Ionic Mobility / Conductivity
Background
Electrical conductivity in electrolytes is controlled by how many charged species are present and how quickly these ions move in an electric field.
\[\kappa = F\cdot 1000\cdot c\left(\nu_+|z_+|u_+ + \nu_-|z_-|u_-\right)\]
This relation combines concentration, charge number, and ionic mobility into a practical prediction for \(\kappa\) in \(\mathrm{S/m}\).
- higher mobility \(u\) \(\Rightarrow\) higher conductivity
- more ions per formula unit \(\nu\) \(\Rightarrow\) stronger contribution
- higher charge number \(|z|\) \(\Rightarrow\) larger transport effect per ion
Crosslink: For activity corrections, see Activity Coefficient (γ).
Formulas (MathJax)
\[\kappa = F\cdot 1000\cdot c\left(\nu_+|z_+|u_+ + \nu_-|z_-|u_-\right)\]
\[u_+ = \frac{\frac{\kappa}{F\cdot 1000\cdot c} - \nu_-|z_-|u_-}{\nu_+|z_+|}\]
\[t_+ = \frac{\nu_+|z_+|u_+}{\nu_+|z_+|u_+ + \nu_-|z_-|u_-},\quad t_- = 1-t_+\]
Formula symbol legend
- \(\kappa\): conductivity of the solution [S/m]
- \(F\): Faraday constant \(96485\,\mathrm{C/mol}\)
- \(c\): concentration [mol/L]
- \(\nu_+,\nu_-\): ionic stoichiometric factors
- \(z_+,z_-\): ionic charge numbers
- \(u_+,u_-\): ionic mobilities [m²/(V·s)]
- \(t_+,t_-\): relative conductivity shares
Detailed examples
Example 1 (1:1 electrolyte, \(\kappa\) prediction): \(c=0.010\,\mathrm{mol/L}\), \(u_+=5.2\cdot10^{-8}\), \(u_-=7.9\cdot10^{-8}\,\mathrm{m^2/(V\cdot s)}\), \(\nu_+=\nu_-=|z_+|=|z_-|=1\). Then \(\kappa\approx0.126\,\mathrm{S/m}\).
Example 2 (mobility back-calculation): Measured \(\kappa=0.126\,\mathrm{S/m}\), known \(u_-\) and composition as above. Back-calculation gives \(u_+\approx5.2\cdot10^{-8}\,\mathrm{m^2/(V\cdot s)}\).
Example 3 (contribution analysis): Using the same data gives about \(t_+\approx39.7\%\) and \(t_-\approx60.3\%\), showing the more mobile anion contributes more strongly.
Practical value: Coupling mobility and conductivity helps in sensors, quality control, process monitoring (e.g., rinse water, plating baths, electrolytes), and plausibility checks for conductivity measurements.
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