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Kohlrausch’s Law is a key topic in electrochemistry, particularly for Class 12 students. It offers a fundamental understanding of the conductivity of electrolytic solutions. In this article, we’ll delve into the **formula**, **derivation**, and a set of **practice problems** to help solidify your understanding of this critical concept. Whether you’re preparing for board exams or competitive exams, this comprehensive guide will walk you through everything you need to know about **Kohlrausch Law**.

What is Kohlrausch’s Law?

Kohlrausch’s Law, also known as the **Kohlrausch Law of Independent Migration of Ions**, states that the molar conductivity of an electrolyte at infinite dilution can be expressed as the sum of the individual contributions of the anion and cation of the electrolyte. It shows that each ion contributes independently to the total conductivity of the solution, regardless of the nature of the other ion in the electrolyte.

In simple terms, Kohlrausch’s Law provides a framework to calculate the **limiting molar conductivity** (also called **molar conductivity at infinite dilution**) of an electrolyte, which is the molar conductivity when the concentration of the electrolyte approaches zero.

**Key Definition:**

**Limiting Molar Conductivity**: The conductivity of a solution when its concentration tends towards zero, where interactions between ions are negligible.

The formula of Kohlrausch’s Law

The mathematical expression of Kohlrausch’s Law is as follows:

Λm0=λ+0+λ−0\Lambda^0_m = \lambda^0_+ + \lambda^0_-Λm0=λ+0+λ−0

Where:

**Λₘ⁰**is the limiting molar conductivity of the electrolyte.**λ⁰₊**is the limiting molar conductivity of the cation.**λ⁰₋**is the limiting molar conductivity of the anion.

This formula shows that the total limiting molar conductivity of an electrolyte is simply the sum of the contributions from its cations and anions.

**Explanation of Terms:**

**Cation**: The positively charged ion in the electrolyte (e.g., Na⁺ in NaCl).**Anion**: The negatively charged ion in the electrolyte (e.g., Cl⁻ in NaCl).**Molar Conductivity (Λ)**: The conductance of an electrolyte solution per unit concentration, represented as ΛmΛ_mΛm.

Derivation of Kohlrausch’s Law

Kohlrausch’s Law is derived from experimental data regarding the conductance of strong electrolytes at infinite dilution. Here’s the step-by-step breakdown of the derivation:

**Step 1: Conductance at Infinite Dilution**

At infinite dilution, the electrolyte dissociates completely into its constituent ions. The molar conductivity of an electrolyte can be expressed as the sum of the ionic conductivities of the cations and anions:

Λm0=λ+0+λ−0\Lambda^0_m = \lambda^0_+ + \lambda^0_-Λm0=λ+0+λ−0

**Step 2: Independence of Ion Migration**

Kohlrausch found that the molar conductivity at infinite dilution does not depend on the interaction between the cation and anion but on their independent movement through the solution. Therefore, the cation’s and anion’s contributions can be added to calculate the total limiting molar conductivity.

**Step 3: Ionic Contribution**

The individual ionic conductivities, **λ⁰₊** and **λ⁰₋**, depend on the nature of the ion, its charge, and its mobility in the solution. Since each ion migrates independently, its conductivities can be summed up to give the total limiting molar conductivity.

Thus, the law is expressed as:

Λm0=λ+0+λ−0\Lambda^0_m = \lambda^0_+ + \lambda^0_-Λm0=λ+0+λ−0

**Application to Strong and Weak Electrolytes**

For **strong electrolytes**, this law holds since they dissociate completely at infinite dilution. However, **weak electrolytes** only partially dissociate, and Kohlrausch’s Law helps estimate their limiting molar conductivity by extrapolating data from dilute solutions.

Importance of Kohlrausch’s Law

Kohlrausch’s Law has various applications in **electrochemistry** and **analytical chemistry**:

**Determination of Limiting Molar Conductivity**: For weak electrolytes, the limiting molar conductivity cannot be measured directly because they do not dissociate completely in solution. Kohlrausch’s Law allows the calculation of**Λₘ⁰**by summing the contributions of the ions.**Calculation of Degree of Dissociation (α)**: The law helps calculate the degree of dissociation of weak electrolytes by comparing their molar conductivity at a given concentration to the limiting molar conductivity:

α=ΛmΛm0\alpha = \frac{\Lambda_m}{\Lambda^0_m}α=Λm0Λm

**Determination of Solubility**: By applying Kohlrausch’s Law, the solubility of sparingly soluble salts can be determined from their limiting molar conductivity.**Analysis of Electrolyte Behavior**: The law helps analyze how electrolytes behave in solution, particularly how different ions contribute to the overall conductivity.

Practice Problems on Kohlrausch’s Law

**Problem 1: Calculation of Limiting Molar Conductivity**

**Question**: The limiting molar conductivities of Na⁺ and Cl⁻ are 50.11 S cm² mol⁻¹ and 76.34 S cm² mol⁻¹, respectively. Calculate the limiting molar conductivity of NaCl.

**Solution**: Using Kohlrausch’s Law:

Λm0=λ+0+λ−0\Lambda^0_m = \lambda^0_+ + \lambda^0_-Λm0=λ+0+λ−0

Substitute the values:

Λm0=50.11+76.34=126.45 S cm² mol⁻¹\Lambda^0_m = 50.11 + 76.34 = 126.45 \text{ S cm² mol⁻¹}Λm0=50.11+76.34=126.45 S cm² mol⁻¹

Thus, the limiting molar conductivity of NaCl is 126.45 S cm² mol⁻¹.

**Problem 2: Degree of Dissociation**

**Question**: The molar conductivity of acetic acid at a concentration of 0.01 M is 4 S cm² mol⁻¹, and its limiting molar conductivity is 390.7 S cm² mol⁻¹. Calculate the degree of dissociation of acetic acid.

**Solution**: The degree of dissociation (**α**) is given by:

α=ΛmΛm0\alpha = \frac{\Lambda_m}{\Lambda^0_m}α=Λm0Λm

Substitute the values:

α=4390.7=0.0102\alpha = \frac{4}{390.7} = 0.0102α=390.74=0.0102

Thus, the degree of dissociation of acetic acid is 0.0102, or 1.02%.

**Problem 3: Solubility of a Sparingly Soluble Salt**

**Question**: The limiting molar conductivity of Ag⁺ is 61.92 S cm² mol⁻¹, and that of Cl⁻ is 76.34 S cm² mol⁻¹. If the molar conductivity of a saturated solution of AgCl is found to be 1.225 S cm² mol⁻¹, calculate the solubility of AgCl.

**Solution**: The limiting molar conductivity of AgCl is:

Λm0=λ+0+λ−0\Lambda^0_m = \lambda^0_+ + \lambda^0_-Λm0=λ+0+λ−0 Λm0=61.92+76.34=138.26 S cm² mol⁻¹\Lambda^0_m = 61.92 + 76.34 = 138.26 \text{ S cm² mol⁻¹}Λm0=61.92+76.34=138.26 S cm² mol⁻¹

The solubility (**S**) of AgCl can be found using the relation:

S=ΛmΛm0S = \frac{\Lambda_m}{\Lambda^0_m}S=Λm0Λm

Substitute the values:

S=1.225138.26=8.86×10−3 mol/LS = \frac{1.225}{138.26} = 8.86 \times 10^{-3} \text{ mol/L}S=138.261.225=8.86×10−3 mol/L

Thus, the solubility of AgCl is 8.86×10−3 mol/L8.86 \times 10^{-3} \text{ mol/L}8.86×10−3 mol/L.

Conclusion

Kohlrausch’s Law is an indispensable concept in the study of **electrolytic conductance**, providing a deeper understanding of how individual ions contribute to the total conductivity of an electrolyte. Through the formula, derivation, and practice problems, it becomes evident that this law is crucial for calculating limiting molar conductivity, analyzing electrolyte behaviour, and determining solubility. By mastering Kohlrausch’s Law, students gain a valuable tool for tackling various electrochemical problems in both theoretical and practical settings.