7.2.3 Half-Life Method for Determining Rate Constant

This topic explains the half-life method for determining the rate constant of a reaction. It defines half-life and shows how it relates to the rate constant for different orders of reaction.

Definition

Half-Life (t₁/₂): The time it takes for the concentration of a reactant to be reduced to half of its initial value.

Main Idea

The half-life (t₁/₂) of a reaction is the time it takes for the concentration of a reactant to be reduced to half. The relationship between half-life and rate constant depends on the order of the reaction. By knowing the half-life and the order of the reaction, the rate constant can be calculated.

Core Concept

The relationship between half-life and rate constant depends on the order of the reaction:

• [A₀] = initial concentration of reactant
• k = rate constant

Detailed Explanation

What is Half-Life?

• The half-life (t₁/₂) of a reaction is the time it takes for the concentration of a reactant to be reduced to half.
• The relationship between half-life and rate constant depends on the order of the reaction.

Half-Life Equations for Different Orders

For zero order (n = 0): t₁₂ = [A₀]2k

For first order (n = 1): t₁₂ = 0.693k

For second order (n = 2): t₁₂ = 1k[A₀]

Example: First Order Reaction

Consider the following reaction that follows first-order kinetics, its half-life is 30 s.

A + 2B → C

Rate constant for this reaction can be determined as follows:

t₁₂ = 0.693k

30 = 0.693k

k = 0.69330

k = 0.0231 s^-1

Therefore, the rate constant for the reaction is: k = 0.0231 s^-1

Summary Table

Intermediate Level

Half-Life Equations

Example

• First order reaction
• t₁/₂ = 30 s

Required: - Rate constant (k)

Solution: t₁₂ = 0.693k 30 = 0.693k k = 0.0231 s^-1

**Final Answer:** k = 0.0231 s⁻¹

Key Points

• For first order: t₁/₂ independent of [A₀].
• For zero order: t₁/₂ depends on [A₀].
• For second order: t₁/₂ inversely proportional to [A₀].

Important Points

• 1. Half-life: Time for concentration to reduce to half.
• 2. Zero order: t₁₂ = [A₀]/2k
• 3. First order: t₁₂ = 0.693k
• 4. Second order: t₁₂ = 1k[A₀]
• 5. First order half-life independent of [A₀].
• 6. Example: t₁/₂ = 30 s → k = 0.0231 s⁻¹.

Remember

• - Half-life: Time for concentration to halve
• - Zero order: t₁₂ = [A₀]/2k
• - First order: t₁₂ = 0.693k
• - Second order: t₁₂ = 1k[A₀]
• - **Example:** t₁/₂ = 30 s → k = 0.0231 s⁻¹

Common Mistakes

• 1. Using the wrong equation for the order: Check the order first.
• 2. Forgetting that first order half-life is independent of [A₀]: It is constant.
• 3. Mixing up zero and second order equations: Zero = [A₀]/2k; Second = 1k[A₀].
• 4. Forgetting units: k = s⁻¹ for first order.

Quick Revision

• Half-life: Time for concentration to reduce to half.
• Zero order: t₁₂ = [A₀]/2k
• First order: t₁₂ = 0.693k
• Second order: t₁₂ = 1k[A₀]
• First order half-life independent of [A₀].
• Example: t₁/₂ = 30 s → k = 0.0231 s⁻¹.

60-Second Revision

• Half-life: Time for concentration to halve
• Zero order: t₁₂ = [A₀]/2k
• First order: t₁₂ = 0.693k
• Second order: t₁₂ = 1k[A₀]
• **Example:** t₁/₂ = 30 s → k = 0.0231 s⁻¹

QUALITY VALIDATION

• Topic: 7.2.3 Half-Life Method for Determining Rate Constant
• Chapter: 7 (Chemical Kinetics)

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