Question

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

Short answer

The Ksp for AgCl at 10°C is approximately $3.95\times {10}^{-11}$.

Step-by-step solution

Calculate the molar concentration of AgCl

First, we need to convert the mass of AgCl into moles. The molar mass of AgCl is 143.32 g/mol (107.87 for Ag and 35.45 for Cl). Given that 9.0 x 10^-4 g of AgCl dissolves in 1 L of water, we can calculate the molar concentration: Molar concentration of AgCl = (9.0 x 10^-4 g) / (143.32 g/mol) = $6.28\times {10}^{-6}$ mol/L

Determine the molar concentration of Ag+ and Cl- ions

For every mole of AgCl that dissolves, one mole of Ag+ and one mole of Cl- are produced. Therefore, the molar concentration of Ag+ and Cl- ions in the solution is equal to the molar concentration of AgCl: [Ag+] = $6.28\times {10}^{-6}$ mol/L [Cl-] = $6.28\times {10}^{-6}$ mol/L

Calculate Ksp

Now that we have the molar concentration of Ag+ and Cl- ions in the saturated solution, we can calculate Ksp using the formula: Ksp = [Ag+] [Cl-] Ksp = $6.28\times {10}^{-6}$ mol/L * $6.28\times {10}^{-6}$ mol/L = $3.95\times {10}^{-11}$ So the Ksp for AgCl at 10°C is approximately $3.95\times {10}^{-11}$.

Key concepts

Silver Chloride Dissolution

Silver chloride (AgCl) is a sparingly soluble salt, known for its formation of a precipitate in water. When it dissolves, a small number of its ions separate into the surrounding water. This dissolution happens in a dynamic balance between the solid salt and dissolved ions. In a saturated solution of silver chloride at equilibrium, the rate at which AgCl dissolves is equal to the rate at which it precipitates back to solid form. This property is key in understanding its solubility, a crucial concept for calculating the solubility product constant, or Ksp.

Molar Concentration

Molar concentration is a measure of the amount of a solute present in a given volume of solution. It is expressed in moles per liter (mol/L). For solutions containing sparingly soluble substances like silver chloride, knowing the molar concentration of the dissolved ions allows us to understand how much of the substance has actually dissolved. In this case, we calculated that the molar concentration of AgCl was approximately $6.28\times {10}^{-6}mol/L$. This required converting the mass of AgCl into moles using its molar mass, which helps chemists to determine the concentration of ions in the solution. Understanding molar concentrations is essential for further applications, such as calculating equilibrium constants.

Chemical Equilibrium

Chemical equilibrium is a state in a reversible reaction where the concentrations of reactants and products remain constant over time. This occurs when the rate of the forward reaction equals the rate of the reverse reaction, creating a balanced system. In the context of silver chloride dissolution, the chemical equilibrium involves the solid AgCl and its corresponding ions in solution. The dissolution and precipitation reactions balance each other, maintaining a consistent concentration. This concept is foundational for calculating the solubility product constant $K_{sp}$, which quantifies the equilibrium between the solid salt and its ions in a saturated solution.

Temperature Effect on Solubility

The solubility of substances like silver chloride can be affected by temperature. For many solids, as temperature increases, solubility usually increases as well. However, this is not always the case for every substance. For example, silver chloride exhibits only a slight change in solubility with variations in temperature, indicating limited sensitivity to temperature. When calculating $K_{sp}$, knowing the temperature at which solubility measurements are taken is vital, as the equilibrium position can shift with a change in temperature. Consequently, the value of $K_{sp}$ is specific to the temperature under which the dissolution was measured, which, in this problem, was 10°C.