Ions in solution do not behave as if they were alone: every ion drags an atmosphere of counter-ions that shields it and discounts its effective concentration. The activity coefficient γ is that discount factor, and this Activity Coefficient Calculator computes it from ionic strength, ion charge and ion size using the three classical Debye–Hückel-family equations for water at 25 °C — the limiting law, the extended law and the Davies equation — side by side.
Worked Example: A Divalent Ion at I = 0.01 M
For Ca²⁺ (z = 2, å ≈ 6 Å) in a solution of ionic strength 0.01 M, the calculator returns γ ≈ 0.626 by the limiting law, 0.676 by the extended Debye–Hückel equation, and 0.662 by the Davies equation. The spread is the lesson: all three share the core term −A·z²·√I = −0.509 × 4 × 0.1 = −0.2036 for log₁₀ γ, and differ only in how they correct it as concentration grows. The limiting law applies no correction and is trustworthy only below about 0.001–0.005 M; the extended law divides by (1 + B·å·√I), modeling the ion’s finite size, and stays reasonable to roughly 0.1 M; Davies adds the empirical −0.3·I term inside the bracket, needs no ion-size parameter, and is the workhorse up to about 0.5 M. That a doubly charged ion at a mere 0.01 M already loses a third of its effective concentration — activity 0.0066 where concentration says 0.01 — is why equilibrium calculations in real electrolytes ignore γ at their peril.
Ionic Strength: The Input That Runs Everything
Ionic strength I = ½ Σ cᵢzᵢ² totals every ion’s contribution, weighted by the square of its charge — which is why 0.01 M CaCl₂ has I = ½(0.01 × 4 + 0.02 × 1) = 0.03 M, triple the salt’s molarity, and why a pinch of a trivalent salt dominates a solution’s electrostatics. Every equation on this page is a function of I and z only (plus å for the extended law): γ falls as I rises (more shielding), and falls with z², so trivalent ions at moderate I can have γ below 0.2. The constants A = 0.509 and B = 0.328 are properties of water at 25 °C (they drift with temperature and solvent — A rises as temperature climbs, and in ethanol both are several times larger, which is why salts misbehave so much more in mixed solvents). Above roughly 0.5 M the whole Debye–Hückel family yields to ion-interaction models (Pitzer, SIT): the calculator’s three curves remain formally computable there, but their authority has expired — a limit printed here deliberately rather than discovered by the user in a failed equilibrium.
Where Gamma Changes Real Answers
Solubility: the thermodynamic Ksp is written in activities, so a salt’s molar solubility in a salty medium exceeds the pure-water prediction — the “salt effect” that this page quantifies (and the reason the common-ion calculation done in concentrations alone undershoots). pH and buffers: electrodes respond to hydrogen-ion activity, and buffer pKa values shift with ionic strength through exactly these γ ratios — physiological saline at I ≈ 0.15 M is far outside concentration-land. Geochemistry and oceanography live at I ≈ 0.7 M in seawater, territory where Davies is already straining and Pitzer models take over. Equilibrium constants generally: any K measured at one ionic strength transfers to another only through γ corrections of reactants and products, each raised to its stoichiometric power — a correction that grows with the charge imbalance of the reaction, and that this calculator supplies ion by ion.
Common Mistakes
Computing ionic strength as the sum of concentrations and forgetting both the ½ and the z² weighting. Entering a signed charge — the equations use z², so sign is irrelevant, but entering z = 0 for a neutral species is correct behavior (γ = 1 from these ion-atmosphere formulas; neutral-species salting effects are a separate, smaller phenomenon). Using the limiting law at 0.1 M because it is the simplest line — its error there is tens of percent. Applying water-at-25 °C constants to hot or mixed-solvent systems. And reporting a mean ionic activity coefficient γ± as if it were a single ion’s γ: experiments measure the mean over cation and anion, while this page computes the individual-ion quantity the Debye–Hückel theory actually predicts.
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Frequently Asked Questions
What is an activity coefficient?
The factor γ converting concentration to activity, a = γ × c — an ion’s effective concentration after its ionic atmosphere’s shielding is accounted. γ approaches 1 at infinite dilution and falls as ionic strength and charge rise.
How is ionic strength computed?
I = ½ Σ cᵢzᵢ² over all ions. Example: 0.01 M CaCl₂ gives I = ½(0.01 × 2² + 0.02 × 1²) = 0.03 M — charge weighting makes multivalent ions count far beyond their molarity.
Which equation should I use?
Limiting law below about 0.005 M; extended Debye–Hückel (with a realistic ion size) to about 0.1 M; Davies to about 0.5 M without needing å. At I = 0.01 for Ca²⁺ they give 0.626, 0.676 and 0.662 — Davies and extended agree best where both are valid.
What are A and B in the equations?
Solvent-and-temperature constants: for water at 25 °C, A = 0.509 and B = 0.328 (in molal/L units with å in angstroms). Both change with temperature and solvent dielectric constant.
Why does charge matter so much?
Because log γ scales with z²: doubling the charge quadruples the effect. A trivalent ion at I = 0.01 already shows a γ near 0.4 by Davies, versus about 0.90 for a monovalent ion in the same solution.
Can gamma exceed 1?
Within these dilute-solution equations, no — they only model shielding. At high concentrations real γ values turn around and can exceed 1 (hydration and excluded-volume effects), which is Pitzer-model territory beyond this page’s range.