Question

\(\mathrm{pH}\) of \(0.005 \mathrm{M}\) calcium acetate (pKa of \(\mathrm{CH}_{3} \mathrm{COOH}\) \(=4.74\) ) is: (a) \(7.37\) (b) \(9.37\) (c) \(9.26\) (d) \(8.37\)

Short answer

The pH of the solution is \(9.37\) (Option b).

Step-by-step solution

Determine the Reaction

Calcium acetate is a salt of a weak acid, acetic acid (\(\text{CH}_3\text{COOH}\)). When dissolved in water, it partially dissociates into calcium ions and acetate ions \(\text{CH}_3\text{COO}^-\) which can slightly react with water to reform acetic acid and hydroxide ions \(\text{OH}^-\). This makes the solution basic.

Write the Hydrolysis Equation

The hydrolysis of acetate ions can be written as: \(\text{CH}_3\text{COO}^- + \text{H}_2\text{O} \rightleftharpoons \text{CH}_3\text{COOH} + \text{OH}^-\). The equilibrium constant expression for this would involve the concentration of \(\text{CH}_3\text{COOH}\), \(\text{OH}^-\), and \(\text{CH}_3\text{COO}^-\).

Calculate the Base Ionization Constant (Kb)

Using the relations: \(\text{pK}_w = \text{pK}_a + \text{pK}_b\), where \(\text{pK}_w = 14\), we get: \(\text{pK}_b = 14 - 4.74 = 9.26\). Convert to \(K_b\) using \(K_b = 10^{-\text{pK}_b}\).

Set Up ICE Table

Initial concentration of \(\text{CH}_3\text{COO}^-\) is approximately \(0.005 \ M\). Assume an initial change of \(x\) for the ionization: the equation at equilibrium is \([\text{CH}_3\text{COOH}] = x, [\text{OH}^-] = x, [\text{CH}_3\text{COO}^-] = 0.005 - x\).

Solve the Equilibrium Expression

Using \(K_b = \frac{[\text{CH}_3\text{COOH}][\text{OH}^-]}{[\text{CH}_3\text{COO}^-]}\), substitute known values and solve for \(x\). With \(K_b\) being very small, the approximation \(0.005 - x \approx 0.005\) simplifies solving: \(x^2 = 0.005 K_b\).

Calculate \([OH^-]\) and \(pOH\)

With \(x\) as the concentration of \([\text{OH}^-]\), compute \([\text{OH}^-]\) and solve for \(\text{pOH} = -\log(\text{[OH]}^-)\).

Determine the pH

Convert \(\text{pOH}\) to \(\text{pH}\) using \(\text{pH} = 14 - \text{pOH}\). The calculated \(\text{pH}\) of the solution should be close to one of the given options.

Key concepts

Ionic Equilibrium

Ionic equilibrium in a solution refers to the state where the rate of formation of ions from molecules equals the rate at which ions recombine to form molecules. This concept plays a crucial role in understanding the behavior of substances like calcium acetate in water. Calcium acetate, being a salt of a weak acid (acetic acid), dissociates into acetate ions and calcium ions when it dissolves in water. The acetate ion (\(\text{CH}_3\text{COO}^-\)) shows a tendency to interact with hydrogen ions (\(\text{H}^+\)) from water, shifting the equilibrium slightly back towards the molecular form of acetic acid. This interaction is a classic example of ionic equilibrium. Keep in mind:

• The balance between ionized and unionized particles affects the pH of the solution significantly.
• Ionic equilibrium is dynamic, continuously adjusting as ions interact and form new compounds.
• This concept is pivotal when calculating pH in solutions containing ions that can recombine to form weak acids or bases.

Hydrolysis of Salts

Hydrolysis of salts is a process where an ion from a dissolved salt reacts with water, leading to the formation or release of an acid or a base. In our example, calcium acetate undergoes hydrolysis due to the presence of acetate ions, which react with water. The hydrolysis reaction for acetate ions is written as: \(\text{CH}_3\text{COO}^- + \text{H}_2\text{O} \rightleftharpoons \text{CH}_3\text{COOH} + \text{OH}^-\). Understanding this reaction helps in predicting whether a salt solution will be acidic, neutral, or basic. Since acetate ion is the conjugate base of a weak acid, its hydrolysis results in a basic solution, which is why the pH of calcium acetate solution is greater than 7.Here's what you need to remember:

• Not all salts undergo hydrolysis. Only those formed from weak acids or weak bases typically do.
• The direction and extent of the shift in equilibrium during hydrolysis determine the nature of the solution.
• Hydrolysis is the key to understanding why solutions of salts from weak acids or bases can be basic or acidic.

Equilibrium Constant Expressions

The equilibrium constant expression is crucial for quantifying the position of equilibrium in chemical reactions. For the hydrolysis of acetate ions, the equilibrium constant, known as the base ionization constant (\(K_b\)), helps determine the concentration of different species at equilibrium.The expression is given by:\[K_b = \frac{[\text{CH}_3\text{COOH}][\text{OH}^-]}{[\text{CH}_3\text{COO}^-]}\]This formula shows the ratio of product concentrations to reactant concentrations. Knowing \(K_b\) allows us to find the concentration of hydroxide ions, and thus, calculate the \(pH\) of the solution.Key insights:

• The equilibrium constant expression simplifies complex equilibria into manageable calculations.
• Understanding \(K_b\) or \(K_a\) (for acids) is essential for predicting the behavior of weak acids or bases in water.
• Equilibrium constants are temperature-dependent, meaning equilibrium positions can shift with changing temperatures.

Base Ionization Constant

The base ionization constant (\(K_b\)) measures the extent to which a base can produce hydroxide ions in a solution. It is derived from the fundamental concept of equilibrium constants, specifically tailored for bases. For the acetate ion in calcium acetate, we connect this to the base ionization constant provided by its conjugate acid’s dissociation constants – a linkage often expressed as:\[pK_w = pK_a + pK_b\]where\(pK_w\) is the negative logarithm of the ionic product of water.Using this relation, we derive \(pK_b\) and subsequently \(K_b = 10^{-pK_b}\). This allows us to find how strongly the acetate ion hydrolyzes to form \(OH^-\) in water. Calculating \(K_b\) provides insight into the basicity of the solution.Important to remember:

• The smaller the \(K_b\), the weaker the base and the less it will dissociate in water.
• \(pK_b\) is a useful scale for comparing base strengths.
• This constant is crucial in calculating pH for solutions involving weak bases.