Activity Coefficient Calculator (Debye–Hückel, extended DH and Davies equations)

Understanding the activity coefficient calculator — concept and principle

In real salt solutions ions do not behave as if they were alone, so their effective concentration (activity) is lower than their actual concentration. This calculator finds the activity coefficient γ with the Debye–Hückel limiting law, the extended Debye–Hückel equation and the Davies equation, and gives the activity.

Ion screening makes the ion act as if only about 68% as concentrated; the limiting law is out of its range here.

How it is calculated

For water at 25 °C, with A = 0.509 and B = 0.00328 pm⁻¹ (both per √(mol/L)):

Limiting law: log γ = −A z² √I (valid for I up to about 0.001 M).
Extended Debye–Hückel: log γ = −A z² √I ÷ (1 + B α √I), with ion size α in pm (up to about 0.1 M).
Davies: log γ = −A z² (√I ÷ (1 + √I) − 0.3 I) (rough up to about 0.5 M).
Activity a = γ × c.

The principle

Each ion attracts a cloud of oppositely charged ions around it. That ionic atmosphere partly screens its charge, lowering its chemical "effectiveness". Debye and Hückel modelled this screening and found that log γ depends on the ionic strength and on the square of the ion's charge. Because of the z² term, a doubly charged ion is affected four times as much as a singly charged one. The extended form adds the finite size of the ion; Davies adds an empirical term for higher ionic strengths.

Worked example

Default inputs: I = 0.01 M, z = 2, α = 600 pm, c = 0.0033 M.

√I = 0.1; A z² √I = 0.509 × 4 × 0.1 = 0.2036.
Extended: log γ = −0.2036 ÷ (1 + 0.00328 × 600 × 0.1) = −0.2036 ÷ 1.1968 ≈ −0.1701, so γ ≈ 0.6759.
Activity a = 0.6759 × 0.0033 ≈ 0.00223 mol/L.
Davies: log γ = −2.036 × (0.1 ÷ 1.1 − 0.003) ≈ −0.179, γ ≈ 0.6622.
Limiting law: γ = 10^−0.2036 ≈ 0.6257. Suitable models at this ionic strength: extended Debye–Hückel or Davies.

Each point is the calculator re-run with only “Charge of the ion z” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Charge of the ion z	Activity coefficient γ (extended Debye–Hückel)
0.5	0.9758
1	0.9067
1.5	0.8022
2	0.6759
2.5	0.5422
3	0.4142
4	0.2087

How to read the result

γ near 1 means the solution behaves almost ideally; smaller values mean stronger ionic interactions. Use activities in equilibrium constants, solubility products and pH calculations when accuracy matters. The limiting law underestimates γ here because 0.01 M is outside its range.

Common mistakes

Using concentration instead of ionic strength (I = ½ Σ ci zi²).
Using the limiting law at high ionic strength.
Using ion size in ångströms with B in pm⁻¹ (1 Å = 100 pm).
Applying 25 °C constants at very different temperatures.

Keep in mind: These models become inaccurate in concentrated solutions. For precise work use validated data or models such as Pitzer equations.

Frequently asked questions

What is an activity coefficient?

The factor relating activity to concentration, a = γc; it is 1 in an ideal solution.

Why do higher charges lower γ more?

log γ is proportional to z², so a 2+ ion's deviation is four times that of a 1+ ion.

Which equation should I use?

Limiting law below about 0.001 M, extended Debye–Hückel up to about 0.1 M, Davies roughly up to 0.5 M.

Sources: Harris, D. C., Quantitative Chemical Analysis (activity coefficients, Kielland table) · Kielland, J. (1937). Individual Activity Coefficients of Ions in Aqueous Solutions. Journal of the American Chemical Society, 59(9).

Checked 9 Nov 2026. P. Debye & E. Hückel, Phys. Z. 24, 185 (1923); C. W. Davies, Ion Association (1962); ion-size parameters from J. Kielland, J. Am. Chem. Soc. 59, 1675 (1937). Constants A = 0.509 and B = 0.328 Å⁻¹ for water at 25 °C are standard values (analytical-chemistry texts, e.g. Harris, Quantitative Chemical Analysis).

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