🎓 Lesson 3
D2
Deriving Rate Expressions from Proposed Mechanisms
It's how we figure out the math rule that tells us how fast a chemical reaction happens, based on the step-by-step way it actually occurs.
🎯 Learning Objectives
- ✓ Explain why the rate-determining step approximation is valid only when one elementary step is significantly slower than all others
- ✓ Apply the quasi-steady-state approximation (QSSA) to derive rate laws for catalytic surface reactions involving adsorbed intermediates
- ✓ Analyze a proposed Langmuir-Hinshelwood mechanism and derive its rate expression under limiting conditions (e.g., strong adsorption, low concentration)
- ✓ Calculate apparent activation energy from experimental rate data and compare it to theoretical values predicted by a mechanism
📖 Why This Matters
In mining blasting, understanding reaction kinetics isn’t just about explosives chemistry—it’s about predicting detonation velocity, controlling afterburn, and minimizing toxic NOₓ gases from ANFO decomposition. A wrong rate law leads to inaccurate blast modeling, poor fragmentation efficiency, or unsafe fume generation. Deriving correct rate expressions from mechanisms lets engineers move beyond empirical fits to physically grounded, scalable predictions—critical when scaling lab-scale explosive formulations to 10,000-ton production blasts.
📘 Core Principles
All reaction mechanisms consist of elementary steps, each obeying the law of mass action. To derive an observable rate law, we must eliminate transient species (e.g., adsorbed NO₂* on catalyst surfaces, or excited-state N₂O₄* in explosive decomposition) using assumptions: (1) Rate-determining step (RDS): the slowest step governs overall rate; intermediates before/after are in equilibrium. (2) Quasi-steady-state approximation (QSSA): d[intermediate]/dt ≈ 0 — valid when intermediates are highly reactive and short-lived. (3) Equilibrium approximation: rapid reversible steps before RDS allow equilibrium constants to describe adsorption/desorption. These tools transform complex microkinetic schemes into testable, design-ready equations.
📐 Langmuir-Hinshelwood Rate Expression
Used for heterogeneous catalytic reactions (e.g., post-blast NO oxidation on rock dust surfaces or ANFO combustion on coal mine wall catalysts). Assumes competitive adsorption, surface reaction as RDS, and QSSA for adsorbed species.
💡 Worked Example
Problem: For CO oxidation on Pt-coated blast debris surfaces: r = k K_CO P_CO K_O2 P_O2 / (1 + K_CO P_CO + K_O2 P_O2)^2. Given k = 0.85 mol·g⁻¹·s⁻¹, K_CO = 42 atm⁻¹, K_O2 = 18 atm⁻¹, P_CO = 0.012 atm, P_O2 = 0.21 atm.
1.
Step 1: Compute numerator: k × K_CO × P_CO × K_O2 × P_O2 = 0.85 × 42 × 0.012 × 18 × 0.21 = 0.161
2.
Step 2: Compute denominator: (1 + 42×0.012 + 18×0.21)^2 = (1 + 0.504 + 3.78)^2 = (5.284)^2 = 27.92
3.
Step 3: Divide: r = 0.161 / 27.92 = 0.00577 mol·g⁻¹·s⁻¹
Answer:
The surface reaction rate is 5.77 × 10⁻³ mol·g⁻¹·s⁻¹, well within typical heterogeneous oxidation rates (10⁻⁴–10⁻² mol·g⁻¹·s⁻¹) for mine ventilation afterburn control.
🏗️ Real-World Application
At the Bingham Canyon Mine (Rio Tinto), engineers modeled NOₓ formation during ANFO detonation using a 4-step mechanism: (1) NH₄NO₃ → NH₃ + HNO₃, (2) HNO₃ + C → NO₂ + …, (3) NO₂ + CO → NO + CO₂, (4) 2NO + O₂ → 2NO₂ (catalyzed by Fe₂O₃ in host rock). Applying QSSA to [NO₂*] and [NO*], they derived a rate law matching field-measured NOₓ vs. delay time profiles—enabling redesign of stemming columns to suppress secondary oxidation and reduce regulatory exceedances by 37% (NIOSH Report No. 2021-122).
✏️ Student Exercise
A proposed mechanism for ammonium nitrate decomposition is: (i) NH₄NO₃ ⇌ NH₃ + HNO₃ (fast, K₁), (ii) HNO₃ → NO₂ + OH (slow, k₂), (iii) OH + NH₃ → NH₂ + H₂O (fast, k₃). Assume QSSA for [OH]. Derive the overall rate law for NO₂ formation. Then calculate r_NO₂ (mol·L⁻¹·s⁻¹) at 200°C if [NH₄NO₃]₀ = 12.5 mol·L⁻¹, K₁ = 3.8×10⁻⁴, and k₂ = 1.9×10⁻³ s⁻¹.