In an ideal solution every dissolved particle behaves independently, so its chemical effect is set by its concentration alone. Real ions are surrounded by a loose “atmosphere” of oppositely charged ions that partly shields their charge, so they act as if they were less concentrated than they are. The activity coefficient γ is the correction factor: the effective concentration, or activity, equals γ times the actual concentration. A value of 1 means ideal behaviour; values below 1 mean the ion is shielded by its neighbours.
Peter Debye and Erich Hückel derived the first theory for this effect in 1923. Their limiting law says that log γ falls with the square of the ion charge and with the square root of the ionic strength, which is why a triply charged ion such as Al³⁺ or PO₄³⁻ deviates far more than Na⁺ at the same ionic strength. The law is accurate only for very dilute solutions (I below about 0.01 mol/kg), so this calculator also offers the extended equation, which adds the size of the hydrated ion, and the empirical Davies equation, which stays useful up to roughly 0.5 mol/kg.
Activity coefficients matter whenever you use an equilibrium constant, a solubility product or the Nernst equation with real concentrations instead of activities.
How to use the Activity Coefficient Calculator
- Equation
- Charge number of the ion (z) — Use the signed charge, e.g. 2 for Ca²⁺ or −1 for Cl⁻; only z² matters.
- Ionic strength
- Ionic strength (I)
- Salt concentration
- Cation charge
- Cations per formula unit
- Anion charge (magnitude)
- Anions per formula unit
- Constant A (kg^½·mol^−½) — 0.509 for water at 25 °C (log₁₀ form).
- Constant B (kg^½·mol^−½·Å⁻¹) — 0.328 for water at 25 °C.
- Effective (hydrated) ion size å — About 9 Å for H⁺ and Al³⁺, 6 Å for Ca²⁺, 4 Å for Na⁺ and SO₄²⁻, 3 Å for K⁺ and Cl⁻.
- Concentration of the ion (for its activity)
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
The calculator first finds the ionic strength, either directly from your input or as ½ Σ cᵢzᵢ² for a fully dissociated salt. It then evaluates the chosen equation for log₁₀ γ with the constants A (and B for the extended form), raises 10 to that power to get γ, and multiplies γ by the ion concentration to give the activity.
Worked example
Find the activity coefficient of a triply charged ion at an ionic strength of 0.09 mol/kg in water at 25 °C. The limiting law gives log₁₀ γ = −0.509 × 3² × √0.09 = −0.509 × 9 × 0.3 = −1.374, so γ = 10^−1.374 ≈ 0.042. The Davies equation gives log₁₀ γ = −0.509 × 9 × (0.3/1.3 − 0.3 × 0.09) = −0.933, so γ ≈ 0.117 – almost three times larger. At this ionic strength the limiting law overstates the effect, which is why the extended forms exist.
For a 2+ ion at I = 0.01 mol/kg the four equations agree much better: 0.626 (limiting), 0.676 (extended, å = 6 Å), 0.653 (Güntelberg) and 0.662 (Davies).
Typical ion sizes for the extended equation
| Ion | Effective size å (Å) |
|---|---|
| H⁺, Al³⁺, Fe³⁺ | 9 |
| Mg²⁺ | 8 |
| Li⁺, Ca²⁺, Cu²⁺, Zn²⁺, Fe²⁺ | 6 |
| Sr²⁺, Ba²⁺ | 5 |
| Na⁺ | 4–4.5 |
| CO₃²⁻ | 4.5 |
| SO₄²⁻ | 4 |
| OH⁻, F⁻ | 3.5 |
| K⁺, Cl⁻, Br⁻, I⁻, NO₃⁻ | 3 |
Values from Kielland’s classic table of hydrated ion sizes, rounded; they are fitted parameters, not exact radii.
Tips
- Use the final volume of the solution, not the volume of solvent added, for molarity.
- For dilutions, C₁V₁ = C₂V₂ only holds when both concentrations use the same unit.
- Add acid to water, never water to acid.
- Everything is calculated in your browser.
Common mistakes to avoid
- Mixing w/w, w/v and v/v percentages.
- Forgetting that pH is logarithmic – one unit is a tenfold change in [H⁺].
- Using pKa where Ka is asked for (or the other way round).
Inputs explained
| Field | What to enter |
|---|---|
| Equation | Choose one: Debye–Hückel limiting law (I ≲ 0.01), Extended Debye–Hückel (I ≲ 0.1), Güntelberg (I ≲ 0.1), Davies (I ≲ 0.5) |
| Charge number of the ion (z) | A number. Use the signed charge, e.g. 2 for Ca²⁺ or −1 for Cl⁻; only z² matters. |
| Ionic strength | Choose one: I know the ionic strength, Calculate it from a dissolved salt |
| Ionic strength (I) | Units: mol/kg, mmol/kg, mol/L (M), mmol/L (mM) |
| Salt concentration | Units: mol/kg, mmol/kg, mol/L (M), mmol/L (mM) |
| Cation charge | A number |
| Cations per formula unit | A number |
| Anion charge (magnitude) | A number |
| Anions per formula unit | A number |
| Constant A (kg^½·mol^−½) | A number. 0.509 for water at 25 °C (log₁₀ form). |
| Constant B (kg^½·mol^−½·Å⁻¹) | A number. 0.328 for water at 25 °C. |
| Effective (hydrated) ion size å | Units: Å, pm, nm. About 9 Å for H⁺ and Al³⁺, 6 Å for Ca²⁺, 4 Å for Na⁺ and SO₄²⁻, 3 Å for K⁺ and Cl⁻. |
| Concentration of the ion (for its activity) | Units: mol/kg, mmol/kg, mol/L (M), mmol/L (mM) |
What each result means
- Activity coefficient γ = 10^[log₁₀ γ]
- Activity (effective concentration) = [Activity coefficient γ]×[Concentration of the ion (for its activity)]
- Deviation from ideal behaviour = (1 – [Activity coefficient γ])×100
Formula
Limiting law: log₁₀ γ = −A·z²·√I · Extended: log₁₀ γ = −A·z²·√I ÷ (1 + B·å·√I) · Güntelberg: log₁₀ γ = −A·z²·√I ÷ (1 + √I) · Davies: log₁₀ γ = −A·z²·(√I ÷ (1 + √I) − 0.3·I) · I = ½·Σ cᵢ·zᵢ² · activity a = γ·cWorked example
Inputs — Equation: Debye–Hückel limiting law (I ≲ 0.01); Charge number of the ion (z): 2; Ionic strength: I know the ionic strength; Ionic strength (I): 0.01 mol/kg; Constant A (kg^½·mol^−½): 0.509; Concentration of the ion (for its activity): 0.01 mol/kg.
Results — Ionic strength used = 0.01 mol/kg; log₁₀ γ = -0.2036; Activity coefficient γ = 0.62575; Activity (effective concentration) = 0.00626 mol/kg; Deviation from ideal behaviour = 37.43 %; Range check = The ionic strength is inside the range where this equation works well.
Key terms
- Activity
- Effective concentration of a species, a = γ·c/c°, which replaces concentration in exact equilibrium expressions.
- Activity coefficient (γ)
- Ratio of activity to concentration; 1 for ideal behaviour.
- Ionic strength (I)
- ½ Σ cᵢzᵢ², a measure of the total charge concentration in a solution.
- Ionic atmosphere
- The cloud of oppositely charged ions that surrounds each ion and screens its charge.
Frequently asked questions
What is an activity coefficient?
It is the factor that turns a concentration into an activity (effective concentration). γ = 1 for an ideal solution; in real ionic solutions it is usually below 1, for example about 0.63 for a 2+ ion at I = 0.01 mol/kg.
How do I calculate ionic strength?
Add cᵢ·zᵢ² for every ion and halve the sum. For 0.01 mol/kg CaCl₂: ½ × (0.01 × 2² + 0.02 × 1²) = 0.03 mol/kg.
Which equation should I use?
The limiting law below about 0.01 mol/kg, the extended Debye–Hückel or Güntelberg equation up to about 0.1 mol/kg, and the Davies equation up to about 0.5 mol/kg. Above that, specific-ion models such as Pitzer equations are needed.
Why does the charge matter so much?
The charge enters as z², so at the same ionic strength log γ for a 3+ ion is nine times larger than for a 1+ ion. At I = 0.09 the limiting law gives γ ≈ 0.70 for z = 1 but only 0.042 for z = 3.
What is the value of A at 25 °C?
A ≈ 0.509 kg^½·mol^−½ for water at 25 °C when the equation uses log₁₀. It rises slowly with temperature (about 0.49 at 0 °C).
Can the activity coefficient be greater than 1?
Yes – at high ionic strength (above roughly 1 mol/kg for many salts) γ passes through a minimum and rises again, sometimes above 1. The Davies equation shows this upturn because of its +0.3·I term.
What is the mean activity coefficient?
For a salt MₚXq it is γ± = (γ₊ᵖ·γ₋^q)^(1/(p+q)), the quantity that measurements actually give, since single-ion activities cannot be measured separately.