Kohlrausch’s Law

1.0Introduction

Kohlrausch’s Law, formulated by Friedrich Kohlrausch in the late 19th century, explains how the limiting molar conductivity of an electrolyte at infinite dilution can be calculated by adding the individual contributions of its cation and anion.

This law is crucial in:

• Calculating limiting molar conductivity (Λ⁰)
• Determining the degree of dissociation (α) and the dissociation constant (Kₐ) of weak electrolytes
• Estimating the solubility of sparingly soluble salts

2.0Statement of Kohlrausch’s Law

At infinite dilution, the limiting molar conductivity of an electrolyte is the sum of the limiting molar conductivities of its constituent ions.

${\Lambda }_{\text{electrolyte}}^{\circ }={\Lambda }_{\text{cation}}^{\circ }+{\Lambda }_{\text{anion}}^{\circ }$

Example

For sodium chloride (NaCl):

${\Lambda }_{\text{NaCl}}^{\circ }={\Lambda }_{{\text{Na}}^{+}}^{\circ }+{\Lambda }_{{\text{Cl}}^{-}}^{\circ }$

At infinite dilution, both ions move independently and contribute separately to the total conductivity.

Molar Conductivity

Molar conductivity (Λₘ) is a measure of how well an electrolyte conducts electricity in solution per mole of solute. It represents the conducting power of all the ions produced by one mole of the electrolyte in a given volume of solution.

• Symbol: Λₘ
• Unit: Siemens meter² per mole (S·m²·mol⁻¹)

Molar conductivity is defined as the conductivity (κ) of a solution divided by the molar concentration (c) of the electrolyte:

${\Lambda }_m=\frac{\kappa }{c}$

Behavior with Dilution

As the concentration of the solution decreases, molar conductivity increases. This is because:

• The ions experience less inter-ionic attraction,
• They move more freely through the solvent,
• There is greater ionic mobility in more dilute solutions.

Limiting Molar Conductivity (Λ⁰ₘ)

At infinite dilution, where the electrolyte is completely dissociated and ion interactions are negligible, the molar conductivity reaches a maximum value known as the limiting molar conductivity, denoted by Λ⁰ₘ.

This value is crucial in:

• Applying Kohlrausch's Law,
• Determining the degree of dissociation and dissociation constant of weak electrolytes.

3.0Formula of Kohlrausch’s Law

Kohlrausch’s Law, also called the Law of Independent Migration of Ions, states that:

At infinite dilution, when the electrolyte is completely dissociated, the molar conductivity of an electrolyte is equal to the sum of the individual contributions of its cation and anion.

It can be mathematically expressed as:

$\Lambda eq\circ =\lambda cation\circ +\lambda anion\circ {\Lambda }_{\text{eq}}^{\circ }={\lambda }_{\text{cation}}^{\circ }+{\lambda }_{\text{anion}}^{\circ }\Lambda eq\circ =\lambda cation\circ +\lambda anion\circ$

Where:

• Λ°eq = Molar conductivity at infinite dilution
• λ°cation = Limiting molar conductivity of the cation
• λ°anion = Limiting molar conductivity of the anion

4.0Graphical Representation of Kohlrausch’s Law

Kohlrausch’s Law can be illustrated by plotting molar conductivity (Λₘ) of an electrolyte against the square root of its concentration (√c).

• This plot typically yields a straight line.
• The intercept of the line on the y-axis gives the limiting molar conductivity (Λ⁰ₘ).
• The slope of the line is negative and denoted by –A, where A is a constant that depends on:
• • the nature of the electrolyte
  • the temperature of the solution

This graphical method is especially helpful in determining Λ⁰ₘ for weak electrolytes, as they do not fully dissociate even at low concentrations. Therefore, their limiting molar conductivity cannot be measured directly and must be extrapolated using data from strong electrolytes.

Kohlrausch’s Law Graph Behavior

• Strong Electrolytes: Show a near-linear decrease in Λₘ with increasing √c due to ion-ion interactions.
• Weak Electrolytes: Exhibit a sharp rise in Λₘ at lower concentrations because ionization increases significantly with dilution.

Hence, direct extrapolation for weak electrolytes is not reliable. Instead, Kohlrausch’s Law is used to compute Λ⁰ₘ indirectly.

Calculation of Degree of Dissociation (α)

For weak electrolytes, once Λ and Λ⁰ are known, the degree of dissociation (α) can be calculated using:

$\alpha =\Lambda \circ \alpha =\frac{\Lambda }{{\Lambda }^{\circ }}$

Where:

• α = Degree of dissociation
• Λ = Molar conductivity at a given concentration
• Λ⁰ = Limiting molar conductivity

5.0Applications of Kohlrausch’s Law

Kohlrausch’s Law has several important applications in electrochemistry, particularly in understanding the behavior of electrolytes in dilute solutions. Here are some of its key uses:

• Calculation of the Dissociation Constant (Kₐ) of Weak Electrolytes

Kohlrausch’s Law helps determine the dissociation constant by using the degree of dissociation (α) and limiting molar conductivity (Λ⁰). This is crucial for analyzing the strength and ionization extent of weak acids and bases.

•  Estimation of Limiting Molar Conductivity (Λ⁰)

For weak electrolytes, direct measurement of limiting molar conductivity is not feasible. Using Kohlrausch’s Law, Λ⁰ can be calculated by summing the individual contributions of the ions at infinite dilution.

•  Determination of the Degree of Dissociation (α)

By comparing the molar conductivity (Λ) at a given concentration with Λ⁰, the degree of dissociation (α) can be calculated using the formula:

$\alpha =\frac{\Lambda }{{\Lambda }^{\circ }}$

This helps quantify how much of the electrolyte has dissociated into ions in solution.

• Calculation of Solubility of Sparingly Soluble Salts

Kohlrausch’s Law is also useful for determining the solubility of salts like AgCl or BaSO₄. Once the limiting molar conductivity is known, the solubility can be back-calculated using conductivity data.

• Determination of Transport Numbers of Ions

The law aids in calculating the transport number (or transference number) of ions—i.e., the fraction of total current carried by each ion in an electrolyte. This is vital in understanding current flow and ion movement in electrochemical cells.

6. Assessment of Ionic Strength

Kohlrausch’s Law can be indirectly used to evaluate the ionic strength of a solution—a factor affecting activity coefficients and electrochemical equilibria.

6.0Solved Example 

1. Calculate the molar conductance of 0.02 M solution of an electrolyte which has a resistance of 310 ohm at 298 K. Cell constant is 0.68 cm-1.

Solution

To calculate the molar conductance (Λₘ) of a solution, we use the following steps

Given:

• Concentration (c) = 0.02 M
• Resistance (R) = 310 ohm
• Cell constant (l/A) = 0.68 cm⁻¹
• Temperature = 298 K

Calculate Conductivity (κ)

κ = Cell constant​ R

κ = 0.68t​310   ==0.0021935  S cm−1

Calculate Molar Conductivity (Λₘ)

Rounding to a reasonable number of significant figures 

The molar conductance of the solution is approximately 109.7 S cm² mol⁻¹

2.Determine the limiting molar conductivity of sodium sulfate (Na₂SO₄), given that the limiting molar conductivity of Na⁺ is 50.1 S·cm²/mol and that of ½ SO₄²⁻ is 80.0 S·cm²/mol.

Solution

Given:

$$\begin{gathered} {\lambda }_{{\text{Na}}^{+}}^{\circ }=50.1S\cdot c{m}^2\text{mol} \\ \frac{1}{2}{\lambda }_{{\text{SO}}_4^{2-}}^{\circ }=80.0S\cdot c{m}^2\text{mol} \end{gathered}$$

Using Kohlrausch’s Law:

$l{\lambda }_{{\text{Na}}_2{\text{SO}}_4}^{\circ }=2{\lambda }_{{\text{Na}}^{+}}^{\circ }+{\lambda }_S^{\circ }{O}_4^{2-}=2(50.1)+2(80.0)=260.2S\cdot c{m}^2\text{mol}$

Answer:

$l\boxed{260.2S\cdot c{m}^2\text{mol}}$