R2.3.2 The equilibrium law and constant

Exploring the Equilibrium Constant $K$

1. You're designing a chemical process to produce ammonia in the Haber process.
2. You know the reaction is reversible, meaning it won't go entirely in one direction.
3. At equilibrium, the forward and reverse reactions balance each other out. But how much ammonia can you expect at equilibrium?

This is where the equilibrium constant $K$ becomes invaluable.

What is the Equilibrium Constant?

• It is derived from the stoichiometry of the reaction and remains constant as long as the temperature is unchanged.
• Consider the general reaction:

$$aA+bB\rightleftharpoons cC+dD$$

• At equilibrium, the concentrations of $A$, $B$, $C$, and $D$ remain constant. The equilibrium constant $K$ is expressed as:

$$K=\frac{[C{]}^c[D{]}^d}{[A{]}^a[B{]}^b}$$

where:

• $[X]$ represents the equilibrium concentration of species $X$ in ${\text{mol dm}}^{-3}$.
• $a,b,c,d$ are the stoichiometric coefficients from the balanced chemical equation.

Deriving the Expression for $K$: Step-by-Step

Let’s illustrate this with an example: the synthesis of ammonia.

$$N_2(g)+3H_2(g)\rightleftharpoons 2N{H}_3(g)$$

1. Identify the Reactants and Products:
   • Reactants: $N_2$ and $H_2$
   • Product: $N{H}_3$
2. Write the General Form of $K$:
   According to the equilibrium law, the concentrations of the products are placed in the numerator, while the concentrations of the reactants are placed in the denominator. Each species is raised to the power of its stoichiometric coefficient:

$$K=\frac{[N{H}_3{]}^2}{[N_2][H_2{]}^3}$$

1. Interpret the Expression:
   • If $K$ is large ($K>1$), the numerator dominates, indicating that the equilibrium mixture contains more products than reactants.
   • If $K$ is small ($K<1$), the denominator dominates, meaning the equilibrium mixture contains more reactants than products.

Properties of the Equilibrium Constant

1. Temperature Dependence:
   • $K$ remains constant only at a fixed temperature. Altering the temperature shifts the equilibrium position and changes the value of $K$.
2. No Units:
   • $K$ is often treated as a dimensionless quantity because the units of concentration cancel out when the expression is written correctly.
3. Homogeneous vs. Heterogeneous Equilibria:
   • For homogeneous equilibria (all species in the same phase), all reactants and products appear in the $K$ expression.
   • For heterogeneous equilibria (species in different phases), the concentrations of pure solids and liquids are omitted because they are constant.

Interpreting $K$: What Does It Tell Us?

The value of $K$ provides insight into the extent of the reaction at equilibrium:

1. $K>1$:
   • Products are favored.
   • The reaction proceeds significantly in the forward direction.
2. $K<1$:
   • Reactants are favored.
   • The reaction does not proceed significantly in the forward direction.
3. $K=1$:
   • Neither reactants nor products are favored.
   • The concentrations of reactants and products are comparable.

Reflection and Practice