Debye-Hückel Limiting Law
The Debye-Hückel limiting law predicts the activity coefficients of ions in dilute electrolyte solutions.
Explains why real solutions deviate from ideal behavior.
The Formula
I = ½ Σ(ci × zi²)
Developed by Peter Debye and Erich Hückel in 1923, this theory explains why ionic solutions deviate from ideal behavior. At infinite dilution, ions behave ideally (γ± = 1). As concentration increases, each ion is surrounded by an "ionic atmosphere" of oppositely charged ions. This atmosphere lowers the effective activity of each ion, reducing γ± below 1. The law is valid only for dilute solutions (I < 0.01 mol/L).
Variables
| Symbol | Meaning | Value/Unit |
|---|---|---|
| γ± | Mean activity coefficient (1 = ideal; <1 = real) | dimensionless |
| A | Debye-Hückel constant (depends on solvent and temperature) | 0.509 for water at 25°C |
| z+, z− | Charges of cation and anion | dimensionless |
| I | Ionic strength of the solution | mol/L |
| ci | Molar concentration of ion i | mol/L |
| zi | Charge number of ion i | dimensionless |
Example 1 — NaCl Solution
Calculate the mean activity coefficient of 0.01 mol/L NaCl at 25°C.
NaCl dissociates as Na&sup+ (z=+1) and Cl⊃− (z=−1)
I = ½(cNa+ × 1² + cCl− × 1²) = ½(0.01 + 0.01) = 0.01 mol/L
log γ± = −0.509 × |1 × 1| × √0.01 = −0.509 × 0.1 = −0.0509
γ± = 10^(−0.0509) = 0.889 — NaCl ions are about 11% less active than an ideal solution
Example 2 — Multivalent Electrolyte
Calculate γ± for 0.001 mol/L CaCl&sub2; (z+ = +2, z− = −1).
Concentration: Ca²&sup+ = 0.001 M, Cl⊃− = 0.002 M
I = ½(0.001 × 4 + 0.002 × 1) = ½(0.004 + 0.002) = 0.003 mol/L
log γ± = −0.509 × |2 × 1| × √0.003 = −0.509 × 2 × 0.05477 = −0.0558
γ± = 10^(−0.0558) = 0.879 — multivalent ions deviate more from ideal than monovalent ions
When to Use It
Use the Debye-Hückel limiting law when:
- Correcting electrode potential calculations for real solution activity
- Calculating accurate equilibrium constants in dilute ionic solutions
- Predicting solubility changes due to the ionic strength (salting-in effect)
- Electrochemistry research and pH meter calibration
- Environmental chemistry — predicting speciation of metal ions in natural waters
For more concentrated solutions (I > 0.1 mol/L), extended versions such as the Davies equation or Pitzer model are needed.