Chemical equilibrium and the equilibrium constant K
A chemical equilibrium is reached when the rates of the forward and reverse reactions are equal and macroscopic concentrations no longer change. This is a dynamic, not static, equilibrium.
For the general reaction: a A + b B ⇌ c C + d D
The thermodynamic equilibrium constant K is defined as:
K = ∏ (a_i)^{ν_i}
where a_i is the activity of species i and ν_i is its stoichiometric coefficient (positive for products, negative for reactants).
Activity conventions: - Gas: a_i = P_i / P° (partial pressure relative to P° = 100 kPa). - Solute: a_i = [i] / c° (concentration relative to c° = 1 mol·L⁻¹). - Pure liquid or solid: a_i = 1.
K is dimensionless and depends only on T.
Reaction quotient Q and predicting direction
The reaction quotient Q has the same expression as K but is evaluated at any concentrations (not necessarily equilibrium ones):
- Q < K: reaction proceeds in the forward direction (→ towards products).
- Q = K: system at equilibrium.
- Q > K: reaction proceeds in the reverse direction (← towards reactants).
This is Le Chatelier's principle restated in thermodynamic terms: any imbalance (Q ≠ K) drives the system to evolve until Q = K.
The K–ΔG° link: the fundamental equation
The relation between K and the standard Gibbs energy is:
ΔG° = −RT ln K ↔ K = exp(−ΔG°/RT)
where R = 8.314 J·mol⁻¹·K⁻¹ and T is the absolute temperature.
- Very negative ΔG° → K ≫ 1: essentially complete reaction.
- Very positive ΔG° → K ≪ 1: reaction barely occurs.
- ΔG° = 0 → K = 1: perfectly symmetric equilibrium.
The variation of ΔG with extent of reaction ξ is:
ΔG = ΔG° + RT ln Q
At equilibrium (minimum of G), ΔG = 0 and Q = K, which recovers ΔG° = −RT ln K.
Temperature dependence — van't Hoff equation
K depends on T according to the van't Hoff equation (integrated form):
ln(K₂/K₁) = −(ΔH°/R)(1/T₂ − 1/T₁)
or in differential form: d(ln K)/dT = ΔH°/RT²
- Exothermic reaction (ΔH° < 0): K decreases as T increases (shift towards reactants).
- Endothermic reaction (ΔH° > 0): K increases with T.
A plot of ln K vs 1/T (van't Hoff plot) is a straight line with slope −ΔH°/R — an experimental method to determine ΔH° and ΔS°.
Gas-phase and solution equilibria
In the gas phase: K_p (pressure basis) and K_c (concentration basis) are related by:
K_p = K_c · (RT/P°)^{Δn_gas}
In aqueous solution, acid-base equilibria are a special case: - Ka (weak acid): HA + H₂O ⇌ H₃O⁺ + A⁻ - Kb (weak base): B + H₂O ⇌ BH⁺ + OH⁻ - Ka · Kb = K_w = 10⁻¹⁴ at 25 °C (ionic product of water)
For a buffer solution (Henderson-Hasselbalch equation):
pH = pKa + log([A⁻]/[HA])
The thermodynamics of equilibria thus unifies the chemistry of gases, solutions, and heterogeneous reactions.