Equilibrum Expressions and the Equilibrium Constant, Kc

This lesson covers: 

1. What the equilibrium constant (K_c) is
2. How K_c indicates the position of equilibrium
3. How to write an expression for K_c
4. Calculations involving K_c
5. The effect of changing conditions on the equilibrium constant (K_c)

Introducing the equilibrium constant

When a reversible reaction reaches a state of dynamic equilibrium, we can calculate a value called the equilibrium constant (K_c) using the molar concentrations of the reactants and products at equilibrium.

K_c gives us a quantitative measure of where the equilibrium lies - whether there are more products or more reactants present at equilibrium.

• A large K_c value indicates the equilibrium position favours the products.
• A small K_c value indicates the equilibrium position favours the reactants.

A K_c value of 1 indicates that the reaction is at equilibrium, and the concentrations of reactants and products are equal when raised to their respective stoichiometric coefficients.

Writing an expression for K_c

For the general equilibrium reaction:

aA + bB ⇌ dD + eE

The equilibrium constant K_c is given by:

Kc ​=AaBbDdEe​

Where the lower case letters represent the coefficients in the balanced chemical equation.

For example, for the reaction:

H_2(g) + I_2(g) ⇌ 2HI_(g)

The K_c expression would be:

Kc ​=[H2​][I2​][HI]2​

Calculating values for K_c

There are 3 worked examples below demonstrating various equilibrium calculations involving the equilibrium constant (K_c).

If we know the equilibrium concentrations of all reactants and products, we can substitute them into the K_c expression to calculate a value for K_c.

The units for K_c vary, so we must determine the units after each calculation.

Worked example 1 - Determining the equilibrium constant (K_c)

For the reaction H_2(g) + I_2(g) ⇌ 2HI_(g) at 580 K, the equilibrium concentrations are:

[HI] = 0.60 mol dm⁻³

[H₂] = 0.20 mol dm⁻³

[I₂] = 0.20 mol dm⁻³

Determine the equilibrium constant (K_c).

Step 1: Write the equilibrium constant (K_c) expression

Kc​=[H2​][I2​][HI]2​

Step 2: Substitution and correct evaluation

Kc​=0.20×0.20(0.60)2​=9

In some cases, we may need to calculate some equilibrium concentrations before we can find K_c.

Worked example 2 - Determining the equilibrium constant (K_c)

0.25 moles of PCl₅ decomposes at 650 K in a 4.0 dm³ vessel. At equilibrium, 0.10 moles of Cl₂ is present.

Determine the equilibrium constant K_c for the reaction:

PCl_5(g) ⇌ PCl_3(g) + Cl_2(g)

Step 1: Deduce moles of PCl₃ formed at equilibrium

Moles of PCl₃ = moles of Cl₂ = 0.10 mol

Step 2: Calculate remaining moles of PCl₅ at equilibrium

Moles of PCl₅ decomposed = 0.10 mol

Moles of PCl₅ at equilibrium = 0.25 - 0.10 = 0.15 mol

Step 3: Determine equilibrium concentrations

[PCl5​] =4.00.15​=0.0375 mol dm−3

[PCl3​] =4.00.10​=0.025 mol dm−3

[Cl2​] =4.00.10​=0.025 mol dm−3

Step 4: Write the equilibrium constant (K_c) expression

Kc​=[PCl5​][PCl3​][Cl2​]​

Step 5: Substitution and correct evaluation

Kc​=0.0375(0.025×0.025)​=0.017 mol dm⁻³

If we know the equilibrium constant (K_c) value and some equilibrium concentrations, we can use the K_c expression to determine an unknown equilibrium concentration.

Worked example 3 - Determining equilibrium concentrations

When propanoic acid was allowed to reach equilibrium with ethanol at 30°C, it was found that the equilibrium mixture contained 1.8 mol dm^-3 propanoic acid and 3.0 mol dm^-3 ethanol.

If the K_c value is 3.5 at 30°C, determine the concentrations of the other components at equilibrium. The balanced equation is:

CH₃CH₂COOH_(aq) + C₂H₅OH_(aq) ⇌ CH₃CH₂COOC₂H_5(aq) + H₂O_(l)

Step 1: Write the equilibrium constant (K_c) expression

Kc​=[CH3​CH2​COOH][C2​H5​OH][CH3​CH2​COOC2​H5​][H2​O]​

Step 2: Substitute the known equilibrium concentrations into the K_c expression

3.5=1.8×3.0[CH3​CH2​COOC2​H5​][H2​O]​

Step 3: Rearrange to calculate [CH₃CH₂COOC₂H₅][H₂O]

[CH₃CH₂COOC₂H₅][H₂O] = 3.5 x 1.8 x 3.0 = 18.9

Step 4: Set [CH₃CH₂COOC₂H₅] = [H₂O] = x

From the balanced equation, [CH₃CH₂COOC₂H₅] = [H₂O] at equilibrium

x² = 18.9

x =√18.9=4.3 mol dm−3

Therefore, [CH₃CH₂COOC₂H₅]_equil = [H₂O]_equil = 4.3 mol dm^-3

Temperature changes affect K_c

The equilibrium constant (K_c) is influenced solely by temperature. Changes in concentration or pressure have no impact on K_c.

When the position of equilibrium is altered due to a temperature shift:

• A decrease in the amount of product at equilibrium causes K_c to decrease.
• An increase in the amount of product at equilibrium causes K_c to increase.

This ensures that the value of K_c aligns with the new position of equilibrium.

Example:

2SO_2(g) + O_2(g) ⇌ 2SO_3(g),  ΔH = -197 kJ mol^-1

Raising the temperature causes the equilibrium to shift left towards the reactants, absorbing heat and resulting in less SO₃ at the new equilibrium position.

With a decrease in SO₃:

• The proportion of products reduces.
• Consequently, K_c (the ratio of products to reactants) decreases.

The impact of temperature changes on exothermic and endothermic reactions is summarised in the table below:

Concentration and pressure changes do not affect K_c

While shifts in concentration or pressure can move the equilibrium position, altering the quantities present to counteract the change, K_c remains unchanged at a specific temperature.

For example:

N_2(g) + 3H_2(g) ⇌ 2NH_3(g)

• Adding more N₂ leads to an increased formation of NH₃ at the new equilibrium. However, the equilibrium ratio of [NH₃]² to [N₂][H₂]³ remains equal to the constant value of K_c.
• Thus, changes in concentration and pressure do not directly influence K_c - its value is only determined by temperature.

Catalysts do not change K_c or equilibrium position

Catalysts increase the rate of both the forward and reverse reactions, reducing the time needed to reach equilibrium following a change.

However, catalysts neither alter the equilibrium position nor the value of K_c.

The table below summarises the effects of changing reaction conditions on the position of equilibrium and the value of K_c: