Problem 74 - A Complete Resource Book in Chemistry for JEE Main

Question

Solution of \(0.1 \mathrm{~N} \mathrm{NH}_{4} \mathrm{OH}\) and \(0.1 \mathrm{~N} \mathrm{NH}_{4} \mathrm{Cl}\) has \(\mathrm{pH}\) \(9.25\), then find out \(\mathrm{pKb}\) of \(\mathrm{NH}_{4} \mathrm{OH}\) (a) \(9.25\) (b) \(4.75\) (c) \(3.75\) (d) \(8.25\)

Step-by-Step Solution

Answer

The pKb of \(\mathrm{NH}_4\mathrm{OH}\) is 4.75.

Step 1: Understanding the Given Buffer System

We are given a buffer solution comprising of 0.1 N NH₄OH and 0.1 N NH₄Cl with a pH of 9.25. This is a classic case of a weak base (NH₄OH) and its salt (NH₄Cl) buffer.

Step 2: Use Henderson–Hasselbalch Equation for Base Buffers

For a basic buffer system, the Henderson–Hasselbalch equation is:\[ \text{pH} = \text{pKa} + \log \left( \frac{[\text{Base}]}{[\text{Salt}]} \right) \]Here, pKa is for NH₄OH, and because \( [\text{Base}] = [\text{Salt}] \), the log term becomes 0.

Step 3: Calculate pKa

The equation simplifies to \( \text{pH} = \text{pKa} \). We are given pH = 9.25, so: \\[ \text{pKa} = 9.25 \]

Step 4: Relate pKa and pKb

For the relationship between pKa and pKb of a conjugate acid-base pair: \[ \text{pKa} + \text{pKb} = 14 \]Using \( \text{pKa} = 9.25 \), we solve for pKb:\[ \text{pKb} = 14 - 9.25 \]

Step 5: Solve for pKb

Compute the value:\[ \text{pKb} = 4.75 \]

Key Concepts

pH calculation

In chemistry, understanding the concept of pH is crucial for analyzing the acidity or basicity of a solution. pH is a logarithmic scale used to specify the acidity or basicity of an aqueous solution. It is defined mathematically as:\[ \text{pH} = -\log_{10} [H^+] \]where \([H^+]\) denotes the hydrogen ion concentration. The pH scale ranges from 0 to 14. A pH of 7 is neutral, meaning that the concentration of hydrogen ions is equal to the concentration of hydroxide ions. Values lower than 7 indicate acidic solutions, whereas values higher than 7 point to basic solutions. Given the pH of a buffer solution, like the one in the exercise where the pH is 9.25, it signifies a basic environment. This pH value indicates that there is a lower concentration of hydrogen ions compared to hydroxide ions, typical of a basic solution. Observing the pH allows us to comprehend the behavior of the solution in different chemical environments and is essential for further calculations like finding the pKa or pKb of substances in the solution.

buffer solution

A buffer solution is pivotal in maintaining a stable pH in a chemical system, even when small amounts of acid or base are added. This property is vital in many biological and chemical processes. Buffers are typically composed of a weak acid and its conjugate base or a weak base and its conjugate acid. The buffer mentioned in the original exercise is made of ammonium hydroxide (NH₄OH), a weak base, and ammonium chloride (NH₄Cl), the corresponding salt. This represents a basic buffer system. The function of this buffer is to resist changes in pH, thereby keeping the environment within a narrow pH range, which is particularly important in many chemical reactions and physiological processes. A buffer's capacity, or its ability to maintain pH stability, depends largely on the concentrations of the weak acid/base and its conjugate. In our example, both the NH₄OH and NH₄Cl are present in equal concentrations, which is a classic setup for a buffer solution. Calculations regarding the buffer’s pH often involve the Henderson–Hasselbalch equation, which is integral to understanding buffer systems.

pKa and pKb relationship

The relationship between pKa and pKb is fundamental in understanding the acid-base chemistry of compounds. For a conjugate acid-base pair, the relationship is mathematically given as:\[ \text{pKa} + \text{pKb} = 14 \]This formula arises from the water ion product constant, wherein the sum of the pKa of an acid and the pKb of its conjugate base equals 14 at 25°C. Knowing this relationship helps us transition between acid and base dissociation constants effortlessly.In the exercise, we know the pKa for NH₄OH is 9.25. By utilizing the above relationship, we can find the pKb by rearranging the formula to:\[ \text{pKb} = 14 - \text{pKa} \]Thus calculating the pKb of NH₄OH gives us 4.75, signifying its basic nature. This relationship is incredibly valuable in predicting how substances will behave in various chemical environments and is often crucial in developing buffer solutions with the desired characteristics.