Activity Coefficient Calculator

Find an ion's activity coefficient from its charge and the solution's ionic strength, using the Debye-Huckel limiting law or the extended form with an ion-size parameter.

⚗️ Activity Coefficient Calculator

log₁₀(γ) = −A z² √I  (A = 0.509, water, 25°C)

log₁₀(γ) = −A z² √I ÷ (1 + B a √I)  (A = 0.509, B = 0.328)

integer
mol/L
Å
Activity coefficient γ
—
log₁₀(γ)
—
Estimated activity
—
Step-by-step working

⚗️ What is an Activity Coefficient?

The activity coefficient (γ) is a correction factor used in solution thermodynamics to account for the fact that ions in a real electrolyte solution do not behave exactly like an ideal, infinitely dilute solution would predict. Because ions of opposite charge attract and ions of like charge repel, each ion is surrounded by a slightly asymmetric cloud of neighboring ions, called the ionic atmosphere, which lowers its effective concentration, or activity, below its actual molar concentration. Activity and concentration are related by activity = γ × concentration.

The Debye-Huckel theory, developed by Peter Debye and Erich Huckel in 1923, gives a theoretical prediction of γ from first principles of electrostatics. Its simplest form, the limiting law, is log₁₀(γ) = −A z² √I, where A is a constant depending on the solvent and temperature (0.509 for water at 25°C), z is the ion's charge, and I is the solution's ionic strength. This form is exact only as ionic strength approaches zero, which is why it is called a limiting law.

For solutions with slightly higher ionic strength, up to roughly 0.1 mol/L, the extended Debye-Huckel equation adds an ion-size parameter a, representing the effective hydrated radius of the ion: log₁₀(γ) = −A z² √I ÷ (1 + B a √I), with B another solvent- and temperature-dependent constant (0.328 for water at 25°C). This extended form accounts for the finite size of real ions, which the point-charge limiting law ignores, and gives noticeably better agreement with measured activity coefficients.

This calculator supports both the limiting law for very dilute solutions and the extended form for moderately dilute solutions, showing the full step-by-step calculation and a chart of how the activity coefficient changes with ionic strength for the ion charge you specify.

📐 Formula

log₁₀(γ)  =  −A z² √I   (Debye-Huckel limiting law)
γ = activity coefficient of the ion (dimensionless)
A = Debye-Huckel constant, ≈ 0.509 (mol/L)−0.5 for water at 25°C
z = charge of the ion (dimensionless integer)
I = ionic strength of the solution (mol/L)
Extended form: log₁₀(γ) = −A z² √I ÷ (1 + B a √I), with B ≈ 0.328 Å−1(mol/L)−0.5 and a = ion-size parameter (Å)
Example: z = 1, I = 0.01 mol/L: log₁₀(γ) = −0.509 × 1 × 0.1 = −0.0509, so γ = 10−0.0509 = 0.88941

📖 How to Use This Calculator

Steps

1
Choose the model. Select Limiting law for very dilute solutions (I below about 0.01 mol/L), or Extended for higher ionic strength up to about 0.1 mol/L.
2
Enter charge and ionic strength. Type the ion charge magnitude |z| and the solution's ionic strength I in mol/L. For the extended model, also enter the ion-size parameter a in angstroms.
3
Read the activity coefficient. Click Calculate to see gamma, log10(gamma), and the estimated activity, with full step-by-step working.

💡 Example Calculations

Example 1 — Monovalent Ion, Dilute Solution

z = 1, I = 0.01 mol/L, limiting law

1
√I = √0.01 = 0.1
2
log₁₀(γ) = −0.509 × 1² × 0.1 = -0.0509
3
γ = 10-0.0509 = 0.88941
γ = 0.88941
Try this example →

Example 2 — Divalent Ion, More Dilute Solution

z = 2, I = 0.001 mol/L, limiting law

1
√I = √0.001 = 0.03162
2
log₁₀(γ) = −0.509 × 2² × 0.03162 = -0.0644
3
γ = 10-0.0644 = 0.86222
γ = 0.86222
Try this example →

Example 3 — Divalent Ion, Extended Model

z = 2, I = 0.05 mol/L, a = 6 Å, extended model

1
√I = √0.05 = 0.2236
2
log₁₀(γ) = −0.509 × 4 × 0.2236 ÷ (1 + 0.328 × 6 × 0.2236) = -0.3161
3
γ = 10-0.3161 = 0.48290, noticeably lower than the limiting law would give at this higher ionic strength
γ = 0.48290
Try this example →

❓ Frequently Asked Questions

What is an activity coefficient?+
An activity coefficient (gamma) is a correction factor that relates an ion's actual concentration to its thermodynamically effective concentration, called its activity. In real electrolyte solutions, electrostatic interactions between ions make them behave as if less concentrated than they really are, so gamma is usually less than 1, and activity = gamma x concentration.
What is the Debye-Huckel limiting law?+
The Debye-Huckel limiting law estimates an ion's activity coefficient from its charge and the solution's ionic strength: log10(gamma) = -A z^2 sqrt(I), where A is a solvent- and temperature-dependent constant (0.509 for water at 25 degrees C), z is the ion's charge, and I is the ionic strength in mol/L. It is called the limiting law because it becomes exact only in the limit of very low ionic strength.
What is ionic strength?+
Ionic strength I measures the total concentration of ions in a solution, weighted by the square of their charges: I = 0.5 x sum(ci x zi^2) over all ions. It accounts for the fact that a doubly or triply charged ion contributes more to electrostatic interactions than a singly charged ion at the same concentration.
When should I use the extended Debye-Huckel equation instead of the limiting law?+
Use the extended Debye-Huckel equation, log10(gamma) = -A z^2 sqrt(I) / (1 + B a sqrt(I)), when ionic strength exceeds roughly 0.01 mol/L, since the simple limiting law starts to noticeably overestimate the deviation from ideal behavior above that. The extended form remains reasonably accurate up to about 0.1 mol/L by including the ion-size parameter a.
What is the ion-size parameter a?+
The ion-size parameter a, sometimes called the distance of closest approach, represents the effective hydrated radius of the ion in angstroms, accounting for the water molecules bound to it in solution. Typical published values range from about 3 angstroms for small ions like K+ or Cl- up to around 9 angstroms for smaller, more strongly hydrated ions like Li+ or Mg2+.
Why does activity coefficient depend on charge squared?+
The electrostatic interaction energy between ions in solution scales with the product of their charges, and averaging over the surrounding ionic atmosphere brings in a factor of z^2 for a given ion. This is why the Debye-Huckel equations use z^2, and why multiply-charged ions like Ca2+ or Al3+ show much larger deviations from ideal behavior than singly charged ions like Na+ at the same ionic strength.
What values do A and B take for water at 25 degrees C?+
For water at 25 degrees C, A is approximately 0.509 (mol/L)^-0.5 and B is approximately 0.328 angstrom^-1 (mol/L)^-0.5 when ionic strength is expressed in mol/L. Both constants depend on the solvent's dielectric constant and density and on temperature, so they must be looked up separately for other solvents or temperatures.
Is the activity coefficient always less than 1?+
Within the range where Debye-Huckel theory applies, yes, both the limiting law and the extended form always predict gamma less than 1, because the electrostatic screening between ions of opposite charge lowers the free energy relative to an ideal solution. At much higher ionic strengths, beyond where Debye-Huckel is valid, activity coefficients can actually rise above 1 due to other effects such as ion pairing and hydration changes.
What is the difference between activity and concentration?+
Concentration is the actual amount of a species per unit volume, while activity is the thermodynamically effective concentration that appears correctly in equilibrium constant expressions and chemical potential equations. Activity equals gamma times concentration, and the two converge (gamma approaches 1) only in the limit of infinite dilution.
Why does Debye-Huckel theory break down at high ionic strength?+
Debye-Huckel theory assumes ions are point charges that interact only through long-range electrostatic screening, an approximation that works well when ions are far apart on average. At higher ionic strength, ions get closer together, short-range effects like ion size, ion pairing, and changes in the solvent's structure become significant, and neither the limiting law nor the extended form captures these accurately.
What is the activity of an ion, and how is it calculated here?+
The activity of an ion equals its activity coefficient gamma multiplied by its molar concentration: activity = gamma x c. This calculator estimates activity using the ionic strength I as a stand-in for concentration to illustrate the correction, giving activity = gamma x I in mol/L; for a specific species, substitute that ion's own molar concentration instead of I.
Tags: Activity Coefficient Calculator Debye Huckel Ionic Strength Calculator Limiting Law Extended Debye Huckel Gamma Coefficient Electrolyte Solution Calculator Formula Free Online

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What is an activity coefficient?

An activity coefficient (gamma) is a correction factor that relates an ion's actual concentration to its thermodynamically effective concentration, called its activity. In real electrolyte solutions, electrostatic interactions between ions make them behave as if less concentrated than they really are, so gamma is usually less than 1, and activity = gamma x concentration.

What is the Debye-Huckel limiting law?

The Debye-Huckel limiting law estimates an ion's activity coefficient from its charge and the solution's ionic strength: log10(gamma) = -A z^2 sqrt(I), where A is a solvent- and temperature-dependent constant (0.509 for water at 25 degrees C), z is the ion's charge, and I is the ionic strength in mol/L. It is called the limiting law because it becomes exact only in the limit of very low ionic strength.

What is ionic strength?

Ionic strength I measures the total concentration of ions in a solution, weighted by the square of their charges: I = 0.5 x sum(ci x zi^2) over all ions. It accounts for the fact that a doubly or triply charged ion contributes more to electrostatic interactions than a singly charged ion at the same concentration.

When should I use the extended Debye-Huckel equation instead of the limiting law?

Use the extended Debye-Huckel equation, log10(gamma) = -A z^2 sqrt(I) / (1 + B a sqrt(I)), when ionic strength exceeds roughly 0.01 mol/L, since the simple limiting law starts to noticeably overestimate the deviation from ideal behavior above that. The extended form remains reasonably accurate up to about 0.1 mol/L by including the ion-size parameter a.

What is the ion-size parameter a?

The ion-size parameter a, sometimes called the distance of closest approach, represents the effective hydrated radius of the ion in angstroms, accounting for the water molecules bound to it in solution. Typical published values range from about 3 angstroms for small ions like K+ or Cl- up to around 9 angstroms for smaller, more strongly hydrated ions like Li+ or Mg2+.

Why does activity coefficient depend on charge squared?

The electrostatic interaction energy between ions in solution scales with the product of their charges, and averaging over the surrounding ionic atmosphere brings in a factor of z^2 for a given ion. This is why the Debye-Huckel equations use z^2, and why multiply-charged ions like Ca2+ or Al3+ show much larger deviations from ideal behavior than singly charged ions like Na+ at the same ionic strength.

What values do A and B take for water at 25 degrees C?

For water at 25 degrees C, A is approximately 0.509 (mol/L)^-0.5 and B is approximately 0.328 angstrom^-1 (mol/L)^-0.5 when ionic strength is expressed in mol/L. Both constants depend on the solvent's dielectric constant and density and on temperature, so they must be looked up separately for other solvents or temperatures.

Is the activity coefficient always less than 1?

Within the range where Debye-Huckel theory applies, yes, both the limiting law and the extended form always predict gamma less than 1, because the electrostatic screening between ions of opposite charge lowers the free energy relative to an ideal solution. At much higher ionic strengths, beyond where Debye-Huckel is valid, activity coefficients can actually rise above 1 due to other effects such as ion pairing and hydration changes.

What is the difference between activity and concentration?

Concentration is the actual amount of a species per unit volume, while activity is the thermodynamically effective concentration that appears correctly in equilibrium constant expressions and chemical potential equations. Activity equals gamma times concentration, and the two converge (gamma approaches 1) only in the limit of infinite dilution.

Why does Debye-Huckel theory break down at high ionic strength?

Debye-Huckel theory assumes ions are point charges that interact only through long-range electrostatic screening, an approximation that works well when ions are far apart on average. At higher ionic strength, ions get closer together, short-range effects like ion size, ion pairing, and changes in the solvent's structure become significant, and neither the limiting law nor the extended form captures these accurately.

What is the activity of an ion, and how is it calculated here?

The activity of an ion equals its activity coefficient gamma multiplied by its molar concentration: activity = gamma x c. This calculator estimates activity using the ionic strength I as a stand-in for concentration to illustrate the correction, giving activity = gamma x I in mol/L; for a specific species, substitute that ion's own molar concentration instead of I.

📌 Quick Tips

💡The Debye-Huckel limiting law is only accurate for very dilute solutions, roughly I below 0.01 mol/L. Above that, the extended form with an ion-size parameter gives better results.
💡A = 0.509 and B = 0.328 are specific to water at 25 degrees C. Both constants change with solvent and temperature, so do not reuse them for non-aqueous or high-temperature systems.
💡Higher charge ions deviate from ideal behavior faster than lower charge ions, since gamma depends on z squared. A divalent ion (z=2) has 4 times the log(gamma) deviation of a monovalent ion at the same ionic strength.
💡The activity coefficient gamma is always less than 1 in these models, reflecting that ion-ion electrostatic interactions reduce an ion's effective (thermodynamically active) concentration below its actual molar concentration.
💡The ion-size parameter a in the extended law represents the effective hydrated radius of the ion in angstroms; typical values range from about 3 for small monatomic ions up to 9 for larger or more strongly hydrated ions.