Question

The resistance of a column formed by a 0.10 mol L^{-1} concentrated solution is 6.5 × 10^3 ohm. Its diameter is 1 cm and length is 50 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: \[ R = \frac{\rho L}{A}
\] Where: - \( R \) is the resistance,
- \( \rho \) is the resistivity,
- \( L \) is the length of the column,
- \( A \) is the cross-sectional area of the column.
We are given: - Resistance \( R = 6.5 \times 10^3 \, \Omega \),
- Length \( L = 50 \, \text{cm} = 0.50 \, \text{m} \),
- Diameter \( d = 1 \, \text{cm} = 0.01 \, \text{m} \), Step 1: Calculate the cross-sectional area.
The cross-sectional area \( A \) of the column is given by the area of a circle: \[ A = \pi \left(\frac{d}{2}\right)^2 = \pi \left(\frac{0.01}{2}\right)^2 = 7.854 \times 10^{-5} \, \text{m}^2
\] Step 2: Calculate the resistivity.
Rearranging the formula for resistance, we can solve for resistivity: \[ \rho = \frac{R A}{L} = \frac{(6.5 \times 10^3) (7.854 \times 10^{-5})}{0.50} = 1.02 \, \Omega \, \text{m}
\] Thus, the resistivity is \( \rho = 1.02 \, \Omega \, \text{m} \). Step 3: Calculate the conductivity.
Conductivity \( \sigma \) is the reciprocal of resistivity: \[ \sigma = \frac{1}{\rho} = \frac{1}{1.02} = 0.980 \, \text{S/m}
\] Thus, the conductivity is \( \sigma = 0.980 \, \text{S/m} \). Step 4: Calculate the molar conductivity.
Molar conductivity \( \Lambda_m \) is given by: \[ \Lambda_m = \frac{\kappa}{C}
\] Where: - \( \kappa \) is the conductivity,
- \( C \) is the concentration of the solution in mol/L.
Given \( C = 0.10 \, \text{mol/L} \), we can calculate the molar conductivity: \[ \Lambda_m = \frac{0.980}{0.10} = 9.80 \, \text{S·m}^2/\text{mol}
\] Thus, the molar conductivity is \( \Lambda_m = 9.80 \, \text{S·m}^2/\text{mol} \).