Problem 273 - A Complete Resource Book in Chemistry for JEE Main

Question

The pKa of a weak acid, HA is \(4.80\). The \(\mathrm{pK}_{b}\) of a weak base, \(\mathrm{BOH}\) is \(4.78\). The \(\mathrm{pH}\) of an aqueous solution of the corresponding salt, BA will be \([\mathbf{2 0 0 8}]\) (a) \(9.58\) (b) \(4.79\) (c) \(7.01\) (d) \(9.22\)

Step-by-Step Solution

Answer

The pH of the solution is 4.79, corresponding to option (b).

Step 1: Understand the Relationship Between pKa and pKb

A salt of a weak acid and a weak base, such as BA, has the pH of its aqueous solution determined by both the acid and base equilibria. The pH of the solution is influenced by the hydrolysis of the salt, which in turn depends on the pKa of the acid and the pKb of the base.

Step 2: Use Hydrolysis and pH Equation

For a salt of a weak acid and weak base, the pH can be estimated using: \[ \text{pH} = \frac{1}{2} (\text{pK}_a + \text{pK}_b) \]Given pKa = 4.80 and pKb = 4.78, this equation will help us find the pH of the solution.

Step 3: Calculate using Given Values

Substitute the given pKa and pKb values into the equation:\[ \text{pH} = \frac{1}{2} (4.80 + 4.78) \]This simplifies to:\[ \text{pH} = \frac{1}{2} \times 9.58 = 4.79 \]

Step 4: Choose the Correct Answer from Options

The calculated pH of the solution is 4.79. Comparing this with provided options, the correct answer is option (b) 4.79.

Key Concepts

Understanding Weak Acid

A weak acid is a type of acid that does not fully dissociate in water. This means that, when it is dissolved, only a small fraction of its molecules release protons (H⁺ ions) into the solution, unlike strong acids which dissociate completely. This partial dissociation is due to the strength of the bond between the hydrogen ion and the rest of the molecule, which is not as easily broken.

The extent of this dissociation can be quantified using the acid dissociation constant, known as \(K_a\). The pKa is simply the negative logarithm of this constant, \(pK_a = -\log(K_a)\). A higher pKa value indicates a weaker acid, as it suggests that only a small amount of the acid molecules have dissociated. Comprehending weak acids is vital for predicting the behavior of acidic solutions.

Exploring Weak Base

A weak base is similar to a weak acid in that it does not completely ionize in solution. Weak bases partially accept protons when dissolved in water, forming only a small amount of hydroxide ions (OH⁻). The equilibrium for this dissociation is characterized by the base dissociation constant, \(K_b\).

Correspondingly, \(pK_b\) is defined as the negative log of \(K_b\), where \(pK_b = -\log(K_b)\). Similar to acids, a higher \(pK_b\) indicates a weaker base. When combined with weak acids, they can form salts that influence the overall pH of a solution through a process called hydrolysis.

Diving into pKa and pKb

The concepts of pKa and pKb are crucial in understanding acid-base chemistry. Simply put, pKa describes the strength of an acid, while pKb refers to the strength of a base. These are logarithmic measures, making them more intuitive and easier to work with, particularly when comparing relative strengths of different acids and bases.

A key relationship between these values is that the sum of pKa and pKb for a conjugate acid-base pair is often close to 14 in water, reflecting a balance of hydrogen and hydroxide ions. When calculating the pH of a solution containing the salt of a weak acid and a weak base, this balance helps predict the resulting acidity or alkalinity of your solution.

Understanding Salt Hydrolysis

Salt hydrolysis refers to the reaction of salt with water to form either an acidic or basic solution. When a salt composed of a weak acid and a weak base is dissolved in water, it undergoes hydrolysis. This process causes the water to break down into ions which then interact with the ions of the salt, influencing the solution's pH.

The resultant pH can typically be calculated using the relationship between pKa of the weak acid and pKb of the weak base. For example, the pH of such a solution can be estimated using the equation \[ \text{pH} = \frac{1}{2} (\text{pK}_a + \text{pK}_b) \]. This formula shows that hydrolysis tends to produce a neutral solution, but it is the difference in the relative strengths (pKa and pKb) that shifts the pH towards acidity or basicity. Understanding salt hydrolysis is essential in predicting how salts influence the pH of their solutions.