Question

The resistance of a column formed by a \(0.10\ \mathrm{mol\ L^{-1}}\) concentrated solution is \(6.5\times10^{3}\ \Omega\). Its diameter is \(1\ \text{cm}\) and length is \(50\ \text{cm}\). Calculate its resistivity, conductivity and molar conductivity.

Hint:

When calculating resistance, conductivity, and molar conductivity, always remember the relationship between them. Molar conductivity depends on both the concentration and conductivity of the solution.

Answer 1

The formula for resistance (R) of a solution is given by:

R = ρL / A

Where:

• R is the resistance,
• ρ is the resistivity,
• L is the length of the column,
• A is the cross-sectional area of the column.

We are given:

• Resistance R = 6.5 × 10³ Ω,
• Length L = 50 cm = 0.50 m,
• Diameter d = 1 cm = 0.01 m.

Step 1: Calculate the cross-sectional area.

The cross-sectional area A of the column is given by the area of a circle:

A = π (d/2)² = π (0.01/2)² = 7.854 × 10^-5 m²

Step 2: Calculate the resistivity.

Rearranging the formula for resistance, we can solve for resistivity:

ρ = (R × A) / L = (6.5 × 10³ × 7.854 × 10^-5) / 0.50 = 1.02 Ω·m

Thus, the resistivity is ρ = 1.02 Ω·m.

Step 3: Calculate the conductivity.

Conductivity σ is the reciprocal of resistivity:

σ = 1 / ρ = 1 / 1.02 = 0.980 S/m

Thus, the conductivity is σ = 0.980 S/m.

Step 4: Calculate the molar conductivity.

Molar conductivity Λ_m is given by:

Λ_m = κ / C

Where:

• κ is the conductivity,
• C is the concentration of the solution in mol·L⁻¹.

Given C = 0.10 mol·L⁻¹, we can calculate the molar conductivity:

Λ_m = 0.980 / 0.10 = 9.80 S·m²/mol

Thus, the molar conductivity is Λ_m = 9.80 S·m²/mol.