How to use the ionic strength and activity coefficient calculator
- Choose the activity model: the Debye-Huckel limiting law, the extended Debye-Huckel law or the Davies equation.
- For the extended model, enter an ion size in angstroms. The page starts with 4 as an example, and the same value is used for every ion.
- List each ion in the solution with its concentration in moles per liter and its charge, for example 1 for sodium and -1 for chloride.
- Read the ionic strength, then the activity coefficient and the activity of each ion.
Formula
Ionic strength: I = 1/2 x sum of c x z^2 over every ion, with c in mol/L
Limiting law: log10(gamma) = -0.509 z^2 sqrt(I)
Extended law: log10(gamma) = -0.509 z^2 sqrt(I) / (1 + 0.328 a sqrt(I)), with the ion size a in angstroms
Davies: log10(gamma) = -0.509 z^2 (sqrt(I) / (1 + sqrt(I)) - 0.3 I). Activity = gamma x concentration.
The constants 0.509 and 0.328 are the Debye-Huckel values for water at 25 C. At another temperature they change a little.
0.1 M sodium chloride and 0.1 M calcium chloride
For 0.1 M NaCl the ions are Na+ at 0.1 M with charge 1 and Cl- at 0.1 M with charge -1. The ionic strength is half of 0.1 x 1 + 0.1 x 1 = 0.1 mol/L. With an ion size of 4 angstroms, the extended law gives an activity coefficient of 0.76955 for each ion, so the activity of Na+ is 0.076955 mol/L. The limiting law gives 0.6903, and the Davies equation gives 0.7816.
For 0.1 M CaCl2 the ions are Ca2+ at 0.1 M and Cl- at 0.2 M. The ionic strength is half of 0.1 x 4 + 0.2 x 1 = 0.3 mol/L. The extended law gives 0.22445 for Ca2+ and 0.68831 for Cl-, which shows how strongly a double charge is pulled down at the same ionic strength.
Why activities matter
In a dilute solution an ion behaves as if its concentration were its activity. In a salty solution the ions attract the oppositely charged ones around them, and the effective concentration is lower. The activity coefficient measures that drop. It is close to 1 when the ionic strength is very low and falls as the ionic strength rises, faster for ions with a bigger charge.
Equilibrium constants are defined with activities. When you use concentrations instead, solubility and acid strength change with the salt content. See the Ksp solubility calculator and the pH and weak acid calculator.
Where each model works
The limiting law is exact only as the ionic strength goes to zero, and it is usually trusted up to about 0.005 mol/L. The extended law reaches about 0.1 mol/L, and the Davies equation is often used up to about 0.5 mol/L. The page warns you when you are above those ranges, and the values then are rough guides.
List every ion that is present, because all of them count toward the ionic strength. If the charges do not add up to zero, the page tells you that an ion is probably missing. To prepare solutions with a known concentration, use the solution dilution calculator.
Assumptions and limits
- Constants are those for water at 25 C. The ion size is the same for every ion in the extended model.
- Each ion is listed with its own concentration and a non-zero charge.
- Ion pairs and complex ions are not modeled. They lower the free ion concentrations.
- Ranges of use for each model are common guides and not strict limits.
Frequently asked questions
What is ionic strength?
It is a measure of the total charge in a solution: half the sum of each ion's concentration times its charge squared.
What is an activity coefficient?
It is the factor that turns the concentration of an ion into its activity, which is the effective concentration. It is 1 for an ideal solution and below 1 for real salty solutions.
Which model should I use?
Use the limiting law for very dilute solutions, the extended law up to about 0.1 mol/L and the Davies equation for higher values. Beyond that, measured coefficients are better.
Why does a charge of 2 matter so much?
The charge enters as z squared, so an ion with charge 2 feels four times the effect of an ion with charge 1 at the same ionic strength.