8.3 The Law of Mass Action

This topic explains the Law of Mass Action — a general description of the equilibrium state. It covers the statement of the law, the equilibrium constant expression, and the conditions for equilibrium.

Definition

Law of Mass Action: The rate at which a substance reacts is proportional to its active mass and the rate of a chemical reaction is proportional to the product of the active masses of the reacting substances.

Equilibrium Constant (K_c): The ratio of the product of the concentrations of products to the product of the concentrations of reactants, each raised to the power of its stoichiometric coefficient.

Main Idea

Two chemists C.M. Guldberg and P. Waage in 1864 proposed the law of mass action as a general description of the equilibrium state. It states that "the rate at which a substance reacts is proportional to its active mass and the rate of a chemical reaction is proportional to the product of the active masses of the reacting substances." It can also be defined as the rate of chemical reaction is proportional to the product of the molar concentration of each reacting substance raised to a power equal to its stoichiometric coefficient in the balanced chemical equation.

Core Concept

For a general reversible reaction:

aA_(g) + bB_(g) ⇌ cC_(g) + dD_(g)

• Rate of forward reaction: R_f ∝ [A]^a [B]^b = k_f [A]^a [B]^b
• Rate of reverse reaction: R_r ∝ [C]^c [D]^d = k_r [C]^c [D]^d
• At equilibrium: k_f [A]^a [B]^b = k_r [C]^c [D]^d
• Equilibrium constant: K_c = [C]^c [D]^d[A]^a [B]^b

Detailed Explanation

Statement of the Law of Mass Action

• Two chemists C.M. Guldberg and P. Waage in 1864 proposed the law of mass action.
• It states that "the rate at which a substance reacts is proportional to its active mass and the rate of a chemical reaction is proportional to the product of the active masses of the reacting substances."
• It can also be defined as the rate of chemical reaction is proportional to the product of the molar concentration of each reacting substance raised to a power equal to its stoichiometric coefficient in the balanced chemical equation.
• The term active mass means the concentration of the reactants and products in moles dm⁻³ for a dilute solution.

General Reversible Reaction

Consider the following general reversible reaction:

aA_(g) + bB_(g) ⇌ cC_(g) + dD_(g)

Where A, B, C, and D represent chemical species and a, b, c, and d are their coefficients in the balanced equation.

Rate of Forward Reaction

According to the law of mass action:

Rate of forward reaction, R_f ∝ [A]^a [B]^b

R_f = k_f [A]^a [B]^b … (1)

Where k_f is the rate constant for the forward reaction.

Rate of Reverse Reaction

Rate of reverse reaction, R_r ∝ [C]^c [D]^d

R_r = k_r [C]^c [D]^d … (2)

Where k_r is the rate constant for the reverse reaction.

At Equilibrium

At equilibrium state:

Rate of forward reaction = Rate of reverse reaction

k_f [A]^a [B]^b = k_r [C]^c [D]^d

On rearranging:

k_fk_r = [C]^c [D]^d[A]^a [B]^b

K_c = [C]^c [D]^d[A]^a [B]^b … (3)

Where K_c = k_fk_r and is known as the equilibrium constant, and equation (3) is known as the equilibrium constant expression. The square brackets indicate the concentration of the chemical species at equilibrium.

Important Features of the Equilibrium Constant Expression

• K_c applies only at the equilibrium state. The subscript c indicates the concentration of reactants and products in moles per dm³ at equilibrium state.
• K_c does not depend on the initial concentration of reactants and products but depends upon temperature. It has only one value at a given temperature.
• K_c is related to the coefficients of the balanced chemical equation. The concentration of the products is placed in the numerator and those of reactants in the denominator. Each concentration is raised to a power equal to its coefficient in the balanced chemical equation.
• The magnitude of K_c indicates the position of equilibrium. When K_c is less than 1, the denominator is greater in magnitude than the numerator. This means the concentration of the reactants is greater than those of products when the equilibrium is established. When K_c is greater than 1, the numerator is greater in magnitude than the denominator. This means the concentration of the products is greater than those of the reactants at equilibrium.

Problem Solving Strategy

• Write the products in the numerator and the reactants in the denominator within square brackets.
• Raise each concentration to the power that corresponds to the coefficient of each species in the balanced chemical equation.

Example 8.1

N_2(g) + 3H_2(g) ⇌ 2NH_3(g)

K_c = [NH₃]²[H₂]³[N₂]

Example 8.2

K_c = [CO₂][CO][O₂]²

Concept Assessment Exercise 8.1

• The following equations represent various industrial reactions at equilibrium. Write K_c expression for each of these reactions. Do not forget to balance the equations.
• Give the balanced equations that correspond to following equilibrium expressions.

K_c = [CH₃OH][CO][H₂]²

K_c = [N₂][H₂O]²[NO]²[H₂]²

Example 8.3: Calculating K_c

The following equilibrium concentrations were observed for the reaction at 500°C.

A_2(g) + B_2(g) ⇌ 2AB_(g)

Given: [A] = 0.399 M, [B] = 1.197 M, [AB] = 0.203 M

Required: K_c

Solution:

Here [A₂] = 0.399 mol dm^-3, [B₂] = 1.197 mol dm^-3, [AB] = 0.203 mol dm^-3

K_c = [AB]²[A₂][B₂]

K_c = (0.203 mol dm^-3)²(0.399 mol dm^-3)(1.197 mol dm^-3)

K_c = 0.086

Intermediate Level

Law of Mass Action

• Proposed by Guldberg and Waage (1864).
• Rate ∝ active mass.
• At equilibrium: K_c = [C]^c [D]^d[A]^a [B]^b

Equilibrium Constant (K_c)

Example 8.1

N₂ + 3H₂ ⇌ 2NH₃ \] \[ K_c = [NH₃]²[H₂]³[N₂]

Example 8.3

• [A] = 0.399 M
• [B] = 1.197 M
• [AB] = 0.203 M

Solution: K_c = (0.203)²(0.399)(1.197) = 0.086

Final Answer: K_c = 0.086

Important Points

• 1. Law of Mass Action: Rate ∝ active mass.
• 2. Proposed by Guldberg and Waage (1864).
• 3. K_c = [C]^c [D]^d[A]^a [B]^b
• 4. K_c depends only on temperature.
• 5. K_c < 1: More reactants.
• 6. K_c > 1: More products.
• 7. Example 8.3: K_c = 0.086.

Remember

• - Law of Mass Action: Rate ∝ active mass
• - Proposed by: Guldberg and Waage (1864)
• - K_c: [C]^c [D]^d[A]^a [B]^b
• - K_c < 1: More reactants
• - K_c > 1: More products
• - Example 8.3: K_c = 0.086

Common Mistakes

• 1. Forgetting to balance the equation: Essential for correct K_c.
• 2. Placing reactants in numerator: Products go in numerator.
• 3. Forgetting exponents: Use coefficients as exponents.
• 4. Thinking K_c depends on initial concentrations: It depends only on temperature.

Quick Revision

• Law of Mass Action: Rate ∝ active mass.
• Proposed by Guldberg and Waage (1864).
• K_c = [C]^c [D]^d[A]^a [B]^b
• K_c depends only on temperature.
• K_c < 1: More reactants.
• K_c > 1: More products.
• Example 8.3: K_c = 0.086.

60-Second Revision

• Law of Mass Action: Rate ∝ active mass
• Proposed by: Guldberg and Waage (1864)
• K_c: [C]^c [D]^d[A]^a [B]^b
• K_c < 1: More reactants
• K_c > 1: More products
• Example 8.3: K_c = 0.086

QUALITY VALIDATION

• Topic: 8.3 The Law of Mass Action
• Chapter: 8 (Chemical Equilibrium)

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