Engineering & Fluid Mechanics
Davies Equation Formula: Aqueous Ion Activity Coefficient Up to I = 0.5 mol/L
Extend Debye-Hückel electrolyte thermodynamics up to moderate ionic strength (I <= 0.5 M) without ion-size parameters using Cecil W. Davies’s 1962 equation log10(gamma_i) = -A * z_i² * (SQRT(I)/(1 + SQRT(I)) - 0.3 * I).
Why geochemical and water-chemistry models (PHREEQC, MINTEQ) use Davies up to I = 0.5 M!
The limiting Debye-Hückel law log10(gamma) = -A * z^2 * SQRT(I) fails above I ≈ 0.005 M, while extended Debye-Hückel requires an empirical hydrated-radius parameter a_i for every ion species! Cecil W. Davies (1938; revised 0.3*I coefficient in Ion Association, 1962) added a linear salting-out / short-range interaction term to create a universal parameter-free activity model for natural waters, buffers, and brines up to I = 0.5 mol/L: log10(gamma_i) = -A * z_i^2 * (SQRT(I) / (1 + SQRT(I)) - 0.3 * I), where A = 0.509 (mol/L)^(-1/2) in water at 25 °C!
Notice the z_i^2 valence sensitivity: at ionic strength I = 0.08 M (25 °C, A = 0.509), a divalent ion such as Ca2+ or SO4^2- (|z| = 2) experiences log10(gamma) = -0.4000 -> gamma = 10^(-0.4000) = 0.3981 (effective chemical activity < 40% of molarity!), whereas a monovalent ion Na+ or Cl- (|z| = 1) has gamma = 0.7943!
| Aqueous Ion Species at I = 0.08 mol/L (25 °C) | Charge Squared z² | Davies log10(gamma_i) | Activity Coefficient gamma_i |
|---|---|---|---|
| Divalent Ion Ca²⁺ / CO₃²⁻ (|z| = 2) | 4 | -0.4000 | 0.3981 (Strong electrostatic shielding) |
| Monovalent Ion Na⁺ / HCO₃⁻ (|z| = 1) | 1 | -0.1000 | 0.7943 |
| Concentrated Brine I = 0.85 M (> 0.5 M) | Outside Davies Window | "Outside Davies Range (0 <= I <= 0.5 M)" | Requires Pitzer / SIT Model |
Davies Single-Ion Activity Coefficient gamma_i = 10^(-A * z² * (SQRT(I)/(1+SQRT(I)) - 0.3*I))
Computes dimensionless molar activity coefficient gamma_i across 0 <= I <= 0.5 mol/L.
=IF(OR(B2 = 0, B3 < 0, B3 > 0.5, B4 <= 0), "Outside Davies Range (0 <= I <= 0.5 M)", ROUND(10 ^ (-B4 * (B2 ^ 2) * (SQRT(B3) / (1 + SQRT(B3)) - 0.3 * B3)), 4))Davies Base-10 Log Activity Coefficient log10(gamma_i)
Evaluates -A * z^2 * (SQRT(I)/(1 + SQRT(I)) - 0.3*I) directly for pK_a’ conditional equilibrium shifts.
=IF(OR(B2 = 0, B3 < 0, B3 > 0.5, B4 <= 0), "Outside Davies Range (0 <= I <= 0.5 M)", ROUND(-B4 * (B2 ^ 2) * (SQRT(B3) / (1 + SQRT(B3)) - 0.3 * B3), 4))Try the example
Use the original values or change them to test your understanding.