also known as molarity, is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute dissolved per liter of solution (mol/L). It allows for the quantification of solute concentration and plays a crucial role in calculating the solubility product constant.

In practice, to calculate the molar concentration, you would divide the mass of the solute by its molar mass (the sum of the individual atomic masses in g/mol), and then again by the volume of the solution in liters. For instance, with silver chloride:
\[ \text{Molar concentration} = \frac{9.0 \times 10^{-4} \,\text{g}}{143.32 \,\text{g/mol}} \]
This formula shows that if you have 9.0 x 10⁻⁴ grams of AgCl in one liter of water, and the molar mass of AgCl is 143.32 g/mol, the molarity of the solution is 6.28 x 10⁻⁶ mol/L.

Molar concentration is critical when discussing the solubility product constant (Ksp) since Ksp is calculated using the molar concentrations of the constituent ions at equilibrium. Understanding molarity thus provides a foundation for predicting and explaining solubility behaviors in different ionic compounds.