The conductivity of an electrolytic solution is measured with a conductivity cell, which has two electrodes a fixed distance l apart, each with an area of cross-section A. The resistance R of the cell is given by R = ρl/A = l/(κA), where ρ is the resistivity and κ the conductivity of the solution.
The quantity l/A is constant for a given cell; it is denoted by G* and is called the cell constant. It is found by measuring the resistance of the cell filled with a KCl solution whose conductivity is known accurately at various concentrations: G* = l/A = Rκ.
The molar conductivity of a solution, Λm, is its conductivity divided by its molar concentration (Λm = κ/c). Its SI unit is S m2 mol−1 when c is expressed in mol m−3. Conductivity always decreases with decrease in concentration, for both strong and weak electrolytes, because the number of ions per unit volume that carry the current in the solution decreases.
A conductivity cell filled with 1.0 mol L−1 KCl solution, whose conductivity is 11.13 S m−1, has a resistance of 10 Ω at 298 K. The same cell filled with a 0.005 mol L−1 solution of an electrolyte X has a resistance of 1113 Ω at 298 K. What is the molar conductivity of this solution of X?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2.0 × 10−2 S m2 mol−1
The cell constant G* is found from the KCl solution of known conductivity, G* = Rκ. The same cell then gives the conductivity of solution X from its resistance, κ = G*/R. The molar conductivity is Λm = κ/c, with c in mol m−3 when κ is in S m−1.
Cell constant: \( G^* = R\kappa = 10\ \Omega \times 11.13\ \mathrm{S\,m^{-1}} = 111.3\ \mathrm{m^{-1}} \)
Conductivity of X: \( \displaystyle \kappa = \frac{G^*}{R} = \frac{111.3\ \mathrm{m^{-1}}}{1113\ \Omega} = 0.100\ \mathrm{S\,m^{-1}} \)
Concentration: \( c = 0.005\ \mathrm{mol\,L^{-1}} = 0.005 \times 1000\ \mathrm{mol\,m^{-3}} = 5\ \mathrm{mol\,m^{-3}} \)
Molar conductivity: \( \displaystyle \Lambda_m = \frac{\kappa}{c} = \frac{0.100\ \mathrm{S\,m^{-1}}}{5\ \mathrm{mol\,m^{-3}}} = 2.0 \times 10^{-2}\ \mathrm{S\,m^2\,mol^{-1}} \)
This is the same as 200 S cm2 mol−1, since 1 S m2 mol−1 = 104 S cm2 mol−1.