Page 27 - MODERN ELECTROCHEMISTRY

xxvi CONTENTS

4.4. A Simple Atomistic Picture of Ionic Migration ........... 442
4.4.1. Ionic Movements under the Influence of an Applied Electric Field .... 442
4.4.2. Average Value of the Drift Velocity .................... 443
4.4.3. Mobility of Ions............................... 444
4.4.4. Current Density Associated with the Directed Movement of Ions in Solution,
in Terms of Ionic Drift Velocities ..................... 446
4.4.5. Specific and Equivalent Conductivities in Terms of Ionic Mobilities . . . 447
4.4.6. The Einstein Relation between the Absolute Mobility and the Diffusion
Coefficient ................................. 448
4.4.7. Drag (or Viscous) Force Acting on an Ion in Solution ........... 452
4.4.8. The Stokes–Einstein Relation ....................... 454
4.4.9. The Nernst–Einstein Equation ....................... 456
4.4.10. Some Limitations of the Nernst–Einstein Relation ............ 457
4.4.11. The Apparent Ionic Charge ......................... 459
4.4.12. A Very Approximate Relation between Equivalent Conductivity and
Viscosity: Walden’s Rule .......................... 461
4.4.13. The Rate-Process Approach to Ionic Migration .............. 464
4.4.14. The Rate-Process Expression for Equivalent Conductivity ........ 467
4.4.15. The Total Driving Force for Ionic Transport: The Gradient of the Electro-
chemical Potential ............................. 471
Further Reading .................................... 476
4.5. The Interdependence of Ionic Drifts ................. 476
4.5.1. The Drift of One Ionic Species May Influence the Drift of Another . . . 476
4.5.2. A Consequence of the Unequal Mobilities of Cations and Anions, the
Transport Numbers ............................. 477
4.5.3. The Significance of a Transport Number of Zero ............. 480
4.5.4. The Diffusion Potential, Another Consequence of the Unequal Mobilities
of Ions .................................... 483
4.5.5. Electroneutrality Coupling between the Drifts of Different Ionic Species 487
4.5.6. How to Determine Transport Number ................... 488
4.5.6.1. Approximate Method for Sufficiently Dilute Solutions ...... 488
4.5.6.2. Hittorf’s Method.................................. 489
4.5.6.3. Oliver Lodge’s Experiment. . . . . . . . . . . . . . ....... 493
4.5.7. The Onsager Phenomenological Equations ................ 494
4.5.8. An Expression for the Diffusion Potential ................. 496
4.5.9. The Integration of the Differential Equation for Diffusion Potentials: The
Planck–Henderson Equation ........................ 500
4.5.10. A Bird’s Eye View of Ionic Transport ................... 503
Further Reading .................................... 505
4.6. Influence of Ionic Atmospheres on Ionic Migration ........ 505
4.6.1. Concentration Dependence of the Mobility of Ions ............ 505
4.6.2. Ionic Clouds Attempt to Catch Up with Moving Ions ........... 507
4.6.3. An Egg-Shaped Ionic Cloud and the “Portable” Field on the Central Ion . 508