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In real solutions, ions are surrounded by a cloud of oppositely charged ions that partly screens their charge, so they behave as if they were less concentrated. The effective concentration is the activity, a = γc, where γ is the activity coefficient. The ionic strength, I = ½Σcᵢzᵢ², measures how crowded with charge a solution is, and the Debye–Hückel equation predicts how γ falls as I rises. Exact equilibrium constants are written with activities, which explains why adding an unrelated salt can change solubility and pH.
Calculate ionic strength for several solutions, noticing how the z² term makes a 2+ or 3+ ion count heavily. Then use the Debye–Hückel limiting law to estimate γ for ions of charge 1, 2 and 3 at the same ionic strength. Finish by redoing a Ksp calculation with activities and comparing the solubility with the concentration-only answer.
In introductory equilibrium problems you write Ksp, Ka and other constants with molar concentrations. That works well for very dilute solutions. In more concentrated ionic solutions, it starts to fail, and careful analytical work has to correct for it.
The reason is electrostatic. Each ion in solution attracts ions of opposite charge and repels ions of the same charge. On average, a cation sits in a slight excess of anions, called its ionic atmosphere. This atmosphere partly cancels the cation's charge, so the ion interacts less strongly than its concentration suggests. Chemists account for this with the activity, a = γc. The activity coefficient γ equals 1 in the limit of infinite dilution and falls below 1 as the solution becomes more crowded with charge.
The measure of crowding is the ionic strength:
I = ½ Σ cᵢzᵢ²
where cᵢ is the molar concentration of each ion and zᵢ its charge. Squaring the charge makes highly charged ions dominate. A 0.010 M NaCl solution has I = 0.010 M, but a 0.010 M MgCl₂ solution has I = 0.030 M.
Peter Debye and Erich Hückel derived how γ depends on I in 1923. For water at 25 °C, their limiting law is
log γ = −0.51 z² √I
It is accurate only at low ionic strength, roughly up to 0.01 M. An extended form adds a term for ion size and works to about 0.1 M. Two patterns follow from the equation. γ falls as ionic strength rises, and it falls much faster for ions of higher charge.
Activities matter because true equilibrium constants are written with them. For a sparingly soluble salt, Ksp = a(Ag⁺) × a(Cl⁻) = γ₊[Ag⁺] × γ₋[Cl⁻]. If you add an inert salt such as KNO₃, the γ values fall. To keep the product of activities fixed, the concentrations must rise, so more AgCl dissolves. This is the salt effect, and it acts in the opposite direction from the common-ion effect.
Activities also define measurements. The pH is formally −log a(H⁺), and a pH electrode responds to activity, not concentration. Ion-selective electrodes behave the same way, which is why analysts add an ionic-strength adjustment buffer so samples and standards share the same γ.