Equilibrium Law and the Equilibrium Constant

Cato Maximilian Guldberg (1836–1902) and Peter Waage (1833–1900) first proposed the equilibrium law in 1864.

Equilibrium law - the mathematical description of a chemical system at equilibrium. 

When a mixture of reactants and products of a reaction reaches equilibrium at a given temperature, its equilibrium constant always has the same value.

The equilibrium law is sometimes called the law of mass action.

Equilibrium Constant

A general equation for a reversible reaction may be written as follows:

mA + nB + ⇌ xC + yD

Forward rate: k_f [A]^m[B]ⁿ              Reverse rate: k_r [C]^x[D]^y

The square brackets indicate the concentrations of the chemical entities.

We can write the reaction quotient (Q) for this equation.

When evaluated using concentrations, it is called Q_c.

Q_c = [C]^x[D]^y / [A]^m[B]ⁿ

The reaction quotient is equal to the molar concentrations of the products of the chemical equation (multiplied together) over the reactants (also multiplied together), with each concentration raised to the power of the coefficient of that substance in the balanced chemical equation.

For example, the reaction quotient for the reversible reaction 2NO₂ (g) ⇌ N₂O₄ (g) is given by this expression:

Q_c = [N₂O₄] / [NO₂]²

The numeric value of reaction quotient (Q_c) for a given reaction varies -

Q_c depends on the concentrations of products and reactants present at the time when Q_c is determined.

When pure reactants are mixed, Q_c is initially zero because there are no products present at that point.

As the reaction proceeds, the value of Q_c increases as the concentrations of the products increase and the concentrations of the reactants simultaneously decrease.

When the reaction reaches equilibrium, the value of the reaction quotient no longer changes because the concentrations no longer change.

At equilibrium,

Foward rate = Reverse rate

k_f [A]^m[B]ⁿ = k_r [C]^x[D]^y

The ratio of rate constants is another constant.

The equilibrium constant, K_eq is the forward rate constant, kf, divided by the reverse rate constant, kr.

Q_c at equilibrium = k_f / k_r = K_eq = [C]^x[D]^y / [A]^m[B]ⁿ

The equilibrium constant, K, can be used to determine the equilibrium position of a chemical reaction system.

Homogeneous and Heterogeneous Equilibria

Homogeneous equilibrium - a chemical equilibrium system in which all reactants and products are in the same state of matter, such as the gas state.

Heterogeneous equilibrium - a chemical equilibrium system in which the reactants and products are present in at least two different states, such as gases and solids.

Calcium carbonate, CaCO₃ (s), can undergo a reversible decomposition reaction to form solid calcium oxide, CaO(s), and carbon dioxide gas:

CaCO₃ (s) ⇌ CaO (s) + CO₂ (g)

s - solid, g - gas

This reaction is used in the commercial preparation of calcium oxide (also known as lime) from calcium carbonate sources, such as limestone. Straightforward application of the equilibrium law leads to the equation:

K = [CaO (s)][ CO₂ (g)] / [CaCO₃ (s)]

However, the equilibrium position of a heterogeneous equilibrium does not depend on the quantities of pure solids or liquids.

The fundamental reason for this behaviour is that the concentrations of pure solids and liquids cannot change.

In the reversible decomposition reaction of calcium carbonate, the concentrations of solid calcium carbonate and solid calcium oxide remain constant.

Heterogeneous equilibrium of calcium carbonate

 If we represent these constant concentrations by the symbols C1 and C2 respectively, we can write the equilibrium law equation as

K = [ CO₂ (g) ] C1 / C2

Since C1 and C2 are constants, and K is also a constant, we can simplify the equilibrium law equation by including all the constants together. We can then write the equilibrium law equation for this reaction as

K = [ CO₂ (g) ]

If pure solids or pure liquids are involved in a chemical equilibrium system, their concentrations are not included in the equilibrium law equation for the reaction system.

We cannot simplify the equilibrium law expression for solutions or gases, since the concentrations of substances in these states can vary.

The Magnitude of the Equilibrium Constant, K

The magnitude of an equilibrium constant is a measure of the yield of a reaction when it reaches equilibrium.

A large value for K_eq indicates that equilibrium is attained only after the reactants have been largely converted into products.

A small value of K_eq — much less than 1 — indicates that equilibrium is attained when only a small proportion of the reactants have been converted into products.

Relationship between the Magnitude of the Equilibrium Constant, K, and Equilibrium Position