4.4.5 Equilibrium Constant & Entropy

• The equation for calculating the total entropy change is:

ΔS^ꝋ_total = ΔS ^ꝋ_sys + ΔS^ꝋ_surr

(sys = system and surr = surroundings) 

• Remember that ΔS^ꝋ_total is positive for all spontaneous changes
• There is very little change in ΔS^ꝋ_sys with a change in temperature unless there is a change in the state of one of the reactants or products
• There will be a significant change in ΔS^ꝋ_surr however
• The entropy change of the surroundings during a chemical reaction is

• Where ΔH is the enthalpy change and T is the absolute temperature (measured in kelvin)
• We can use this information to determine whether a reaction is spontaneous at a given temperature

   Answer 1: at 293 K (20°C)

• ΔS_surroundings =  = -607.2 J K^-1 mol^-1
• ΔS_total(293K) = (+160.4 - 607.2) = -446.8 J K^-1 mol^-1
• The decomposition of calcium carbonate is not spontaneous at 293K

   Answer 2: at 1173K (900°C)

• ΔS_surroundings = = -151.7 J K^-1 mol^-1
• ΔS_total(1173) = (+160.4 - 151.7) = +8.7 J K-1 mol-1
• The decomposition of calcium carbonate is spontaneous when heated to 1173K

Relationship between entropy change and equilibrium constant

• For a reversible reaction that can reach equilibrium, the equilibrium position can be reached from either side of the reaction
• This means that both the forward and backward reactions are spontaneous
  • ΔS must be positive in both directions
• For example, consider the following reaction:

N₂O₄ (g) ⇌ 2NO₂ (g)

A graph of entropy against the percentage of NO₂ in a mixture of N₂O₄ and NO₂

• The entropy of N₂O₄ is less than the entropy of the equilibrium mixture
• The change in entropy from pure N₂O₄ to the equilibrium mixture is positive
  • The change is spontaneous
• The entropy change for NO₂ to the equilibrium mixture is also positive 
  • This change is also spontaneous
• The entropy change for a mixture of the gases in any proportions moving towards the equilibrium position is also positive
• Neither the forward nor backward reaction can go to completion as the entropy change from the equilibrium mixture to either the reactants or products is negative
• At equilibrium, the total entropy change is zero

ΔS_total [forward reaction] = ΔS_total [backward reaction]

• The relationship between the total entropy of the reaction and the equilibrium constant (K_c or K_p) is

ΔS_total = R lnK

Using total entropy change to calculate an equilibrium constant

• Rearranging the equation mentioned above we get:

lnK =

• Hence:

K =

Relationship between equilibrium constant and equilibrium position

• There is no hard rule for the relationship between equilibrium constant and the position of equilibrium 
• As a general rule, we can say a very large value of K suggests the equilibrium position is pushed towards the products (right-hand side)
• Similarly, a very small value of K suggests the equilibrium position is pushed towards the reactants (left-hand side)