Ionic Mobility

Picture an ion of charge $ze$ sitting in a solvent under a uniform electric field $E$. The field exerts a constant force $zeE$ on it. If this were the only force, the ion would accelerate forever, but the surrounding solvent molecules resist its motion with a frictional drag force that grows with speed. Treating the ion as a sphere of radius $r$ moving through a continuum fluid of viscosity $\eta$, Stokes' law gives the drag force as $6\pi\eta r v$, proportional to velocity $v$. These two forces oppose each other. As the ion speeds up, drag grows until it exactly cancels the electrical force. That happens almost instantaneously because ionic masses are small and solvent friction is large; the relaxation time is of order $10^{-13}$ s, utterly negligible compared to any timescale we can measure electrically. From that point onward the ion moves at a constant terminal velocity, the drift velocity $v_{\text{drift}}$, superimposed on its random thermal jostling. It is this net drift, not the much faster thermal motion, that carries charge in one direction and constitutes an electric current. Because $v_{\text{drift}}$ is proportional to $E$ over the whole range of fields normally used in the laboratory, we define the ionic mobility $u$ as the proportionality constant: drift velocity per unit field. Mobility is an intrinsic property of the ion-solvent pair, independent of field strength (until fields become extremely large, a regime beyond ordinary conductivity cells). It depends on the ion's effective (hydrodynamic) radius, its charge, and the solvent viscosity, all of which is packed into the Stokes-law force balance. The immediate payoff is a bridge to conductivity. A solution's molar ionic conductivity $\lambda$ describes how well a mole of that ion, per unit concentration, carries current. Since current is charge times drift velocity times ion density, and mobility is drift velocity per field, $\lambda$ must be directly proportional to $u$. This is the link that turns a measurable bulk quantity, conductivity, into a measurable microscopic one, mobility, and vice versa.

where $v_{\text{drift}}$ is the terminal ion speed, $u$ the ionic mobility, $E$ the field strength.

where $z$ is the ion's charge number, $e$ the elementary charge, $\eta$ solvent viscosity, $r$ the effective hydrodynamic radius.

where $\lambda$ is the molar ionic conductivity, $z$ the (unsigned) charge number, $F$ the Faraday constant, $u$ the mobility.

where $\Lambda_m^{\circ}$ is the limiting molar conductivity of the electrolyte, restated from Kohlrausch's law of independent migration of ions.

Example 1 — Foundational: mobility of K⁺ from its known molar conductivity. The limiting molar ionic conductivity of K⁺ in water at 298 K is $\lambda_{\mathrm{K^+}} = 7.35\times 10^{-3}\ \text{S m}^2\ \text{mol}^{-1}$. Find its ionic mobility, and then find its drift velocity in a field of $100\ \text{V m}^{-1}$ (a modest lab field across a few centimetres of solution at a few volts). Rearrange $\lambda = zFu$ with $z=1$, $F = 96485\ \text{C mol}^{-1}$:

This matches the standard tabulated value for K⁺, a good consistency check. Now the drift velocity at $E = 100\ \text{V m}^{-1}$:

That is about 7.6 μm per second — glacially slow compared to the ion's thermal speed (hundreds of m/s), confirming that conduction is a gentle drift riding atop violent random motion. Sanity check: units of $u$ (m² V⁻¹ s⁻¹) times $E$ (V m⁻¹) correctly give m s⁻¹.

Example 2 — Applied: comparing H⁺ and Na⁺ mobilities, and estimating an effective ionic radius. At 298 K, the limiting molar ionic conductivities are $\lambda_{\mathrm{H^+}} = 3.498\times10^{-2}\ \text{S m}^2\text{mol}^{-1}$ and $\lambda_{\mathrm{Na^+}} = 5.01\times10^{-3}\ \text{S m}^2\text{mol}^{-1}$. Water's viscosity at 298 K is $\eta = 8.90\times10^{-4}\ \text{Pa s}$.

(a) Mobilities.

H⁺ is roughly 7 times more mobile than Na⁺, despite being far smaller and, naively, expected to be dragged more by its hydration shell, not less. This anomaly is the classic signature of the Grotthuss mechanism: protons do not physically wade through the solvent but hop along hydrogen-bonded water chains, structurally relaying charge rather than diffusing bodily. (OH⁻ shows a similar, smaller anomaly for the same reason.)

(b) Using Na⁺ to estimate an effective hydrodynamic radius via Stokes' law. Rearranging $u = ze/(6\pi\eta r)$ for $r$, with $z=1$ and $e = 1.602\times10^{-19}\ \text{C}$:

Compute the denominator: $6\pi = 18.85$; $18.85 \times 8.90\times10^{-4} = 1.678\times10^{-2}$; then $1.678\times10^{-2}\times 5.19\times10^{-8} = 8.71\times10^{-10}$.

This "Stokes radius" (184 pm) is noticeably larger than the bare crystallographic Na⁺ radius (about 102 pm), because Na⁺ drags a shell of tightly bound water molecules through the solvent — exactly what conductivity measurements are sensitive to, and exactly why applying naive Stokes' law to H⁺ would fail: proton transport isn't hydrodynamic at all.

Friedrich Kohlrausch, whose independent-migration law is the natural prerequisite to this topic, was the first to systematically measure limiting conductivities precisely enough to extract individual ionic contributions. But the person most associated with connecting mobility to a molecular force balance is George Gabriel Stokes, whose 1851 hydrodynamic drag law, originally derived for falling spheres in viscous fluids, turned out to be exactly the missing piece: it lets chemists convert a measured mobility into an effective ionic radius, and vice versa, giving conductivity data real structural meaning.

Ask the tutor: How does the Debye–Hückel–Onsager equation modify ionic mobility as concentration rises, and why do both the electrophoretic and relaxation effects scale with the square root of concentration?