Problem 235 - A Complete Resource Book in Chemistry for JEE Main

Question

A certain buffer solution contains equal conc. of \(X^{-}\) and HX. The \(K_{n}\) of HX is \(10^{-7}\). The \(\mathrm{pH}\) of the buffer solution is

Step-by-Step Solution

Answer

The pH of the buffer solution is 7.

Step 1: Understand the Relationship

The Henderson-Hasselbalch equation for calculating the pH of a buffer solution is given by:\[\text{pH} = \text{pKa} + \log\frac{[\text{Base}]}{[\text{Acid}]}\]For this solution, the acid-base pair is HX (acid) and \(X^-\) (base). Since the concentrations of \(X^-\) and HX are equal, this simplifies the equation.

Step 2: Calculate pKa

The pKa is related to the acid dissociation constant \(K_a\) by the formula:\[\text{pKa} = -\log(K_a)\]Given \(K_a = 10^{-7}\), we substitute into the formula:\[\text{pKa} = -\log(10^{-7}) = 7\]

Step 3: Apply the Henderson-Hasselbalch Equation

Since \([\text{Base}] = [\text{Acid}]\), we have:\[\text{pH} = \text{pKa} + \log(1)\]As \(\log(1) = 0\), this simplifies to:\[\text{pH} = 7 + 0 = 7\]

Key Concepts

Understanding Buffer Solutions

Buffer solutions are special types of solutions that resist changes in pH when small amounts of acid or base are added. They are crucial in many chemical and biological systems where a stable pH is important. A buffer solution is typically made up of a weak acid and its conjugate base or a weak base and its conjugate acid.
For example, in the exercise, the buffer solution contains the acid-base pair of HX and \(X^-\). This combination allows the solution to neutralize added acids or bases by shifting the equilibrium position gently without causing drastic changes in pH.
A buffer works due to the presence of both components. The weak acid can donate protons (\(H^+\)) if a base is added, while the conjugate base can absorb protons if an acid is added. This dual action helps maintain a relatively constant pH within a specific range.

Exploring Acid-Base Equilibrium

The concept of acid-base equilibrium is essential in understanding how buffer solutions function. It involves the balance between acids and bases in a solution, usually described by the equilibrium constant, \(K_a\), for acids or \(K_b\) for bases.
The acid dissociation constant \(K_a\) gives us insight into the strength of an acid. The smaller the \(K_a\), the weaker the acid because it dissociates less in solution. In the exercise, \(K_a\) for HX is \(10^{-7}\), indicating it's a weaker acid.
In buffer systems, this equilibrium is crucial in the balance between the acid (HX) and its conjugate base (\(X^-\)). Because their concentrations are equal, the equilibrium can easily shift to counteract any disturbances to the pH level, thus stabilizing the solution.

Simplifying pH Calculations with Henderson-Hasselbalch Equation

Calculating the pH of a buffer solution, such as the one presented in the exercise, is simplified by the Henderson-Hasselbalch equation. This equation provides a straightforward method to relate the concentrations of the base and acid to the pH of the solution.
The basic form of the equation is:\[\text{pH} = \text{pKa} + \log\frac{[\text{Base}]}{[\text{Acid}]}\]
In our exercise, it’s given that the concentration of the acid (HX) is equal to the concentration of its conjugate base (\(X^-\)). This equality simplifies the equation since \(\log(1) = 0\). Therefore, the pH is directly equal to the \(\text{pKa}\).
The \(\text{pKa}\) is calculated from the given \(K_a\) value of the weak acid using:\[\text{pKa} = -\log(K_a)\]
For \(K_a = 10^{-7}\), \(\text{pKa}\) ends up being 7, which means the pH of the buffer solution is 7. This simple approach efficiently links the acid’s strength, base's presence, and pH into one memorable formula.