Problem 258 - A Complete Resource Book in Chemistry for JEE Main

Question

The dissociation constant of a substituted benzoic acid is \(1.0 \times 10^{-4}\) at \(25^{\circ} \mathrm{C}\). The \(\mathrm{pH}\) of \(0.01 \mathrm{~m}\) solution of its sodium salt is

Step-by-Step Solution

Answer

The pH of the solution is 8.0.

Step 1: Understand the Problem

We need to find the pH of a 0.01 m solution of the sodium salt of a weak acid, where the dissociation constant \(K_a = 1.0 \times 10^{-4}\). This involves understanding the equilibrium of the acid and its relation to its conjugate base in solution.

Step 2: Identify the Relevant Equations

For a weak acid and its conjugate base, we use the Henderson-Hasselbalch equation: \[ \text{pH} = \text{pK}_a + \log\left(\frac{[A^-]}{[HA]}\right) \] where \([A^-]\) is the concentration of the conjugate base and \([HA]\) is the concentration of the acid. For a solution of the sodium salt of the benzoic acid, \([HA] = 0\).

Step 3: Calculate \(\text{pK}_a\)

First, we calculate the \(\text{pK}_a\) from the dissociation constant \(K_a\). Use the formula: \[ \text{pK}_a = -\log(K_a) \] Substituting the given \(K_a\), we find \(\text{pK}_a\): \[ \text{pK}_a = -\log(1.0 \times 10^{-4}) = 4.0 \]

Step 4: Use Henderson-Hasselbalch Equation

Since we have a solution of the sodium salt (conjugate base \([A^-]\)) with no significant presence of the acid \([HA]\), assume \([A^-] = 0.01\,m\) and \([HA] \approx 0\). Applying the Henderson-Hasselbalch equation: \[ \text{pH} = 4.0 + \log\left(\frac{0.01}{0}\right) \]. Since it’s entirely the conjugate base present, the \(\log\) term simplifies to the concentration of the base.

Step 5: Simplify and Solve

In a solution where \([A^-] = 0.01\,m\) and \([HA] = 0\), the \(\text{pH} = \text{pK}_a + 2\). Simplifying, since there is no presence of \([HA]\), we find using the base concentration: \[ \text{pH} = 4 + 2 = 8.0 \].

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is instrumental in pH calculations involving weak acids and their conjugate bases. This equation is expressed as \( \text{pH} = \text{pK}_a + \log\left(\frac{[A^-]}{[HA]}\right) \). It shows the relationship between the pH of a solution, the acid dissociation constant (\( \text{pK}_a \)), and the ratio of the concentrations of the conjugate base (\([A^-]\)) to the acid (\([HA]\)).

This equation is incredibly useful because it allows us to calculate the pH of a buffer solution. Buffers resist changes in pH when small amounts of acid or base are added to them. The Henderson-Hasselbalch equation provides insight into how well a buffer can maintain its pH and helps us design buffer solutions with a desired pH level.

• When \([A^-] > [HA]\), the pH will be greater than the \( \text{pK}_a \).
• If \([A^-] = [HA]\), the pH equals \( \text{pK}_a \).
• And when \([A^-] < [HA]\), the pH will be less than \( \text{pK}_a \).

Understanding this equation and its applications is crucial when studying the behavior of acids and bases in solution.

Weak Acids

Weak acids are acids that do not completely dissociate in water. This behavior contrasts with strong acids, which donate all their available hydrogen ions in solution. Examples of weak acids include acetic acid and benzoic acid.

In weak acids, only a small fraction of the acid molecules release hydrogen ions into the solution at any given time. This partial dissociation is what characterizes them as weak acids and is why the concentration of ions in the solution is less than the initial concentration of the acid. Because of this, weak acids exhibit a higher pH compared to strong acids at similar concentrations.

For instance, a solution of a weak acid like benzoic acid will have both undissociated molecules and ions in equilibrium. This property is critical when applying the Henderson-Hasselbalch equation, as it allows us to calculate the pH if we know the concentrations of the acid and its conjugate base.

Remember, the degree of dissociation is dependent on the acid's dissociation constant, making it a key factor in determining the behavior and properties of the acid in solution.

Dissociation Constant

The dissociation constant, commonly represented as \( K_a \), is a measure of the extent to which an acid can dissociate in a solution. It is crucial for understanding the strength of an acid. The dissociation constant is especially important for weak acids, as it provides insight into how many acid molecules will release hydrogen ions.

The larger the value of \( K_a \), the stronger the acid because more molecules dissociate to produce hydrogen ions. For a weak acid like benzoic acid with a \( K_a \) of \( 1.0 \times 10^{-4} \), we know that it only slightly dissociates in water, thus showing typical weak acid behavior.

The relationship between \( K_a \) and \( \text{pK}_a \) is given by the equation \( \text{pK}_a = -\log(K_a) \). This transforms the dissociation constant into a more manageable logarithmic scale, which is why biochemists and chemists often prefer using \( \text{pK}_a \) for comparison and calculations. Knowing \( K_a \) or \( \text{pK}_a \) allows us to understand how a weak acid will behave in aqueous solutions and is particularly useful when using the Henderson-Hasselbalch equation to find pH.