Stability constants of complexes — chelate effect

1. What is a stability constant?

WHY stepwise? Because a metal cannot grab all ligands simultaneously in one event; statistically it picks them up sequentially. Each step has its own equilibrium.

$M + L \rightleftharpoons ML, \quad K_1 = \frac{[ML]}{[M][L]}$ $ML + L \rightleftharpoons ML_2, \quad K_2 = \frac{[ML_2]}{[ML][L]}$

HOW are $\beta$ and $K$ related? Multiply the steps. For $ML_2$:

$\beta_2 = \frac{[ML_2]}{[M][L]^2} = \frac{[ML]}{[M][L]}\cdot\frac{[ML_2]}{[ML][L]} = K_1 K_2$

In general:

WHY does $K_i$ usually decrease as $i$ grows? Three reasons:

1. Statistical — fewer empty sites left to fill, more filled sites to lose a ligand from.
2. Electrostatic/steric — each added ligand (often charged) crowds and repels the next.
3. Occasionally reversed by structural change (e.g. high→low spin), giving an anomalous $K_i$.

2. The chelate effect

Classic comparison (Ni²⁺, both have 6 N-donors):

$[\text{Ni(NH}_3)_6]^{2+}: \ \log\beta_6 \approx 8.6$ $[\text{Ni(en)}_3]^{2+}: \ \log\beta_3 \approx 18.3 \quad (\text{en} = \text{H}_2N\text{-CH}_2\text{-CH}_2\text{-NH}_2)$

Same donor atoms (six N), vastly more stable when they come in linked pairs (en is bidentate). WHY?

Derive it from thermodynamics

The driving quantity is Gibbs free energy: $\Delta G^\circ = -RT\ln\beta = \Delta H^\circ - T\Delta S^\circ$

So $\ln\beta = -\dfrac{\Delta H^\circ}{RT} + \dfrac{\Delta S^\circ}{R}$. A larger $\beta$ needs more negative $\Delta G$ ⇒ favoured by negative $\Delta H$ and positive $\Delta S$.

Compare the two reactions (water spectator shown explicitly):

$[\text{M(H}_2O)_6] + 6\,\text{NH}_3 \rightleftharpoons [\text{M(NH}_3)_6] + 6\,\text{H}_2O$ Particle count: 7 → 7 (1 complex + 6 NH₃ → 1 complex + 6 H₂O). No net change in number of free molecules.

$[\text{M(H}_2O)_6] + 3\,\text{en} \rightleftharpoons [\text{M(en)}_3] + 6\,\text{H}_2O$ Particle count: 4 → 7 (1 complex + 3 en → 1 complex + 6 H₂O). Number of free particles increases by 3.

$\Delta H^\circ$ is roughly similar in both cases (same M–N bonds being made), so the entropy term decides.

3. Ring size, denticity & macrocyclic effect

• More chelate rings ⇒ greater effect (EDTA, hexadentate, forms 5 rings ⇒ enormous $\beta$).
• Macrocyclic effect: a pre-organised ring ligand (e.g. crown ether, porphyrin) is even more stable than an open-chain chelate — it pays no entropy/enthalpy penalty to wrap up, because it's already shaped right.

4. Common mistakes (steel-manned)

5. Flashcards

Connections

• Coordination number and geometry
• Crystal Field Theory — CFSE (enthalpy contributions to ΔH°)
• Denticity and ligand classification
• EDTA complexometric titrations
• Gibbs free energy and equilibrium constant
• Entropy and the second law
• Macrocyclic and cryptand ligands

Concept Map

Hinglish (regional understanding)