A Level Chemistry - How to Solve: Orders of Reaction & Rate Equations (Initial Rates, Half-Life, Arrhenius Equation)

How to Solve: Orders of Reaction & Rate Equations (Initial Rates, Half-Life, Arrhenius Equation)

Complete Guide

Introduction

"Mastering orders of reaction and rate equations lets you predict how fast a drug breaks down in your body, how long a glow stick lasts, or even how to slow down food spoilage—plus, it’s worth up to 15% of your A-Level Chemistry exam."

WHAT YOU NEED TO KNOW FIRST

1. Rate of reaction = change in concentration ÷ time (units: mol dm⁻³ s⁻¹).
2. Concentration-time graphs – how to read them (zero, first, second order).
3. Logarithms (ln) – used in the Arrhenius equation (you’ll need a calculator).

KEY TERMS & FORMULAS

1. Rate Equation

Formula: Rate = k [A]ᵐ [B]ⁿ - k = rate constant (units depend on order) - [A], [B] = concentrations of reactants (mol dm⁻³) - m, n = orders of reaction (0, 1, or 2) MEMORISE THIS

2. Overall Order of Reaction

Formula: Overall order = m + n - If m = 1 and n = 2, overall order = 3. MEMORISE THIS

3. Units of k (Rate Constant)

4. Half-Life (t₁/₂)

• Zero order: Half-life decreases over time.
• First order: Half-life is constant (independent of concentration).
• Second order: Half-life increases over time. MEMORISE THIS

5. Arrhenius Equation

Formula: k = A e⁻ᴱᵃ/ᴿᵀ - k = rate constant - A = pre-exponential factor (same units as k) - Eₐ = activation energy (J mol⁻¹) - R = gas constant (8.314 J mol⁻¹ K⁻¹) given on exam sheet - T = temperature (K) MEMORISE THIS

Logarithmic form (for calculations): ln k = ln A – (Eₐ / R) × (1 / T) MEMORISE THIS

STEP-BY-STEP METHOD

Part 1: Finding Orders of Reaction (Initial Rates Method)

Step 1: Write the rate equation: Rate = k [A]ᵐ [B]ⁿ Step 2: Identify experiments where only one concentration changes (keep others constant). Step 3: Compare rates: - If [A] doubles and rate stays the same → order = 0 - If [A] doubles and rate doubles → order = 1 - If [A] doubles and rate quadruples → order = 2 Step 4: Repeat for all reactants. Step 5: Write the full rate equation and calculate k (if needed).

Part 2: Calculating k from Half-Life (First Order Only)

Step 1: Confirm reaction is first order (half-life constant). Step 2: Use the formula: t₁/₂ = ln 2 / k Step 3: Rearrange to find k: k = ln 2 / t₁/₂ Step 4: Plug in t₁/₂ (in seconds) and calculate k.

Part 3: Arrhenius Equation (Finding Eₐ or k)

Step 1: Write the logarithmic form: ln k = ln A – (Eₐ / R) × (1 / T) Step 2: If given two temperatures (T₁, T₂) and two k values (k₁, k₂), use: ln (k₂ / k₁) = – (Eₐ / R) × (1 / T₂ – 1 / T₁) Step 3: Rearrange to solve for Eₐ (or k if Eₐ is known). Step 4: Convert Eₐ to kJ mol⁻¹ (divide by 1000).

WORKED EXAMPLES

Example 1 – Basic: Finding Orders & Rate Equation

Question: The reaction A + B → C was studied. Initial rates are given:

Step 1: Compare Expt 1 & 2 ([B] constant, [A] doubles). - Rate doubles (4.0 → 8.0 × 10⁻⁵) → order w.r.t. A = 1 Step 2: Compare Expt 1 & 3 ([A] constant, [B] doubles). - Rate quadruples (4.0 → 16.0 × 10⁻⁵) → order w.r.t. B = 2 Step 3: Write rate equation: Rate = k [A]¹ [B]² Step 4: Calculate k (using Expt 1):
4.0 × 10⁻⁵ = k × 0.1 × (0.1)² k = 4.0 × 10⁻⁵ / (0.1 × 0.01) = 0.4 dm⁶ mol⁻² s⁻¹

What we did and why: - Compared experiments where only one concentration changed. - Used rate changes to find orders. - Plugged values into the rate equation to find k.

Example 2 – Medium: Half-Life & First Order Reactions

Question: A first-order reaction has a half-life of 30 seconds. Calculate the rate constant (k).

Step 1: Confirm first order (half-life constant). Step 2: Use t₁/₂ = ln 2 / k Step 3: Rearrange: k = ln 2 / t₁/₂ Step 4: k = 0.693 / 30 = 0.0231 s⁻¹

What we did and why: - Recognised first-order reactions have constant half-lives. - Used the half-life formula to find k directly.

Example 3 – Exam-Style: Arrhenius Equation

Question: The rate constant for a reaction is 2.5 × 10⁻⁴ s⁻¹ at 300 K and 1.2 × 10⁻³ s⁻¹ at 320 K. Calculate the activation energy (Eₐ).

Step 1: Write the two-point Arrhenius equation: ln (k₂ / k₁) = – (Eₐ / R) × (1 / T₂ – 1 / T₁) Step 2: Plug in values: k₁ = 2.5 × 10⁻⁴, k₂ = 1.2 × 10⁻³, T₁ = 300 K, T₂ = 320 K, R = 8.314 J mol⁻¹ K⁻¹ ln (1.2 × 10⁻³ / 2.5 × 10⁻⁴) = – (Eₐ / 8.314) × (1/320 – 1/300) Step 3: Simplify: ln (4.8) = – (Eₐ / 8.314) × (–2.08 × 10⁻⁴)
1.569 = Eₐ × 2.50 × 10⁻⁵ Step 4: Solve for Eₐ: Eₐ = 1.569 / (2.50 × 10⁻⁵) = 62,760 J mol⁻¹ = 62.8 kJ mol⁻¹

What we did and why: - Used the two-point Arrhenius equation to avoid needing A. - Converted temperatures to Kelvin (essential!). - Rearranged carefully to isolate Eₐ.

COMMON MISTAKES

1. MISTAKE: Forgetting units for k. WHY IT HAPPENS: Students focus on numbers, not units. CORRECT APPROACH: Always write units (e.g., s⁻¹ for first order).

2. MISTAKE: Assuming all reactions are first order. WHY IT HAPPENS: First order is common, but not universal. CORRECT APPROACH: Check half-life or initial rates to confirm order.

3. MISTAKE: Using °C instead of Kelvin in Arrhenius. WHY IT HAPPENS: Forgetting temperature must be in Kelvin. CORRECT APPROACH: Always convert °C → K (+273).

4. MISTAKE: Misinterpreting "rate doubles when [A] doubles" as second order. WHY IT HAPPENS: Confusing rate changes with order. CORRECT APPROACH: Rate doubles → order = 1; rate quadruples → order = 2.

5. MISTAKE: Not rearranging the Arrhenius equation correctly. WHY IT HAPPENS: Forgetting to take ln of both sides. CORRECT APPROACH: Always use ln k = ln A – (Eₐ / R) × (1 / T).

EXAM TRAPS

1. TRAP: Giving half-life data for a non-first-order reaction. HOW TO SPOT IT: Half-life changes with concentration. HOW TO AVOID IT: Only use t₁/₂ = ln 2 / k for first order.

2. TRAP: Asking for k without specifying temperature. HOW TO SPOT IT: "Calculate k" but no T given. HOW TO AVOID IT: k depends on temperature—check if T is implied (e.g., "at 298 K").

3. TRAP: Mixing up Eₐ units (J vs. kJ). HOW TO SPOT IT: Answer requires kJ mol⁻¹ but you have J mol⁻¹. HOW TO AVOID IT: Always convert Eₐ to kJ mol⁻¹ (÷ 1000).

1-MINUTE RECAP

"Right, listen up—this is your last-minute cheat sheet for orders of reaction and rate equations. First, rate = k [A]^m [B]^n—compare experiments to find m and n. If rate doubles when [A] doubles, order = 1. If it quadruples, order = 2. For half-life, only first order has constant t₁/₂—use k = ln 2 / t₁/₂. For the Arrhenius equation, always use Kelvin, and remember ln k = ln A – (Eₐ / R)(1 / T). If you have two k values at two temperatures, use the two-point formula. Watch out for units—k changes with order, and Eₐ must be in kJ mol⁻¹. Now go smash that exam!"