Q3(e). Debye–Hückel limiting law for an ion i

The equation and its symbols

1. ₁₀γᵢ=-A zᵢ² √I
2. γᵢ: activity coefficient of ion i, defined by aᵢ=γᵢ bᵢ/b°; dimensionless. γᵢ→1 as I→0.
3. zᵢ: charge number of the ion (+1 for Na⁺, -2 for SO₄²⁻); dimensionless. It enters as zᵢ², so the sign does not matter.
4. I: ionic strength, I=12Σⱼ bⱼzⱼ², summed over all ions in the solution; unit mol kg⁻¹ (molality bⱼ). In dilute water molar concentrations may be used.
5. A: a constant for the solvent and temperature, A=1.825×10⁶(εᵣT)^(3/2); for water at 25°C A=0.509 (mol kg⁻¹)^(-1/2). It becomes dimensionless when I is written as I/b°.

Meaning and use

1. Origin: each ion is surrounded by an ionic atmosphere of net opposite charge; the atmosphere lowers the energy (chemical potential) of the ion, so γᵢ<1. The effect grows with charge (z²) and with I (thinner atmosphere).
2. Mean coefficient, the measurable quantity: γ_±=-A |z₊z₋| √I
3. Example, ZnCl₂ at 0.0050 mol kg⁻¹: I=12(0.0050×4+0.010×1)=0.015 mol kg⁻¹; γ(Zn²⁺)=0.563, γ(Cl⁻)=0.866, γ_±=0.750.
4. Limits: valid only for very dilute solutions (point ions). At higher I the extended law γ_±=-A|z₊z₋|√I/(1+B√I) is used.

Application. the law corrects measured cell potentials and equilibrium constants to activities (for example, the standard potential of the calomel electrode is found by extrapolating to √I=0), and it explains the salt effect on the solubility of sparingly soluble salts.