Molar Conductivity

The molar conductivity (Λ_m) of an electrolyte solution is defined as the conductivity (κ) of the solution divided by the molar concentration (c) of the electrolyte.

1.0What Is Molar Conductivity?

Molar conductivity is the property of conductance of a solution containing an electrolyte mole or is a function of the ionic resistance of a solution or concentration of salt. 

Molar conductivity (λ) is a measure of the ability of an electrolytic solution to conduct electricity, determined by the concentration of ions formed when the electrolyte dissolves in the solution. 

To find the molar conductivity, we measure the conductivity of the solution using a conductivity meter and divide it by the concentration of the substance in the solution.    This gives us

λ = κ / c

where κ is the conductivity of the solution in Siemens per centimeter (S/cm), and c is the concentration of the solute in moles per liter (mol/L).

So, molar conductivity shows us how effectively a substance can carry an electrical current when dissolved in water. It depends on the substance itself and how much is dissolved in the water. This property varies with the electrolyte concentration and is not a constant value.

2.0Conductivity and Concentration

Conductivity measures how well a solution conducts electricity. When you dilute a solution by adding more solvent (like water), the concentration of ions (charged particles) per unit volume decreases. Since these ions carry electrical current in a solution, lowering their concentration reduces the solution's ability to conduct electricity. Therefore, conductivity continuously decreases as you dilute a solution, whether a weak or strong electrolyte.

Molar Conductivity and Concentration

Molar conductivity (Λm) refers to the conductivity of a solution containing one mole of electrolyte in a given volume. As you dilute a solution, you increase the total volume of the solution containing that one mole of electrolyte; despite the decrease in conductivity with dilution, molar conductivity increases. This is because the increase in volume (containing the same amount of electrolyte) compensates for the decrease in conductivity.

Molar conductivity (Λm) is the conductance of a solution containing one mole of electrolyte, positioned between two electrodes with a cross-sectional area A and unit distance l 

Given that l = 1 , A=V (the volume containing 1 mole of electrolyte):

Λ_m = κV 

where κ  is the conductivity of the solution.

Molar conductivity increases as concentration decreases because the volume V containing one mole of electrolyte increases. Although κ  decreases with dilution, the increase in volume compensates for this, resulting in an overall increase in Λ_m ​.

Physically, this means that at any concentration, Λ_m​ can be viewed as the conductance of the solution between electrodes spaced unit distance apart, with a cross-sectional area large enough to contain one mole of electrolyte. As concentration approaches zero, molar conductivity reaches a value known as limiting molar conductivity, Λ^°_m​.

The variation of Λ_m  with concentration differs for strong and weak electrolytes.

Molar Conductivity ( Λ_m) for Strong Electrolytes

Equation:  Λ_m = Λ^°_m - A√c  describes how molar conductivity changes with concentration for strong electrolytes, crucial for conductivity predictions 

• Λ_m: Molar conductivity at concentration c.
• Λ^°_m: Limiting molar conductivity at infinite dilution.
• A: Constant specific to the type of electrolyte.
• c: Concentration of the electrolyte.
• Λ_m: increases slowly as concentration decreases.

Significance of A:

• A depends on ion charges upon electrolyte dissociation.
• Each type (1-1, 2-1, 2-2) has a unique A.

1-1 Electrolytes: These have ions with single charges, such as Na⁺ and Cl⁻.

2-1 Electrolytes: These have a doubly charged ion and a singly charged ion, such as Mg²⁺ and Cl⁻.

2-2 Electrolytes: These have ions where both the cation and the anion are doubly charged, such as Mg²⁺ and SO₄²⁻.

A is constant for each type across solvents/temperature

Molar Conductivity ( Λ_m)  for Weak Electrolytes

Dissociation increases as the solution is diluted for weak electrolytes like acetic acid.  This leads to a higher molar conductivity ( Λ_m)  at lower concentrations due to more ions being present in the solution containing 1 mole of electrolyte

Unlike strong electrolytes, the limiting molar conductivity ( Λ^∘_m) for weak electrolytes cannot be accurately extrapolated from measurements at zero concentration because the solution's conductivity is too low at infinite dilution to be measured directly.

Instead,  Λ^∘_m for weak electrolytes is determined using Kohlrausch's law of independent migration of ions. At any concentration c, the degree of dissociation α is approximated by the ratio of Λm​ to Λ^∘_m.

α = Λ ​/ Λ^∘_m.

Thus, the molar conductivity Λm ​ can be expressed as:

Λm = α /Λ^∘_m.

This relationship highlights how Λmchanges with concentration for weak electrolytes, depending on the degree of dissociation α and the limiting molar conductivity Λ^∘_m

3.0Limiting molar conductivity and Kohlrausch's law

Limiting Molar Conductivity:

Limiting molar conductivity (Λ°m) refers to the molar conductivity of an electrolyte when its concentration approaches zero, or equivalently, at infinite dilution. At this state, the ions in the solution are fully dissociated and there are minimal ion-ion interactions.

Kohlrausch's Law:

Kohlrausch's law of independent ion migration states that the limiting molar conductivity of an electrolyte can be calculated by adding the contributions from its individual ions—namely, the cation and the anion.

Kohlrausch's law provides a method to calculate the limiting molar conductivity of strong electrolytes like sodium chloride by summing the contributions from their constituent ions. This law highlights how each ion's ability to conduct electricity independently contributes to the overall conductivity of the solution at extremely dilute concentrations, where ion interactions are negligible.

The Kohlrausch law of independent migration of ions is a fundamental tool in both theoretical and practical aspects of electrolyte behavior. It enables the calculation of  Λ^∘_m. and aids in determining dissociation constants for weak electrolytes.

Application to Sodium Chloride:

For example, let's apply Kohlrausch's law to sodium chloride (NaCl), a strong electrolyte that dissociates completely into sodium ions (Na⁺) and chloride ions (Cl^-) in water.

According to Kohlrausch's law: Λ^∘_{m, NaCl}= Λ^∘_{m, Na⁺} + Λ^∘_{m, Cl⁻}

 where:

Λ^∘_{m, Na⁺} is the limiting molar conductivity of sodium ions.

Λ^∘_{m, Cl⁻}  is the limiting molar conductivity of chloride ions.

Specific Conductivities:

Λ^∘_m,, _Na⁺  and Λ^∘_{m, Cl⁻}  are specific conductivities of Na⁺ and Cl⁻ ions, respectively, under conditions of infinite dilution. These values are determined experimentally.

Experimental Determination:

The specific conductivities of each ion in water at very low concentrations are obtained from experimental measurements or referenced in the scientific literature.

4.0Applications of Molar Conductivity

Molar conductivity has several important applications in chemical and electrochemical fields:

• Determination of Dissociation Behavior: It helps determine the degree of dissociation of electrolytes in solution by measuring molar conductivity across various concentrations.
• Electrochemical Studies: It is crucial in electrochemical studies, including kinetics, analytical techniques, and corrosion analysis.
• Industrial Applications: It is used in designing electrolytic cells for metal deposition, electroplating, and refining processes.
• Pharmaceutical Industry: Molar conductivity is utilized to study drug behavior and interactions in solutions. It aids in drug design and formulation by determining solubility, stability, and absorption properties.
• Environmental Monitoring: It measures the electrical conductance of water samples from rivers, lakes, and other bodies. This helps evaluate water quality and determine the concentration of dissolved salts, pollutants, and contaminants.

5.0Solved Examples on Molar Conductivity

Problem: The molar conductivity at infinite dilution Λ^∘_m of acetic acid is 390.7 S cm²/mol. The molar conductivity of a 0.01 M acetic acid solution is measured to be 4.6 S cm²/mol. Calculate the degree of dissociation (α).

Solution:

 α = Λ ​/ Λ^∘_m.

 α = Λ ​/ Λ^∘_m. =  4.6 S cm²/mol / 390.7 S  cm²/mol.

 α ≈ 0.0118

 α ≈1.18%

Problem: The conductivity (κ) of a 0.20 M KCl solution at 298 K is 0.0248 S cm⁻¹. Calculate its molar conductivity (Λ_m​).

Solution: Given:

• Conductivity, κ=0.0248 S cm⁻¹
• Concentration, M=0.20_M

The formula for molar conductivity (Λ⁰_m) is: Λm = κ×1000 / M

Substituting  the given values: 

Λ_m = 0.0248×1000 / 0.20

Calculate: Λ_m  =  24.80 / 0.20

 Λ_m  = 124 S cm²/mol

Therefore, the molar conductivity of the 0.20 M KCl solution at 298 K is 124 S cm²/mol..

Problem: Calculate the limiting molar conductivity (Λ⁰_m​) of NaCl if the limiting molar conductivities of Na⁺and Cl⁻ ions are 50.1 and 76.3 S cm²/mol, respectively.

Solution:

Λ⁰_m(NaCl) = Λ⁰_m(Na+)+Λ⁰_m(Cl−)

Λ⁰_m(NaCl) = 50.1 S cm²/mol +76.3 S cm²/mol

Λ⁰_m(NaCl) = 126.4 S cm²/mol