About

Electrolyte solutions deviate from ideal behavior. The activity coefficient γ_± quantifies this deviation: a value of 1.0 implies ideality, while real solutions at ionic strength I > 0.001 mol/L exhibit measurable non-ideality. Miscalculating γ propagates errors into solubility predictions, electrode potentials (Nernst equation), and reaction equilibrium constants. This tool computes γ_± via three established models: the Debye-Hückel limiting law (valid below 0.01 M), the Extended Debye-Hückel equation (valid to roughly 0.1 M), and the Davies equation (usable up to approximately 0.5 M). The temperature-dependent constants A and B are recalculated from the dielectric properties of water at each specified temperature.

Limitations apply. All three models assume a primitive electrolyte model (charged hard spheres in a dielectric continuum). Ion pairing, specific ion effects, and concentrations above 1 M require Pitzer or SIT frameworks not covered here. The effective ion size parameter a used in the Extended Debye-Hückel model is an empirical fit, not a true ionic radius. Pro tip: for mixed electrolyte systems, compute ionic strength from all dissolved species, not just the salt of interest.

Formulas

The ionic strength of a solution containing n ionic species is computed as:

I = 12 n∑i=1 c_i z_i²

The Debye-Hückel Limiting Law applies to very dilute solutions:

log₁₀ γ_± = −A |z₊ z₋| √I

The Extended Debye-Hückel Equation adds the ion-size parameter:

log₁₀ γ_± = −A |z₊ z₋| √I1 + B a √I

The Davies Equation extends usability to higher ionic strength:

log₁₀ γ_± = −A |z₊ z₋| ( √I1 + √I − 0.3 I )

Where I = ionic strength (mol/L), c_i = molar concentration of ion i, z_i = charge number, A = Debye-Hückel constant (0.5091 at 25°C in (mol/L)^−½), B = Debye-Hückel constant (0.3283 at 25°C in Å⁻¹(mol/L)^−½), a = effective ion-size parameter (Å), γ_± = mean ionic activity coefficient.

Reference Data

Electrolyte	Formula	ν₊	ν₋	z₊	z₋	a (Å)	γ_± at 0.1 M, 25°C (exp.)
Sodium Chloride	NaCl	1	1	+1	−1	4.0	0.778
Potassium Chloride	KCl	1	1	+1	−1	3.6	0.770
Calcium Chloride	CaCl₂	1	2	+2	−1	6.0	0.518
Magnesium Sulfate	MgSO₄	1	1	+2	−2	5.0	0.150
Sulfuric Acid	H₂SO₄	2	1	+1	−2	4.0	0.265
Hydrochloric Acid	HCl	1	1	+1	−1	9.0	0.796
Sodium Hydroxide	NaOH	1	1	+1	−1	3.5	0.766
Barium Chloride	BaCl₂	1	2	+2	−1	5.0	0.500
Potassium Sulfate	K₂SO₄	2	1	+1	−2	4.0	0.268
Lithium Bromide	LiBr	1	1	+1	−1	5.0	0.796
Sodium Nitrate	NaNO₃	1	1	+1	−1	3.0	0.762
Copper Sulfate	CuSO₄	1	1	+2	−2	5.0	0.150
Aluminum Chloride	AlCl₃	1	3	+3	−1	9.0	0.337
Iron(III) Chloride	FeCl₃	1	3	+3	−1	9.0	0.319
Zinc Sulfate	ZnSO₄	1	1	+2	−2	5.0	0.150

Frequently Asked Questions

The Debye-Hückel limiting law is accurate only below approximately 0.005-0.01 mol/L ionic strength. The Extended Debye-Hückel equation extends validity to about 0.1 mol/L by incorporating the ion-size parameter. The Davies equation is a semi-empirical extension usable to roughly 0.5 mol/L. Beyond 0.5 mol/L, none of these models should be trusted; consider Pitzer equations or Specific Ion Interaction Theory (SIT) for concentrated solutions.

Both constants depend on the dielectric constant (relative permittivity) and density of the solvent. For water, the dielectric constant decreases from approximately 87.7 at 0°C to 55.3 at 100°C. As temperature increases, constant A increases (roughly from 0.491 at 0°C to 0.600 at 100°C), meaning activity coefficients deviate further from unity at higher temperatures. This calculator uses an empirical polynomial fit for the dielectric constant of water to recompute A and B at any temperature between 0°C and 100°C.

The parameter a in the Extended Debye-Hückel equation represents the distance of closest approach between ions in angstroms (Å). It is not the crystallographic ionic radius but an empirical fit. Values range from about 3 Å for large, singly charged ions (K⁺, Cl⁻) to 9 Å for small, highly hydrated ions (H⁺, Al³⁺). The reference table on this page lists commonly accepted values. When uncertain, use 3-4 Å for 1:1 electrolytes.

In real experimental data, activity coefficients for many electrolytes pass through a minimum and then exceed 1.0 at high molality (typically above 1-3 mol/kg). This phenomenon arises from short-range ion-solvent interactions and reduced water activity, which none of the three models here capture. The Davies equation may predict γ > 1 at high I due to its linear correction term, but this is coincidental rather than physically meaningful. For reliable high-concentration predictions, use Pitzer's ion-interaction model.

For an electrolyte dissociating into ν₊ cations and ν₋ anions, the mean activity coefficient is defined as γ± = (γ₊^ν₊ · γ₋^ν₋)^(1/(ν₊+ν₋)). Individual ionic activity coefficients cannot be measured independently (they require a non-thermodynamic assumption). All three models computed here yield the mean activity coefficient directly through the |z₊·z₋| product.

Partially. The ionic strength I should be computed from all ions present in solution, which you can do manually and enter as a custom value. However, the models assume a single electrolyte in a pure solvent. For true multi-component modeling (e.g., seawater with Na⁺, Mg²⁺, Cl⁻, SO₄²⁻ simultaneously), you need the Pitzer formalism with binary and ternary interaction parameters, which is beyond the scope of these three equations.