The Debye-Hückel-Onsager Equation

Consider a single cation drifting under an applied electric field. Around it, Debye-Hückel theory says there is a diffuse "ionic atmosphere" — a spherically symmetric excess of anions — built up by electrostatic attraction and disrupted by thermal motion. At equilibrium (no field), this atmosphere is perfectly symmetric: it pulls equally in all directions, exerting no net force. Switch on the field. The cation moves right; the atmosphere, made of ions with their own finite mobility, cannot instantly reform around the new position. It remains slightly displaced behind the moving ion, like a wake. This asymmetric residual atmosphere pulls the central ion backward — this is the relaxation effect (sometimes called the "asymmetry effect"). The atmosphere needs a characteristic relaxation time $\tau$ to rebuild itself, and this time depends on the mobility of the surrounding ions and on the Debye screening length $1/\kappa$, which itself depends on concentration. There is a second, independent drag. The atmosphere, being charged oppositely to the central ion, migrates in the opposite direction under the same field. Since the atmosphere is dissolved in the same viscous solvent, its motion drags solvent molecules along with it — in the opposite sense to the central ion's motion. The central ion therefore has to swim "upstream" through solvent that the atmosphere is dragging backward. This is the electrophoretic effect, a hydrodynamic (Stokes-law type) drag rather than an electrostatic one. Both effects scale with the thickness of the ionic atmosphere, which is set by the Debye length $1/\kappa \propto 1/\sqrt{c}$. Diluting the solution thins the atmosphere, weakens both drags, and raises the conductivity — but the rate at which it does so, per unit $\sqrt{c}$, is a fixed, calculable constant: the Onsager slope.

where: $\Lambda_{\rm m}^{\circ}$ is the limiting molar conductivity at infinite dilution, $c$ is molar concentration, and $A$, $B$ are the Onsager coefficients (relaxation and electrophoretic, respectively).

where (simplified 1:1 case, $z_+ = |z_-| = z$, $q=1/2$): $\eta_0$ is solvent viscosity, $\varepsilon_r$ is relative permittivity, $F$ is the Faraday constant, and $\lambda_\pm^\circ$ are limiting ionic conductivities.

where: $B$ captures the electrophoretic (Stokes-drag) term, independent of the individual ionic mobilities.

where: $\kappa$ is the Debye screening parameter, $1/\kappa$ the ionic-atmosphere radius, $I$ the ionic strength.

where: these are the standard tabulated numerical values for a 1:1 electrolyte in water at 25 °C, absorbing $\eta_0 = 0.8909$ mPa s, $\varepsilon_r = 78.54$.

Example 1 — Predicting the conductivity of dilute KCl(aq). For KCl in water at 25 °C: $\Lambda_{\rm m}^{\circ} = 149.9\ \text{S cm}^2\text{mol}^{-1}$ (from $\lambda^\circ({\rm K^+}) = 73.5$, $\lambda^\circ({\rm Cl^-}) = 76.4\ \text{S cm}^2\text{mol}^{-1}$). Using $A = 0.2297$ and $B = 60.65$ (both in units of $(\text{mol}^{-1}\text{L})^{1/2}$, consistent with $\sqrt{c}$ in mol L$^{-1}$):

(Note: $B$ is quoted here as $60.65\times10^{-3}$ to give $\Lambda_{\rm m}$ directly in S cm² mol⁻¹ when $\Lambda_{\rm m}^\circ$ is in the same units — this is the standard convention in Atkins' tables.) At $c = 1.00\times10^{-3}\ \text{mol L}^{-1}$, $\sqrt{c} = 0.03162\ (\text{mol L}^{-1})^{1/2}$:

At $c = 1.00\times10^{-2}\ \text{mol L}^{-1}$, $\sqrt{c} = 0.1000$:

Sanity check: experimental values are $\Lambda_{\rm m}(10^{-3}\ {\rm M}) \approx 147.8$ and $\Lambda_{\rm m}(10^{-2}\ {\rm M}) \approx 141.3\ \text{S cm}^2\text{mol}^{-1}$. Agreement is excellent at $10^{-3}$ M (within 1%) but the theory over-predicts $\Lambda_{\rm m}$ at $10^{-2}$ M — exactly as expected, since the Onsager law is a limiting law valid only as $c\to 0$.

Example 2 — Extracting $\Lambda_{\rm m}^{\circ}$ from real conductivity data (harder, applied). Suppose conductivity measurements on aqueous HCl at 25 °C give:

We test the Onsager law and extrapolate to find $\Lambda_{\rm m}^{\circ}$. Compute $\sqrt{c}$: $0.02236$, $0.04472$, $0.07071$. Fit $\Lambda_{\rm m} = \Lambda_{\rm m}^{\circ} - S\sqrt{c}$ using the first and third points as a check (linear regression in practice):

Compare to the literature limiting value, $\Lambda_{\rm m}^{\circ}({\rm HCl}) = \lambda^\circ({\rm H^+}) + \lambda^\circ({\rm Cl^-}) = 349.6 + 76.4 = 426.0\ \text{S cm}^2\text{mol}^{-1}$. Our extrapolated slope-intercept value (427.5) is within about 0.3% — a good fit given only three data points and rounding in the table. Now check whether the fitted slope matches the Onsager prediction:

This predicted slope (≈ 26.1) is far smaller than the fitted experimental slope (≈ 101.3). This flags real, expected chemistry: H⁺ conducts anomalously fast via the Grotthuss (proton-hopping) mechanism, and HCl also shows measurable deviations from ideal limiting-law behavior even at millimolar concentrations because $\lambda^\circ({\rm H^+})$ is unusually large, inflating both the true relaxation drag and higher-order (non-limiting) terms not captured by the simple $A+B\Lambda_{\rm m}^\circ$ formula. The exercise shows why the limiting law must be tested and used only at genuinely low $c$, and why "before quoting a theoretical slope" you should compare against the experimental extrapolation first.

Lars Onsager, working in 1926-27, took the qualitative Debye-Hückel picture of ionic atmospheres — originally developed to explain activity coefficients — and asked a dynamical question: what happens to that atmosphere when the central ion moves? His rigorous combination of electrostatics (relaxation) with hydrodynamics (electrophoresis) produced quantitative, parameter-free coefficients $A$ and $B$ that matched experimental Kohlrausch slopes for simple 1:1 electrolytes remarkably well, transforming an empirical square-root law into a predictive theory built from fundamental constants.

Ask the tutor: How does the Onsager equation change for unsymmetrical electrolytes like CaCl₂, where the charges on cation and anion differ?