15.3 Coupled Equilibria

Solubility and Coupled Equilibria

Coupled equilibria describe what happens when two or more chemical reactions share a common species and influence each other's equilibrium positions. These interconnected reactions help explain real-world observations, like why some minerals dissolve more easily in acidic groundwater or why adding ammonia to a solution can dissolve a precipitate that water alone can't.

Understanding coupled equilibria ties together several concepts you've already learned ($K_{\text{sp}}$, Le Chatelier's principle, acid-base chemistry) and shows how they work together in more complex systems.

Effects of Coupled Equilibria

Coupled equilibria occur when two or more reactions share a common species, so a shift in one equilibrium forces the other to adjust. Le Chatelier's principle is the driving idea here: if a shared ion is added or removed, both equilibria respond.

Common ion effect (decreases solubility): Adding an ion that's already part of a dissolution equilibrium pushes that equilibrium back toward the solid, reducing solubility. For example, dissolving $\text{NaCl}$ in a solution that already contains $\text{HCl}$ means there's extra $\text{Cl}^-$ in solution. That extra chloride shifts the $\text{NaCl}$ dissolution equilibrium to the left, so less $\text{NaCl}$ dissolves than it would in pure water.

Complex ion formation (increases solubility): If a ligand in solution reacts with a dissolved metal ion to form a complex ion, it effectively removes free metal ions from solution. This shifts the dissolution equilibrium to the right, pulling more solid into solution. A classic example: $\text{AgCl}$ is nearly insoluble in water, but adding $\text{NH}_3$ forms the complex ion $\text{Ag(NH}_3)_2^+$, which draws $\text{Ag}^+$ out of the dissolution equilibrium and increases the overall solubility of $\text{AgCl}$.

Calculations for Interconnected Equilibria

When you need to find equilibrium concentrations in a coupled system, follow these steps:

1. Write balanced equations for each equilibrium reaction involved (dissolution, complex ion formation, acid-base, etc.).
2. Identify the common species shared between the reactions. This is the link that couples them.
3. Write the equilibrium constant expression for each reaction ($K_{\text{sp}}$, $K_f$, $K_a$, $K_b$, etc.) and note the given values.
4. Combine the equations if possible. When you add two reactions together, the overall equilibrium constant is the product of the individual constants: $K_{\text{overall}} = K_1 \times K_2$.
5. Set up an ICE table (or system of equations) using the combined reaction and solve for the unknown concentrations.

For example, to find the solubility of $\text{AgCl}$ in $\text{NH}_3$, you'd combine the $K_{\text{sp}}$ expression for $\text{AgCl}$ with the $K_f$ expression for $\text{Ag(NH}_3)_2^+$. Multiplying these gives $K_{\text{overall}}$, which you can use in a single ICE table rather than solving two separate equilibria.

pH Impact on Solubility

pH changes can shift dissolution equilibria by reacting with the anion of a sparingly soluble salt. The key question is: does the anion act as a base that can be protonated?

• Salts with basic anions (like $\text{CO}_3^{2-}$, $\text{PO}_4^{3-}$, $\text{S}^{2-}$, $\text{OH}^-$, $\text{F}^-$) are more soluble in acidic solution. At low pH, $\text{H}^+$ protonates these anions (e.g., $\text{CO}_3^{2-} + \text{H}^+ \rightarrow \text{HCO}_3^-$), removing them from the dissolution equilibrium and shifting it to the right.
• Salts with neutral anions (like $\text{Cl}^-$, $\text{Br}^-$, $\text{NO}_3^-$) are not significantly affected by pH because these anions don't react with $\text{H}^+$.

pH also affects complex ion formation. Ligands that are weak bases, such as $\text{NH}_3$, exist in their useful unprotonated form at higher pH. At low pH, $\text{NH}_3$ gets protonated to $\text{NH}_4^+$, which doesn't act as a ligand, so complex ion formation is less effective.

To analyze pH effects on solubility:

1. Identify whether the anion in the salt is basic (conjugate base of a weak acid) or neutral.
2. Determine how pH changes the protonation state of that anion.
3. Consider whether any complex ion formation is also pH-dependent.
4. Predict the net effect on solubility from these combined shifts.

Related Concepts

• Equilibrium constant ($K$): Quantifies how far a reaction proceeds at equilibrium. Each coupled reaction has its own $K$, and combining reactions means multiplying their constants.
• Complexation: The formation of complex ions from a metal ion and ligands. High $K_f$ values mean the complex is very stable, which strongly drives dissolution of otherwise insoluble salts.
• Buffer solutions: Resist pH changes and can stabilize the solubility of pH-sensitive compounds by keeping the protonation state of anions and ligands constant.