Chemical Equilibrium

1. Chemical Equilibrium — Introduction

• For a reversible reaction R ⇌ P :
• At equilibrium, Rate_forward = Rate_backward.
• Concentrations of reactants and products become constant (not necessarily equal).

Rate vs Time graph :

• R_f decreases and R_b increases until they become equal — this is the only correct rate-vs-time graph at equilibrium.

Concentration vs Time graph :

• [R] decreases and [P] increases, then both become constant. At equilibrium [R] may be > [P], [R] = [P], or [R] < [P] — all three are possible.

2. Law of Mass Action & Equilibrium Constant

• Law of mass action : Rate ∝ [active mass (concentration)]^{stoichiometric coefficient}.

Derivation of K_eq :

Consider: A + 2B ⇌ C

• Rate_forward = k_f [A]¹ [B]²
• Rate_backward = k_b [C]
• At equilibrium, k_f[A][B]² = k_b[C]

Two ways to calculate K_eq :

• (i) K_eq = k_f (forward rate constant)k_b (backward rate constant)
• (ii) K_eq = [Products]^s.c. at equilibrium[Reactants]^s.c. at equilibrium

s.c. = stoichiometric coefficient.

Example :

2A(g) + B(g) ⇌ 3C(g)

3. K_c — Equilibrium Constant in Terms of Concentration

• When K_eq is expressed using molar concentrations of species whose concentrations change during the reaction, it is called K_c.

For A(g) ⇌ B(g) + 2C(g) :

Pure solids & liquids :

• Concentration of a pure solid or liquid = dM = constant, so it is excluded from K_c.

CaCO₃(s) ⇌ CaO(s) + CO₂(g)

Relation between concentration and pressure :

4. Relation between K_p and K_c

• For a general gaseous equilibrium, e.g. A(g) + 2B(g) ⇌ C(g), with Δn_g = 1 − 3 = −2
• Since concentration = nV = PRT  (from PV = nRT)

where Δn_g = (moles of gaseous products) − (moles of gaseous reactants); R = 0.0821 L·atm·mol⁻¹·K⁻¹; T in Kelvin.

• K_p is the equilibrium constant expressed in terms of partial pressures.

When is K_p = K_c ?

• When Δn_g = 0, or
• When RT = 1 (numerically)

Example – PCl₅ decomposition :

• PCl₅(g) ⇌ PCl₃(g) + Cl₂(g)   ;   Δn_g = 2 − 1 = +1

5. Writing K_c Expressions

• Only gaseous and aqueous species appear in the expression; pure solids and pure liquids are omitted (activity = 1).

6. Writing K_p Expressions

Solids are excluded from both K_c and K_p expressions.

7. Types of Chemical Equilibrium

• Homogeneous Equilibrium — all reactants and products in the same phase, e.g. PCl₅(g) ⇌ PCl₃(g) + Cl₂(g)
• Heterogeneous Equilibrium — species in more than one phase
• Multiple Equilibrium
• Simultaneous Equilibrium

8. Factors Affecting K — Stoichiometry

• If a balanced equation is multiplied by a factor n, the new equilibrium constant is raised to the power n.

9. Factors Affecting K — Reversing the Reaction

• If the reaction is written in the reverse direction, the new K is the reciprocal of the original.

10. Equilibrium Constant Manipulation

• When reactions are added, their equilibrium constants are multiplied.

11. Formation Constant

• The overall stability (formation) constant β of a complex is the product of its stepwise stability constants.

12. Reaction Quotient (Q) & Direction of Reaction

• Q has the same expression as K but uses instantaneous (non-equilibrium) concentrations.
• At t = 0 for PCl₅(g) ⇌ PCl₃(g) + Cl₂(g), if only PCl₅ is present: Q = 0.

• Q rises from its initial value and levels off at K as the system reaches equilibrium.

13. Properties of Equilibrium Constant

• K_p (equilibrium constant in terms of partial pressure) depends only on Temperature.
• It is independent of catalyst, total pressure, or initial amounts of reactants.
• When K = 1, the equilibrium concentration of reactants is equal to the concentration of products.

14. Degree of Dissociation (α)

• For reaction PCl₅(g) ⇌ PCl₃(g) + Cl₂(g):

• K_c = [PCl₃][Cl₂][PCl₅] = (x/V)(x/V)(a-x)/V
• Degree of Dissociation (α) = moles dissociated at equilibriuminitial moles = xa

15. Equilibrium Expressions in terms of α and P_T

• General expression for A(g) ⇌ B(g) + C(g): K_p = α²1-α² × P_T
• For N₂O₄(g) ⇌ 2NO₂(g):

16. Degree of Dissociation & Pressure Dependence

• For reaction A(g) ⇌ B(g) + C(g) with total pressure p and degree of dissociation α:
• K_p = [(αp)/(1+α)]² / [(1-α)p/(1+α)] = (α²p) / (1-α²).
• When total pressure p increases, α must decrease to keep K_p constant.

17. Dissociation of AB₂

• For the dissociation 2AB₂(g) ⇌ 2AB(g) + B₂(g), if the degree of dissociation 'x' is small compared to 1 (x << 1):

Where K_p is the equilibrium constant and P is the total pressure.

18. Vapour Density and Degree of Dissociation

• Consider the dissociation equilibrium: A_n ⇌ nA

• Total moles at equilibrium = a(1 − α) + anα = a[1 + (n − 1)α]
• Since Vapour Density (V.D.) is inversely proportional to total moles for a given mass, the ratio of initial V.D. (D) to equilibrium V.D. (d) is:

• Solving for degree of dissociation (α):

where D = initial V.D. (theoretical), d = equilibrium V.D. (observed), and n = total gaseous moles of product per mole of reactant.

• Relation using Molar Masses (M_th = theoretical molar mass, M_obs = observed molar mass):

19. Graphs of Dissociation

• For the dissociation N₂O₄(g) ⇌ 2NO₂(g), the relationship between α and the ratio (d/D) is represented by a straight line.
• The graph shows α = 1 at (d/D) = 0.5 and α = 0 at (d/D) = 1. The degree of dissociation α decreases as the ratio (d/D) increases.

20. Heterogeneous Equilibrium

• Equilibria involving solids are generally heterogeneous.
• Example: CaCO₃(s) ⇌ CaO(s) + CO₂(g)

• Note: The equilibrium pressure of CO₂ depends only on temperature.

21. Heterogeneous Equilibrium (Solid-Gas)

• Whenever a solid alone is dissociated in a closed container producing only gaseous products, the equilibrium constant K_p is determined by the total pressure.

22. Simultaneous Equilibria: Dissociation of Two Solids

• When two solids A and D dissociate simultaneously:

• Let p_C = p. Since C is produced by both reactions, the partial pressures are related by p_B = x/p and p_E = y/p.
• Total pressure (P_T) = p_B + p_C + p_E = xp + p + yp = x + y + p²p.
• C comes from both solids, so p_C = p_B + p_E ⇒ p = x/p + y/p ⇒ p_C = √(x + y) and P_T = 2p_C.

23. Factors Affecting K — Temperature (van 't Hoff Equation)

• Rate constants: k_F = A_F · e^{−(E_a)_F/RT}  ;  k_b = A_b · e^{−(E_a)_b/RT}
• k_Fk_b = A_FA_b · e^−ΔH/RT

Two-temperature form (van 't Hoff) :

Compare with the Arrhenius two-temperature form: log k_T₂k_T₁ = E_a2.303 R [ 1T₁ − 1T₂ ]

24. Effect of Temperature on K — Endothermic vs Exothermic

Endothermic equilibrium (ΔH > 0) :

• Temperature increases → K_T₂ > K_T₁ (K increases significantly — exponential dependence)
• Temperature decreases → K_T₂ < K_T₁

Exothermic equilibrium (ΔH < 0) :

• Temperature increases → K_T₂ < K_T₁
• Temperature decreases → K_T₂ > K_T₁

25. Gibbs Energy & Equilibrium Constant

• At equilibrium: ΔG = 0 and Q = K

• Substituting ΔG° = ΔH° − TΔS° :

Graph of ln K vs 1/T :

• Straight line: y = mx + c form
• Slope = −ΔH°/R
• y-intercept = ΔS°/R

26. Intercepts and Slopes

27. Le Chatelier's Principle

• If a system at equilibrium is subjected to a change of pressure, temperature, or number of moles of substance, there will be a tendency for a net reaction in the direction that tends to reduce the effect of this change.

28. Effect of Concentration Change

• For reaction: PCl₅ (g) ⇌ PCl₃ (g) + Cl₂ (g)
• Increase in reactant concentration: Reaction moves in forward direction.
• Increase in product concentration: Reaction moves in backward direction.
• Key Note: If the reactant or product is a solid (s) or pure liquid (l), change in its concentration has no effect on the equilibrium position.

29. Effect of Pressure Change

• For reaction: PCl₅ (g) ⇌ PCl₃ (g) + Cl₂ (g)
• Reaction quotient Q = [PCl₃][Cl₂][PCl₅] = bca × 1V
• Pressure increases (Volume decreases): Reaction moves backward.
• Pressure decreases (Volume increases): Reaction moves forward.

30. Addition of Inert Gas

Constant Volume :

• Addition of inert gas at constant volume has no effect on the equilibrium state.
• Partial pressure of species: P_(i) = X_(i) × P_(T).

Constant Pressure :

• For reaction: PCl₅ (g) ⇌ PCl₃ (g) + Cl₂ (g)
• Addition of inert gas at constant pressure causes the volume to increase.
• Since volume increases, the reaction moves in the forward direction.

31. Equilibrium under Inert Gas Addition & Pressure Change

• Addition of inert gas at constant Volume: Total pressure increases but partial pressures of reactants/products remain constant; equilibrium does not shift.
• Addition of inert gas at constant Pressure: Volume of the system increases. The equilibrium shifts towards the side with a greater number of gaseous moles.
• Effect of Pressure change:
  • Increasing total pressure shifts the equilibrium towards the side with fewer gaseous moles.
  • Decreasing total pressure shifts the equilibrium towards the side with more gaseous moles.
• Condensed phases (solids/liquids): For processes like ice ⇌ water, increasing pressure favors the formation of the denser phase (Le Chatelier's Principle).

32. Effect of Catalyst

• Addition of a catalyst has no change on the equilibrium position.

33. Effect of Temperature on Equilibrium

• Since K_eq depends exponentially on T, the direction of equilibrium shift depends on the sign of ΔH :

34. Effect of Pressure on Physical Equilibrium

• For any physical equilibrium, increasing the pressure shifts the equilibrium in the direction of the physical state having higher density.

Examples of Physical Equilibria :

• T ↑ : Forward (more ice melts).
• P ↑ : Forward (since density of H₂O(l) > density of H₂O(s), more ice melts).

• T ↑ : Forward.
• P ↑ : Backward.

• T ↑ : Forward (endothermic).
• P ↑ : Forward (diamond is denser than graphite).

35. Quick Formula Revision