Problem 129 - A Complete Resource Book in Chemistry for JEE Main

Question

The conductivity of \(0.01 \mathrm{~mol} / \mathrm{dm}^{3}\) aqueous acetic acid at \(300 \mathrm{~K}\) is \(19.5 \times 10^{-5} \mathrm{ohm}^{-1} \mathrm{~cm}^{-1}\) and limiting molar conductivity of acetic acid at the same temperature is \(390 \mathrm{ohm}^{-1} \mathrm{~cm}^{2} \mathrm{~mol}^{-1} .\) The degree of dissociation of acetic acid is (a) \(0.05\) (b) \(0.5 \times 10^{-2}\) (c) \(5 \times 10^{-7}\) (d) \(5 \times 10^{-3}\)

Step-by-Step Solution

Answer

The degree of dissociation is (a) 0.05.

Step 1: Identify the Formula for Degree of Dissociation

The degree of dissociation \( \alpha \) of an electrolyte is calculated using the formula: \[ \alpha = \frac{\Lambda_{m}}{\Lambda_{m}^{\circ}} \] where \( \Lambda_{m} \) is the molar conductivity of the solution and \( \Lambda_{m}^{\circ} \) is the limiting molar conductivity.

Step 2: Calculate Molar Conductivity

The molar conductivity \( \Lambda_{m} \) is given by the relation: \[ \Lambda_{m} = \frac{K}{C} \] where \( K \) is the conductivity and \( C \) is the concentration. Given \( K = 19.5 \times 10^{-5} \mathrm{~ohm}^{-1} \mathrm{~cm}^{-1} \) and \( C = 0.01 \mathrm{~mol/dm}^{3} = 0.01 \mathrm{~mol/L} = 0.01 \mathrm{~mol/cm}^{3} \), we have: \[ \Lambda_{m} = \frac{19.5 \times 10^{-5}}{0.01} = 19.5 \mathrm{~ohm}^{-1} \mathrm{~cm}^{2} \mathrm{~mol}^{-1} \]

Step 3: Substitute Values into the Formula

Using the values of \( \Lambda_{m} = 19.5 \) and \( \Lambda_{m}^{\circ} = 390 \), substitute into the formula: \[ \alpha = \frac{19.5}{390} = 0.05 \]

Step 4: Determine the Correct Option

Compare the calculated degree of dissociation \( \alpha = 0.05 \) to the provided options. The correct option that matches the calculated result is (a) \( 0.05 \).

Key Concepts

Degree of Dissociation

The degree of dissociation is an intriguing concept that refers to how much an electrolyte separates into ions when it dissolves in a solution. It is denoted by the Greek letter \( \alpha \). This parameter plays a crucial role in understanding the behavior of electrolytes in solutions. For weak electrolytes like acetic acid, not all the molecules dissociate completely. Knowing the degree of dissociation helps in calculating the molar conductivity and understanding the efficiency of the electrolyte in conducting electricity in a solution.To calculate this, we use the formula:

• \( \alpha = \frac{\Lambda_{m}}{\Lambda_{m}^{\circ}} \)

This equation reveals that the degree of dissociation is the ratio of the molar conductivity \( \Lambda_{m} \) to the limiting molar conductivity \( \Lambda_{m}^{\circ} \). Molar conductivity is a measure of how well ions can move in a solution, while limiting molar conductivity represents the conductivity at infinite dilution when the dissociation is complete. Thus, \( \alpha \) provides insight into what fraction of the solute dissociates into ions.

Limiting Molar Conductivity

Limiting molar conductivity is a crucial parameter in the study of electrolytes. It represents the theoretical maximum conductivity of an electrolyte when it is completely dissociated into ions and the concentration approaches zero. The symbol used for this concept is \( \Lambda_{m}^{\circ} \).Limiting molar conductivity is essential for several reasons:

• It serves as a reference point for determining the degree of dissociation.
• It helps to compare the ionization power of different electrolytes.
• It aids in calculating other properties such as the mobility of ions.

The limiting molar conductivity can be determined experimentally or through calculations using known values for ions, but it provides a baseline that shows the ideal behavior of the electrolyte under infinite dilution conditions. This makes it a cornerstone for interpreting real-world data of solutions with finite concentrations.

Electrolyte Dissociation

Electrolyte dissociation refers to the process by which a compound separates into its respective ions when dissolved in water or another solvent. This is a foundational concept in chemistry, especially when studying solutions and conductivity. There are two main types of electrolytes:

• Strong electrolytes: These dissociate completely into ions, regardless of the concentration. Examples include salts like sodium chloride (NaCl).
• Weak electrolytes: These do not dissociate completely and are in a dynamic equilibrium in their solutions. Acetic acid is a prime example, as only a small fraction of its molecules dissociate into ions.

Understanding electrolyte dissociation is critical for many applications, such as:

• Designing pharmaceuticals and chemical reactions.
• Conducting titrations and other analytical chemistry techniques.
• Industrial processes that require specific ion concentrations.

The extent of dissociation impacts not only the molar conductivity but also the overall reactivity and functionality of the solution involved.