Show from combination of the equation for current and the flux expression that the conductivity of an electrolyte solution is given by

\[\kappa=\frac{F^{2}}{R_{\mathrm{G}} T} \sum_{i}^{N} D_{i} z_{i}^{2} C_{i}\]

when there are no concentration gradients in the solution.

Calculate the value for \(0.1-\mathrm{M}\) and \(1-\mathrm{M}\) solutions of \(\mathrm{NaCl}\).

Also show that the divergence of the current is equal to zero. Hence show that the Laplace equation holds for the potential field in such cases with no concentration gradients.

Estimate the electrical conductivity of pure water.