What this calculator does
In an ideal dilute solution, concentration and chemical activity are the same thing. Real solutions, especially at higher ionic strength, are not ideal: electrostatic interactions between ions mean their effective, thermodynamically active concentration is lower than their actual concentration. The activity coefficient γ is the correction factor between the two.
The Debye-Hückel theory predicts this effect from just two things: how highly charged the ion is, and how crowded the solution is with other ions (its ionic strength). A more highly charged ion, or a more concentrated solution, pulls the activity coefficient further below 1, meaning the ion behaves as if it were less concentrated than it actually is.
The formula
Take the square root of the ionic strength, multiply it by −0.51 and the square of the ion's charge, then divide by 1 plus that same square root. That gives log₁₀γ; raising 10 to that power gives the activity coefficient itself.
| Term | Meaning |
|---|---|
| γ (gamma) | The activity coefficient: activity ÷ concentration. Equal to 1 in an ideal solution. |
| z | The magnitude of the ion's charge (1 for Na⁺ or Cl⁻, 2 for Ca²⁺ or SO₄²⁻, and so on). |
| I (ionic strength) | A measure of the total ionic concentration in solution, weighted by the square of each ion's charge. |
The inputs explained
| Field | What to enter |
|---|---|
| Ion charge (z), magnitude | The magnitude of the charge on the ion in question, entered as a positive number regardless of whether the ion is a cation or anion. |
| Ionic strength (I) (mol/L) | The ionic strength of the solution in mol/L, calculated from every ion present, not just the one being examined. |
When to use it
Correcting equilibrium calculations
Equilibrium constants written in terms of concentration are only exact in dilute, ideal solutions; at higher ionic strength, activity coefficients correct concentration-based calculations back toward the true thermodynamic behaviour.
Estimating deviations for multiply-charged ions
Because the formula depends on z², highly charged ions such as Ca²⁺ or Fe³⁺ show much larger deviations from ideal behaviour than singly-charged ions at the same ionic strength.
Working with electrode and potentiometric measurements
Electrodes respond to ion activity, not raw concentration, so converting a measured activity back to concentration (or vice versa) requires an activity coefficient like the one estimated here.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the activity coefficient changes with ionic strength
A doubly-charged ion, across a range of solution ionic strengths.
| Ionic strength | Activity coefficient γ | log₁₀γ |
|---|---|---|
| 0.001 mol/L | 0.8659 | -0.0625 |
| 0.005 mol/L | 0.7333 | -0.1347 |
| 0.01 mol/L | 0.6524 | -0.1855 |
| 0.05 mol/L | 0.4238 | -0.3728 |
| 0.1 mol/L | 0.3235 | -0.4901 |
| 0.5 mol/L | 0.1429 | -0.8450 |
How the activity coefficient changes with ion charge
A fixed ionic strength, across ions of increasing charge.
Questions
Why is the activity coefficient always less than 1 in this formula?
The Debye-Hückel expression as used here always produces a negative log γ for a nonzero ionic strength and charge, which means γ itself is always below 1. This reflects that ionic interactions generally reduce an ion's effective activity relative to its concentration, which is the typical (though not universal) behaviour at low to moderate ionic strength.
How accurate is this at high ionic strength?
The extended Debye-Hückel law used here is reliable roughly up to an ionic strength of about 0.1 mol/L. Beyond that, more elaborate models (such as Davies or Pitzer equations, which are not implemented here) are needed for accurate estimates, since real solutions deviate from this simplified electrostatic picture.
What is the difference between this and the plain Debye-Hückel limiting law?
The original limiting law is log γ = −0.51z²√I with no denominator, and is only valid at very low ionic strength (below about 0.001 mol/L). Adding the 1 + √I denominator, as used here, extends the usable range somewhat further before the approximation breaks down.
Does the activity coefficient depend on which other ions are present?
Only indirectly, through the ionic strength, which sums the contribution of every ion in solution. This formula treats all ions of the same charge and ionic strength identically, which is a simplification; more detailed models account for ion-specific interactions as well.
For the ionic strength figure this calculator takes as an input, see the ionic strength calculator. To go from concentration to pH using activity corrections in a related buffer context, see the Henderson-Hasselbalch calculator.