Activity Coefficient Calculator

Compute ionic activity coefficients (γ_i) and the mean activity coefficient γ_± for aqueous electrolytes using Debye–Hückel (limiting), Davies, or Extended Debye–Hückel at 25 °C. Enter ions (charge and amount), and optionally the ion-size parameter a_i (Å).

Background

Electrolyte solutions deviate from ideal behavior. We account for this using an activity coefficient γ so that a = γ×(concentration). For ionic strength I = ½ Σ m_iz_i² (molality basis), Debye–Hückel-type expressions relate log₁₀γ_i to I, charge z_i, and (for the extended form) an effective ion size a_i.

Enter solution data

Model:

Assumes 25 °C water (A = 0.509, B = 0.328). See model comparison ↓

Input basis:

Molality m (mol·kg⁻¹) Molarity c (mol·L⁻¹, dilute approx)

Electrolyte stoichiometry (for γ_±):

ν₊ (cations) ν₋ (anions) e.g., NaCl → 1 & 1; MgCl₂ → 1 & 2

Show:

Step-by-step working

Round to sensible significant figures

Ions:

Add each ionic species present (not the neutral formula). For Extended DH, include a_i (Å).

Quick picks:

Result:

No results yet. Enter inputs and click Calculate.

How to use this calculator

Pick a model: Davies is a solid default up to moderate ionic strength; Extended DH lets you include ion size a_i (Å).
Choose basis: use molality when possible; molarity is acceptable for dilute aqueous solutions.
Add ions: one row per ionic species with charge z and amount (m or c). For Extended DH, add a_i.
Set ν₊, ν₋: enter electrolyte stoichiometric coefficients to report γ_±.
Use Quick picks to prefill examples; edit any field and click Calculate.

Formula & Equation Used

Ionic strength: I = ½ Σ m_i z_i²

Debye–Hückel (limiting): log₁₀γ_i = −A z_i² √I

Davies: log₁₀γ_i = −A z_i² [ √I/(1+√I) − 0.3 I ]

Extended Debye–Hückel: log₁₀γ_i = −A z_i² [ √I / (1 + B a_i √I ) ]

Mean coefficient: γ_± = (γ₊^ν+ γ₋^ν−)^{1/(ν+ + ν−)}

Here A=0.509, B=0.328 for water at 25 °C.

Which model should I choose?

Debye–Hückel (limiting law) — for very dilute solutions (I ≲ 0.01). It’s the simplest but least accurate beyond trace concentrations.
Davies equation — a great all-around choice for most laboratory salt solutions (0.01 ≲ I ≲ 0.5). It balances simplicity and accuracy and doesn’t require ion-size parameters.
Extended Debye–Hückel — recommended for multivalent or mixed electrolytes. Requires ion-size a_i (Å) but performs more smoothly at higher ionic strengths up to about I ≈ 0.5.
Beyond 0.5 M (e.g., seawater, brines) — Debye–Hückel-type models break down; use Pitzer or SIT equations instead.

These guidelines assume 25 °C water. Acceptable ranges shift slightly with temperature and solvent dielectric constant.

Example Problems & Step-by-Step Solutions

Example 1 — 0.10 m NaCl (Davies)

Ions: Na⁺ (z=+1, m=0.10), Cl⁻ (z=−1, m=0.10).
1) Ionic strength: I = ½[(0.10)(1²) + (0.10)(1²)] = 0.10.
2) √I = 0.316; term = √I/(1+√I) − 0.3I ≈ 0.240 − 0.030 = 0.210.
3) log₁₀γ_Na+ = log₁₀γ_Cl− = −(0.509)(1)(0.210) ≈ −0.107 → γ ≈ 0.78.
4) Mean γ_± (1:1) ≈ 0.78.

Example 2 — 0.010 m MgCl₂ (Davies)

Ions: Mg²⁺ (z=+2, m=0.010), Cl⁻ (z=−1, m=0.020).
1) I = ½[(0.010)(2²) + (0.020)(1²)] = 0.03; √I = 0.173.
2) term ≈ 0.173/(1+0.173) − 0.3×0.03 ≈ 0.148 − 0.009 = 0.139.
3) log₁₀γ_Mg²+ = −0.509×4×0.139 ≈ −0.282 → γ ≈ 0.52;
log₁₀γ_Cl− = −0.509×1×0.139 ≈ −0.071 → γ ≈ 0.85.
4) Mean γ_± (1:2) = (γ₊ γ₋²)^1/3 ≈ (0.52×0.85²)^1/3 ≈ 0.72.

Frequently Asked Questions

Q: Molality or molarity?

Debye–Hückel theory is formally molality-based. For dilute aqueous solutions, molarity is a good approximation.

Q: What a_i should I use?

Use effective hydrated ion sizes (Å). If unknown, values between 3–9 Å are typical; results are not very sensitive at low I.

Q: Temperature?

This calculator assumes 25 °C (A=0.509, B=0.328). At other temperatures, A and B vary with solvent properties.

Molality