Solution Ionic Strength (I) & Debye-Huckel Activity Coefficient

List all ions, including counterions

Concentration-based ionic strength is half the sum of each ion concentration times its squared charge number. Enter individual dissolved species, not undissociated salt concentrations. This example uses mol/L, complete dissociation of illustrative NaCl and MgSO4, and no other ionic species. Speciation or ion pairing requires a separate calculation.

The four rows include Na+, Cl−, Mg2+ and SO4²−. At the starting NaCl concentrations of 0.0001 mol/L each, ionic strength is 0.0001 mol/L. Adding 0.0001 mol/L each of Mg2+ and SO4²− raises it to 0.0005 mol/L. Both ions of each salt must be included.

Concentration-based ionic strength

B2:B5 contains nonnegative concentrations; C2:C5 contains nonzero integer charge numbers. G2 retains the raw sum in mol/L. The arithmetic remains defined for an incomplete list, so the activity calculation separately checks charge balance.

=IF(COUNT(B2:C5)<>8,"Enter numeric ion inputs",IF(OR(MIN(B2:B5)<0,C2<>INT(C2),C3<>INT(C3),C4<>INT(C4),C5<>INT(C5),C2=0,C3=0,C4=0,C5=0),"Invalid ion inputs",0.5*SUMPRODUCT(B2:B5,C2:C5^2)))

Dilute-water limiting-law estimate

G3 evaluates 10^(-0.509 z² SQRT(I)) with target charge z in E2 and numerical ionic strength in mol/L. The coefficient is an approximate dilute-water value at 25 °C under this concentration convention. Do not mix concentration and molality conventions or apply this constant to other solvents and temperatures.

This is a limiting-law model, not an exact finite-concentration result. Single-ion coefficients are model-dependent; independently measurable electrolyte quantities generally involve mean ionic activity coefficients. Thermodynamic activity is dimensionless relative to a standard state, rather than simply a concentration with a smaller numerical value.

The exercise rejects I above 0.001 mol/L and charge imbalance exceeding 1e-9 mol/L of charge. These are explicit teaching guards, not a universal accuracy criterion or chemical validation. At I = 0, the formula gives the infinite-dilution limit 1, not a model of the ion composition of pure water.

=IF(NOT(ISNUMBER(E2)),"Enter numeric target charge",IF(OR(E2=0,E2<>INT(E2)),"Invalid target charge",IF(NOT(ISNUMBER(G2)),G2,IF(ABS(SUMPRODUCT(B2:B5,C2:C5))>0.000000001,"Check charge balance",IF(G2>0.001,"Outside this dilute exercise",10^(-0.509*E2^2*SQRT(G2)))))))

Experiment with concentration and charge

Change E2 from 1 to 2: squared charge makes the reduction stronger. Changing only its sign leaves the estimate unchanged. Add equal Mg2+ and SO4²− concentrations; omitting the counterion triggers the charge-balance message. Raising both NaCl concentrations to 0.02 mol/L still gives an ionic strength but stops the activity estimate.

Further reading

• IUPAC Gold Book — Debye–Hückel equation
• University of Kiel — Debye–Hückel theory