Activity Coefficient Calculator
This activity coefficient calculator works out how far an ion in solution departs from ideal behaviour, using the ion charge and the ionic strength of the solution. Enter the charge on the ion and the ionic strength in moles per litre, and the calculator returns the activity coefficient from the Debye-Huckel limiting law, alongside the value from the Davies equation, which stays accurate to much higher ionic strength, and the base-10 logarithm that the models actually compute. An activity coefficient of 1 means the ion behaves ideally; values below 1 mean that inter-ionic attraction lowers the ion's effective concentration, so its activity is less than its stated molarity. Multiply a concentration by the activity coefficient to get the activity that drives equilibria, cell potentials and reaction rates. Chemistry students use this for equilibrium and electrochemistry problems, where using activities instead of concentrations is what separates a rough answer from a correct one, and analysts use it when ionic strength is high enough that concentration alone would mislead. All results assume aqueous solution at 25 degrees Celsius, where the Debye-Huckel constant A is 0.509. The limiting law is trustworthy only in very dilute solution, so for anything above about 0.001 mol per litre lean on the Davies figure, and treat both as models rather than exact truth once the solution gets concentrated.
Aqueous solution at 25C with A = 0.509. Limiting law: log10(gamma) = -0.509 z^2 sqrt(I), reliable only below about 0.001 mol/L. Davies: log10(gamma) = -0.509 z^2 [sqrt(I)/(1 + sqrt(I)) - 0.3 I], usable to about 0.5 mol/L. Models only.
How it works
The Debye-Huckel limiting law models the electrostatic cloud of counter-ions around each ion, giving log10(gamma) = -A z^2 sqrt(I), where A is 0.509 for water at 25 degrees Celsius, z is the ion charge and I is the ionic strength. Because charge enters as z squared, doubly charged ions deviate far more strongly than singly charged ones. The Davies equation keeps the same leading term but adds a denominator and a linear correction, log10(gamma) = -A z^2 [sqrt(I)/(1 + sqrt(I)) - 0.3 I], which bends the curve back up at higher ionic strength and matches measurements far better beyond the dilute limit. Taking 10 to the power of each log gives the activity coefficient itself.
Worked example
Take a singly charged ion, z = 1, in a solution with ionic strength I = 0.01 mol per litre. The square root of I is 0.1. The limiting law gives log10(gamma) = -0.509 x 1 x 0.1 = -0.0509, so gamma = 10^(-0.0509), which is about 0.889. The Davies equation gives sqrt(I)/(1 + sqrt(I)) - 0.3 I = 0.1/1.1 - 0.003 = 0.0879, so log10(gamma) = -0.509 x 0.0879 = -0.0447 and gamma = 0.902. The two models agree closely here because 0.01 mol per litre is still fairly dilute; they would diverge markedly at 0.5 mol per litre.
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