Problem 176 - A Complete Resource Book in Chemistry for JEE Main

Question

A buffer solution is prepared by mixing \(20 \mathrm{ml}\) of \(0.1 \mathrm{M}\) \(\mathrm{CH}_{3} \mathrm{COOH}\) and \(40 \mathrm{ml}\) of \(0.5 \mathrm{M} \mathrm{CH}_{3} \mathrm{COONa}\) and then diluted by adding \(100 \mathrm{ml}\) of distilled water. The \(\mathrm{pH}\) of resulting buffer solution is (Given \(\mathrm{pKa} \mathrm{CH}_{3} \mathrm{COOH}=4.76\) ) (a) \(5.76\) (b) \(4.67\) (c) \(3.48\) (d) \(5.9\)

Step-by-Step Solution

Answer

The pH of the buffer solution is 5.76 (option a).

Step 1: Calculate Initial Moles of Substances

First, calculate the moles of acetic acid (\(\mathrm{CH}_3\mathrm{COOH}\)) and sodium acetate (\(\mathrm{CH}_3\mathrm{COONa}\)). The moles for \(\mathrm{CH}_3\mathrm{COOH}\) is given by the product of molarity and volume: \(0.1 \ \mathrm{M} \times 0.02 \ \mathrm{L} = 0.002 \ \mathrm{moles}\). For \(\mathrm{CH}_3\mathrm{COONa}\), it is \(0.5 \ \mathrm{M} \times 0.04 \ \mathrm{L} = 0.02 \ \mathrm{moles}\).

Step 2: Calculate Final Concentrations After Dilution

The total volume after adding 100 ml of water is \(20 \ \mathrm{ml} + 40 \ \mathrm{ml} + 100 \ \mathrm{ml} = 160 \ \mathrm{ml} = 0.160 \ \mathrm{L}\). The concentration of \(\mathrm{CH}_3\mathrm{COOH}\) is \( \frac{0.002 \ \mathrm{moles}}{0.160 \ \mathrm{L}} = 0.0125 \ \mathrm{M} \) and for \(\mathrm{CH}_3\mathrm{COONa}\) is \( \frac{0.02 \ \mathrm{moles}}{0.160 \ \mathrm{L}} = 0.125 \ \mathrm{M} \).

Step 3: Apply Henderson-Hasselbalch Equation

Use the Henderson-Hasselbalch equation for the buffer solution: \( \mathrm{pH} = \mathrm{pKa} + \log \left(\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}\right) \). Substitute the given \(\mathrm{pKa} = 4.76\), \([\mathrm{A}^-] = 0.125 \ \mathrm{M}\), and \([\mathrm{HA}] = 0.0125 \ \mathrm{M}\) into the equation: \( \mathrm{pH} = 4.76 + \log \left(\frac{0.125}{0.0125}\right)\).

Step 4: Calculate the pH

Simplify the expression inside the logarithm: \( \frac{0.125}{0.0125} = 10 \). Thus, the equation becomes \( \mathrm{pH} = 4.76 + \log(10)\). Since \(\log(10) = 1\), the pH is \(4.76 + 1 = 5.76\).

Step 5: Verify the Answer

The calculated pH is 5.76. Compare it to the given choices: (a) 5.76, (b) 4.67, (c) 3.48, (d) 5.9. The correct answer is (a) 5.76.

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch Equation is a vital tool in chemistry for calculating the pH of buffer solutions. Buffer solutions are composed of a weak acid and its conjugate base, or a weak base and its conjugate acid. This equation is especially useful because it allows us to predict the pH of a solution containing these chemical pairs. The mathematical form of the equation is:

• \[ \text{pH} = \text{pKa} + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \]

In this formula, \([\text{A}^-]\) represents the concentration of the conjugate base, while \([\text{HA}]\) symbolizes the concentration of the acid. The term \(\text{pKa}\) is a constant that indicates how strong or weak an acid is and is unique for each acid. By using this equation, we can determine how the presence of the acid and base affects the pH of the solution.

Acetic Acid

Acetic Acid, scientifically known as \( \text{CH}_3\text{COOH} \), plays a fundamental role in buffer solutions. It's a weak acid commonly found in vinegar. In its essence, a weak acid only partially ionizes in water, which means it does not completely separate into ions. This partial ionization is what allows Acetic Acid to work effectively in buffers, where it can help maintain a relatively stable pH even when small amounts of acid or base are added.In the context of the buffer solution exercise discussed, Acetic Acid contributes to establishing the acidic nature of the buffer. It allows the buffer to resist drastic changes in pH when other substances are introduced. The initial moles of Acetic Acid in the solution are calculated using its molarity and volume, which are then adjusted for the dilution in water to determine its final concentration in the solution.

Sodium Acetate

Sodium Acetate, or \( \text{CH}_3\text{COONa} \), is the sodium salt of Acetic Acid and acts as the conjugate base in the buffer solution. When it dissolves in water, it forms the acetate ion \( \text{CH}_3\text{COO}^- \), which can neutralize added acids and thus helps stabilize the pH.In the exercise, Sodium Acetate is added in a greater concentration than Acetic Acid, which assists in establishing the buffer system. The concentration of Sodium Acetate is calculated from its initial moles and the total volume after dilution. This conjugate base's concentration in the solution is key in the Henderson-Hasselbalch Equation, balancing the effect of Acetic Acid and maintaining the desired pH level.

pH Calculation

Calculating pH in a buffer solution involves combining knowledge of the components of the solution with the Henderson-Hasselbalch Equation. In our example scenario, after determining the concentrations of the Acetic Acid and Sodium Acetate, the equation is applied to find the pH:

• First, determine the ratio of the concentrations of the conjugate base to the weak acid.
• Next, apply the equation: \[ \text{pH} = \text{pKa} + \log \left( \frac{0.125}{0.0125} \right) \]
• Solve the logarithmic component: \( \log(10) = 1 \).
• The final computation yields a pH of 5.76.

The pH calculation concludes with verifying the obtained value against the provided options, confirming the accuracy of the buffer composition and its capacity to maintain the pH within the expected range.