Question

The column of a solution of 0.05 mol L$^{-1}$ NaOH has diameter 2.0 cm and length 100 cm. The resistance of the column of solution is 5.55 $\times$ 10$^3$ ohm. Calculate its resistivity, conductivity and molar conductivity.

Hint:

Resistivity is inversely related to conductivity. Molar conductivity is directly related to the concentration of the solution.

Answer 1

Step 1: Formula for Resistivity.
The resistivity \( \rho \) of a solution is given by: \[ \rho = R \times \frac{A}{L} \] where: - \( R \) is the resistance of the solution (5.55 × 10$^3$ ohms), - \( A \) is the cross-sectional area of the column, - \( L \) is the length of the column. Step 2: Calculate the Cross-Sectional Area.
The area of the column is given by the formula for the area of a circle: \[ A = \pi r^2 \] where \( r \) is the radius of the column. The diameter is 2.0 cm, so the radius \( r = 1.0 \, \text{cm} = 0.01 \, \text{m} \). Thus, \[ A = \pi \times (0.01)^2 = 3.14 \times 10^{-4} \, \text{m}^2 \] Step 3: Calculate Resistivity.
Now, substitute the values into the resistivity formula: \[ \rho = (5.55 \times 10^3) \times \frac{3.14 \times 10^{-4}}{1.0} = 1.74 \, \Omega \, \text{m} \] Step 4: Calculate Conductivity.
Conductivity \( \kappa \) is the reciprocal of resistivity: \[ \kappa = \frac{1}{\rho} = \frac{1}{1.74} = 0.574 \, \text{S/m} \] Step 5: Calculate Molar Conductivity.
Molar conductivity \( \Lambda_m \) is given by: \[ \Lambda_m = \kappa \times \frac{1000}{C} \] where \( C \) is the molar concentration of NaOH (0.05 mol L$^{-1}$). Substituting the values: \[ \Lambda_m = 0.574 \times \frac{1000}{0.05} = 11.48 \, \text{S} \, \text{m}^2 \, \text{mol}^{-1} \]