Activity coefficients, solubility product, and pH

Real electrolyte solutions are not ideal: ion–ion electrostatic interactions mean the Nernst equation must be written with activities $a_i = \gamma_i c_i$, not raw concentrations, where $\gamma_i$ is the single-ion activity coefficient. Because a single ion's activity coefficient can never be measured alone (you cannot isolate one sign of charge in solution), the experimentally accessible quantity for an electrolyte that dissociates into $\nu_+$ cations and $\nu_-$ anions is the mean ionic activity coefficient:

where: $\gamma_+$ and $\gamma_-$ are the (individually inaccessible) cation and anion activity coefficients, and $\nu_+,\nu_-$ are the numbers of cations and anions produced per formula unit. $\gamma_\pm$ is found from cell EMF by computing the Nernst equation "as if" $\gamma_\pm = 1$ and attributing the gap between that ideal prediction and the measured $E$ to the activity-coefficient correction. Theoretically, dilute-solution electrostatics predicts $\gamma_\pm$ directly via the Debye–Hückel limiting law:

where: $A \approx 0.509\ (\text{mol}^{-1}\text{L})^{1/2}$ for water at $298\ \text{K}$, $z_+,z_-$ are the ionic charges, and $I = \tfrac12\sum_i c_i z_i^2$ is the ionic strength, summed over every ion present in solution (spectators included). The law explains why measured EMF of real electrolyte cells departs from the ideal, concentration-only Nernst prediction, and it is trustworthy only for dilute solutions, roughly $I \lesssim 0.01\ \text{mol L}^{-1}$; beyond that, higher-order corrections are needed. The same inversion trick, applied to a different electrode pair, gives $K_{sp}$. Build a concentration-cell-like arrangement: one half-cell is a metal $M$ dipped in a saturated solution of a sparingly soluble salt $MX$ (so $[\text{M}^{n+}]$ is pinned by the dissolution equilibrium, and is tiny), the other half-cell is the same metal in a reference solution of known $[\text{M}^{n+}]$. The figure below shows this setup for silver and silver chloride, the running example of this article.

Measuring $E_{cell}$ and applying the Nernst equation solves for the unknown $[\text{M}^{n+}]$; the dissolution stoichiometry then gives $[\text{X}^{n-}]$, and $K_{sp} = [\text{M}^{n+}][\text{X}^{n-}]$ follows. The full working is the derivation below.

pH from EMF uses the same inversion once more, this time on an electrode whose potential responds only to $\text{H}^+$ activity: the hydrogen electrode ($E = -0.0592\,\text{pH}$ at $298\ \text{K}$ against SHE, accurate but experimentally fiddly), the quinhydrone electrode (simpler, but unreliable above pH $\approx 8$), and the practical standard, the glass electrode, whose boundary potential is linear in pH, $E = \text{const} - 0.0592\,\text{pH}$ at $298\ \text{K}$, with the constant fixed by a buffer calibration.

Problem. A saturated $\text{AgCl}$ solution is in contact with a silver electrode, forming the anode of a concentration cell against a $\text{Ag}\mid\text{Ag}^+(0.10\ \text{M})$ reference cathode. The measured $E_{cell} = 0.450\ \text{V}$ at $298\ \text{K}$. Find $[\text{Ag}^+]$ in the saturated solution and $K_{sp}(\text{AgCl})$.

Step 1. Apply the Nernst equation from the derivation with $n=1$: $E_{cell} = 0.0592\log_{10}\dfrac{0.10}{[\text{Ag}^+]_{sat}}$, so $0.450 = 0.0592\log_{10}\left(\dfrac{0.10}{[\text{Ag}^+]_{sat}}\right)$.

Step 2. Solve for the log term: $\log_{10}\left(\dfrac{0.10}{[\text{Ag}^+]_{sat}}\right) = \dfrac{0.450}{0.0592} = 7.60$.

Step 3. Invert: $\dfrac{0.10}{[\text{Ag}^+]_{sat}} = 10^{7.60} = 3.98\times10^{7}$, giving $[\text{Ag}^+]_{sat} = \dfrac{0.10}{3.98\times10^{7}} = 2.51\times10^{-9}\ \text{M}$.

Step 4. Apply the 1:1 stoichiometry of $\text{AgCl} \rightleftharpoons \text{Ag}^+ + \text{Cl}^-$: $[\text{Cl}^-] = [\text{Ag}^+]_{sat}$, so $K_{sp} = [\text{Ag}^+][\text{Cl}^-] = (2.51\times10^{-9})^2 = \mathbf{6.3\times10^{-18}}$. (This is smaller than the accepted textbook $K_{sp}(\text{AgCl}) \approx 1.8\times10^{-10}$; the numbers illustrate the method, not a calibrated real system.)

Competitive stretch

Convert the molar solubility into a mass concentration: with molar mass $M(\text{AgCl}) = 143.5\ \text{g/mol}$, the solubility is $s = [\text{Ag}^+]_{sat}\times M = 2.51\times10^{-9}\times143.5 = 3.60\times10^{-7}\ \text{g/L}$, and $\text{pK}_{sp} = -\log_{10}(6.3\times10^{-18}) = \mathbf{17.20}$ — a compact way to report a very small equilibrium constant.

• Using raw molar concentration instead of activity ($a_i = \gamma_i c_i$) once the solution is not dilute or the ions are multiply charged — the Debye–Hückel correction is not optional above a few millimolar.
• Forgetting that ionic strength $I = \tfrac12\sum_i c_i z_i^2$ sums over all ions in solution, including spectator ions, not just the salt of interest.
• Assuming $[\text{cation}] = [\text{anion}]$ in every $K_{sp}$-from-EMF problem — this only holds for 1:1 salts like $\text{AgCl}$; $\text{Ag}_2\text{CrO}_4$ needs $[\text{CrO}_4^{2-}] = \tfrac12[\text{Ag}^+]$.
• Treating a glass electrode's calibration constant as fixed forever — it must be re-established with a buffer for each electrode and each session.

Ask the tutor: if the stoichiometric step for a non-1:1 salt like $\text{Ag}_2\text{CrO}_4$ or $\text{PbCl}_2$ feels shaky, ask the AI tutor to walk through one such example alongside the AgCl calculation above.