Calculate the pH and the percent dissociation of the hydrated cation in the following solutions. See Appendix C for the value of the equilibrium constant. (a) 0.010 M Cr1NO323

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Identify the hydrated cation in the solution. For Cr(NO_3)_3, the hydrated cation is Cr^{3+}.

Write the equilibrium expression for the hydrolysis of Cr^{3+} in water: Cr^{3+} + 3H_2O \(\rightleftharpoons\) Cr(OH)_3 + 3H^+.

Use the equilibrium constant (K_eq) for the hydrolysis reaction from Appendix C to set up the expression: K_eq = [Cr(OH)_3][H^+]^3 / [Cr^{3+}].

Assume that the initial concentration of Cr^{3+} is 0.010 M and that the change in concentration due to hydrolysis is x. Set up the equilibrium concentrations: [Cr^{3+}] = 0.010 - x, [H^+] = 3x, and [Cr(OH)_3] = x.

Substitute the equilibrium concentrations into the K_eq expression and solve for x, which represents the concentration of H^+ ions. Use this to calculate the pH as pH = -\(\log\)[H^+].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

pH Calculation

pH is a measure of the acidity or basicity of a solution, defined as the negative logarithm of the hydrogen ion concentration. To calculate pH, one must first determine the concentration of hydrogen ions in the solution, which can be derived from the dissociation of acids or bases. For weak acids, the pH can be calculated using the equilibrium expression involving the acid dissociation constant (Ka).

Percent Dissociation

Percent dissociation refers to the fraction of a solute that has dissociated into ions in a solution, expressed as a percentage. It is calculated by taking the concentration of dissociated ions at equilibrium, dividing it by the initial concentration of the solute, and multiplying by 100. This concept is particularly important for weak acids and bases, as it indicates the strength of the acid or base in solution.

Equilibrium Constant (K)

The equilibrium constant (K) quantifies the ratio of the concentrations of products to reactants at equilibrium for a given reaction. For acid-base reactions, the equilibrium constant is often represented as Ka for acids and Kb for bases. These constants are crucial for calculating pH and percent dissociation, as they provide the necessary information about the extent of dissociation of the solute in solution.