Chemical reaction equilibrium and equilibrium constant

Why it matters

Kinetics tells you how fast a reactor gets somewhere; equilibrium tells you where it can possibly get. The equilibrium conversion caps the yield of ammonia, methanol, sulfur trioxide or hydrogen from the water-gas shift, and it decides operating temperature, pressure and feed ratio before any reactor is sized. All of it follows from one number per reaction: the equilibrium constant.

Key ideas

Reaction coordinate. For a reaction Σνᵢ Aᵢ = 0 (νᵢ positive for products, negative for reactants), the amount of every species is tied to a single variable, the extent of reaction ε: nᵢ = nᵢ₀ + νᵢε and n = n₀ + νε, with ν = Σνᵢ. Mole fractions are then yᵢ = (nᵢ₀ + νᵢε)/(n₀ + νε). Every equilibrium problem reduces to finding ε.

Criterion of equilibrium. At constant T and P the total Gibbs energy is a minimum with respect to ε, which gives Σνᵢμᵢ = 0.

Standard states and the equilibrium constant. Writing μᵢ = Gᵢ° + RT ln(f̂ᵢ/fᵢ°) gives Π(f̂ᵢ/fᵢ°)^νᵢ = exp(−ΔG°/RT) ≡ K. The standard state is the pure species at the system temperature and a fixed standard pressure P° (1 bar): ideal gas for gases, pure liquid or solid for condensed species. Because the standard state is at fixed pressure, K depends on temperature only, never on pressure or composition. K is dimensionless.

ΔG° from tables. ΔG° = Σνᵢ ΔG°f,ᵢ, using standard Gibbs energies of formation (tabulated at 298.15 K; take them from a data book). ΔG° strongly negative means K ≫ 1 and products favoured; strongly positive means K ≪ 1.

Gas-phase reactions. With f̂ᵢ = yᵢφ̂ᵢP and fᵢ° = P°: Π(yᵢφ̂ᵢ)^νᵢ = K(P/P°)^(−ν). For an ideal-gas mixture φ̂ᵢ = 1: Πyᵢ^νᵢ = K(P/P°)^(−ν). Pressure therefore affects the composition (not K) whenever ν ≠ 0.

Liquid-phase and heterogeneous reactions. For species in a liquid solution the activity is aᵢ = xᵢγᵢ (pressure effect on liquids usually neglected). A pure solid or pure liquid has activity 1, so it drops out of K: for CaCO₃(s) ⇌ CaO(s) + CO₂(g), K = P_CO₂/P° (ideal gas).

Kc, Kp and K. Kp (in pressure units) and Kc (in concentration units) used in general chemistry are related to the dimensionless thermodynamic K through the standard pressure or concentration; in this subject use K with P/P° to avoid unit errors.

Le Chatelier as a check. Raising pressure favours the side with fewer gas moles; adding an inert at constant P acts like lowering P; excess of one reactant pushes the conversion of the other up. These follow directly from the yᵢ expression and are useful sanity checks on the algebra.

Formulas

nᵢ = nᵢ₀ + νᵢ·ε, n = n₀ + ν·ε, ν = Σνᵢ yᵢ = (nᵢ₀ + νᵢ·ε)/(n₀ + ν·ε) Σ νᵢ·μᵢ = 0 (equilibrium) ΔG° = Σ νᵢ·ΔG°f,ᵢ K = exp(−ΔG°/(R·T)), ln K = −ΔG°/(R·T) Π (f̂ᵢ/fᵢ°)^νᵢ = K Π (yᵢ·φ̂ᵢ)^νᵢ = K·(P/P°)^(−ν) (gases) Π yᵢ^νᵢ = K·(P/P°)^(−ν) (ideal gases) Π (xᵢ·γᵢ)^νᵢ = K (liquid solution)

Symbols: νᵢ stoichiometric number (dimensionless); ε extent of reaction (mol); nᵢ₀ initial moles (mol); yᵢ, xᵢ mole fractions; μᵢ chemical potential (J/mol); ΔG° standard Gibbs energy change of reaction (J/mol); ΔG°f standard Gibbs energy of formation (J/mol); K equilibrium constant (dimensionless, function of T only); P (bar); P° = 1 bar; φ̂ᵢ fugacity coefficient in mixture; γᵢ activity coefficient; R = 8.314 J/(mol·K); T (K).

Worked examples

Example 1 (standard): K from Gibbs energies of formation. Given: water-gas shift CO(g) + H₂O(g) ⇌ CO₂(g) + H₂(g) at 298.15 K, with ΔG°f (kJ/mol): CO −137.17, H₂O(g) −228.57, CO₂ −394.36, H₂ 0.

1. ΔG° = Σνᵢ ΔG°f,ᵢ = (−394.36 + 0) − (−137.17 − 228.57) = −28.62 kJ/mol.
2. ln K = −ΔG°/RT = 28 620/(8.314 × 298.15) = 11.546.
3. K = e^11.546 = 1.03 × 10⁵. Answer: ΔG° = −28.6 kJ/mol, K ≈ 1.0 × 10⁵ at 298 K (reaction strongly favoured at room temperature, though too slow without a catalyst).

Example 2 (GATE level): equilibrium conversion and the effect of pressure and feed ratio. (a) N₂O₄(g) ⇌ 2NO₂(g), K = 0.15 at the reaction temperature, pure N₂O₄ feed, ideal gases. Find the fractional dissociation at 1 bar and at 5 bar.

1. Basis 1 mol N₂O₄: n_N₂O₄ = 1 − ε, n_NO₂ = 2ε, n = 1 + ε; ν = +1.
2. Πyᵢ^νᵢ = K(P/P°)^(−ν): [2ε/(1 + ε)]²/[(1 − ε)/(1 + ε)] = K/P, i.e. 4ε²/(1 − ε²) = K/P.
3. Solve: ε² = K/(K + 4P).
4. At 1 bar: ε = √(0.15/4.15) = 0.190. At 5 bar: ε = √(0.15/20.15) = 0.0863. (b) Water-gas shift at a temperature where K = 1.0. Find the CO conversion for feeds of (i) 1 mol CO + 1 mol H₂O, (ii) 1 mol CO + 2 mol H₂O.
5. ν = 0, so pressure has no effect, and total moles cancel: K = ε²/[(1 − ε)(n_H₂O,0 − ε)].
6. (i) ε²/(1 − ε)² = 1, so ε/(1 − ε) = 1 and ε = 0.5.
7. (ii) ε² = (1 − ε)(2 − ε) = 2 − 3ε + ε², so 3ε = 2 and ε = 0.667. Answer: (a) 19.0% dissociated at 1 bar, 8.6% at 5 bar; (b) CO conversion 50% with equimolar feed, 66.7% with 2:1 steam

Common mistakes

• Saying K changes with pressure; it depends on T only. Pressure changes the composition through (P/P°)^(−ν).
• Forgetting that total moles change with ε when ν ≠ 0, and writing yᵢ = nᵢ/n₀.
• Including pure solids or liquids in the K expression instead of setting their activity to 1.
• Sign errors in ΔG° = Σνᵢ ΔG°f,ᵢ (products minus reactants), or using kJ with R in J.
• Mixing up a K written for 2NH₃ with one written for NH₃: halving the stoichiometry takes the square root of K.

For GATE CH

• NAT problems: K from ΔG°f data; equilibrium conversion or mole fractions for ideal-gas reactions at given P; effect of inerts, excess reactant or pressure.
• Heterogeneous equilibria (decomposition of solids) and liquid-phase esterification-type problems with activities.
• Conceptual MCQs on standard states, why K is independent of pressure, and Le Chatelier checks.
• Practise setting up the ε table quickly and reducing it to a single algebraic equation.

Quick check

1. On what does the thermodynamic equilibrium constant depend?
2. For A(g) ⇌ 2B(g), does increasing pressure raise or lower the equilibrium conversion?
3. If ΔG° = 0, what is K?
4. What is the activity of a pure solid?
5. The equilibrium constant for 2A ⇌ B is 16. What is it for A ⇌ ½B?

Answers: 1. Temperature only (for a chosen standard state). 2. Lowers it. 3. 1. 4. 1. 5. 4.