Chemical Kinetics Overview
What is Chemical Kinetics?
Definition: Study of reaction rates and mechanisms
Rate of Reaction: Change in concentration per unit time
Average Rate: Δ[Reactant]/Δt or Δ[Product]/Δt
Instantaneous Rate: Slope of tangent to concentration-time curve
Units of Rate: mol L⁻¹ s⁻¹ (or M s⁻¹)
Key Questions: How fast? What factors affect speed? What is the mechanism?
Historical Timeline
1864: Cato Guldberg & Peter Waage – Law of Mass Action
1884: Svante Arrhenius – Arrhenius Equation
1889: Arrhenius – Activation Energy Concept
1913: Max Trautz & William Lewis – Collision Theory
1935: Henry Eyring – Transition State Theory
1950s: Development of Fast Reaction Techniques
Key Concepts
• Rate Laws and Rate Constants
• Order of Reaction (0, 1, 2, fractional)
• Integrated Rate Equations
• Half-life (t₁/₂) calculations
• Collision Theory & Transition State Theory
• Arrhenius Equation & Activation Energy
• Catalysis (homogeneous & heterogeneous)
• Reaction Mechanisms & Rate-Determining Step
Learning Objectives
✓ Write rate laws from experimental data
✓ Determine order of reaction graphically
✓ Calculate half-life for different orders
✓ Use Arrhenius equation to find Eₐ
✓ Explain effect of temperature, concentration, catalyst
✓ Distinguish between rate law and stoichiometry
✓ Propose reaction mechanisms from rate law
Rate Laws & Reaction Orders
Zero Order Reactions
Rate Law: Rate = k
Differential Form: -d[A]/dt = k
Integrated Form: [A]ₜ = [A]₀ – kt
Half-life: t₁/₂ = [A]₀/2k
Graphs:
- [A] vs t: Straight line with slope = -k
- Rate vs [A]: Horizontal line
Examples: Enzyme-catalyzed reactions at high substrate concentration, Photochemical reactions
First Order Reactions
Rate Law: Rate = k[A]
Differential Form: -d[A]/dt = k[A]
Integrated Form: ln[A]ₜ = ln[A]₀ – kt
Half-life: t₁/₂ = 0.693/k (independent of [A]₀)
Graphs:
- ln[A] vs t: Straight line with slope = -k
- Rate vs [A]: Straight line through origin
Examples: Radioactive decay, Unimolecular decomposition
Second Order Reactions
Type 1: Rate = k[A]²
Type 2: Rate = k[A][B]
Integrated Form (Type 1): 1/[A]ₜ = 1/[A]₀ + kt
Half-life (Type 1): t₁/₂ = 1/k[A]₀
Graphs:
- 1/[A] vs t: Straight line with slope = k
- Rate vs [A]²: Straight line through origin
Examples: Dimerization, Saponification of esters
Determining Order
Method 1: Initial Rates Method
• Measure initial rate at different [A]₀
• Rate ∝ [A]ⁿ → log(rate) vs log[A] gives slope = n
Method 2: Integrated Rate Laws
• Plot [A] vs t (zero order check)
• Plot ln[A] vs t (first order check)
• Plot 1/[A] vs t (second order check)
Method 3: Half-life Method
• t₁/₂ ∝ [A]₀¹⁻ⁿ
• If t₁/₂ constant → First order
• If t₁/₂ ∝ 1/[A]₀ → Second order
Rate Law Formulas Summary
Zero Order: [A] = [A]₀ – kt
First Order: ln[A] = ln[A]₀ – kt
Second Order: 1/[A] = 1/[A]₀ + kt
Half-life (1st order): t₁/₂ = 0.693/k
Arrhenius: k = Ae^(-Eₐ/RT)
Comparison of Reaction Orders
| Order | Rate Law | Half-life | Linear Plot | Units of k |
|---|---|---|---|---|
| 0 | Rate = k | [A]₀/2k | [A] vs t | mol L⁻¹ s⁻¹ |
| 1 | Rate = k[A] | 0.693/k | ln[A] vs t | s⁻¹ |
| 2 | Rate = k[A]² | 1/k[A]₀ | 1/[A] vs t | L mol⁻¹ s⁻¹ |
| 2 | Rate = k[A][B] | Complex | 1/[A] vs t | L mol⁻¹ s⁻¹ |
Reaction Rate Theories
Collision Theory (1918)
Postulates:
- Molecules must collide to react
- Collisions must have sufficient energy (≥ Eₐ)
- Collisions must have proper orientation
Rate Expression:
Rate = Z × f × p
where: Z = collision frequency
f = fraction with E ≥ Eₐ
p = steric factor (0 to 1)
Temperature Effect: f = e^(-Eₐ/RT)
Limitations: Doesn’t explain orientation factor well
Transition State Theory (1935)
Also Called: Activated Complex Theory or Absolute Rate Theory
Key Concepts:
- Reactants form activated complex (transition state)
- Transition state is at energy maximum
- Equilibrium between reactants and transition state
- Rate depends on concentration of transition state
Energy Diagram: Shows reactants → transition state → products
Relation to ΔH: Eₐ(forward) – Eₐ(reverse) = ΔH
Advantage: Explains orientation and entropy effects
Comparison of Theories
| Aspect | Collision Theory | Transition State Theory |
|---|---|---|
| Year | 1918 | 1935 |
| Focus | Collision frequency & energy | Formation of activated complex |
| Orientation | Empirical factor (p) | Explained by entropy of activation |
| Relation to Thermodynamics | Limited | Direct (ΔG‡, ΔH‡, ΔS‡) |
Mathematical Expressions
Collision Theory Rate:
k = pZ e^(-Eₐ/RT)
Transition State Theory:
k = (kₐT/h) e^(-ΔG‡/RT)
where: kₐ = Boltzmann constant
h = Planck’s constant
ΔG‡ = Gibbs free energy of activation
Relationship:
ΔG‡ = ΔH‡ – TΔS‡
ln(k/T) = ln(kₐ/h) + ΔS‡/R – ΔH‡/RT
Arrhenius from TST:
A = (kₐT/h) e^(ΔS‡/R)
Eₐ = ΔH‡ + RT
Activation Energy & Arrhenius Equation
Activation Energy (Eₐ)
Definition: Minimum energy required for reaction to occur
Energy Barrier: Difference between reactants and transition state
SI Unit: J mol⁻¹ or kJ mol⁻¹
Typical Values: 40-250 kJ mol⁻¹
Temperature Dependence: Eₐ is approximately constant over small T range
Relation to ΔH:
For endothermic: Eₐ(forward) > Eₐ(reverse)
For exothermic: Eₐ(forward) < Eₐ(reverse)
ΔH = Eₐ(f) – Eₐ(r)
Arrhenius Equation
Exponential Form:
k = A e^(-Eₐ/RT)
Logarithmic Form:
ln k = ln A – Eₐ/RT
Two-Temperature Form:
ln(k₂/k₁) = (Eₐ/R)(1/T₁ – 1/T₂)
Where:
- k = rate constant
- A = pre-exponential factor (frequency factor)
- Eₐ = activation energy
- R = 8.314 J K⁻¹ mol⁻¹
- T = temperature in Kelvin
Graphical Determination
Arrhenius Plot: ln k vs 1/T
Slope: -Eₐ/R
Intercept: ln A
Steps:
- Measure k at different T
- Calculate ln k and 1/T
- Plot ln k vs 1/T
- Slope = -Eₐ/R → Eₐ = -slope × R
- Intercept = ln A → A = e^(intercept)
Example Calculation:
If slope = -5000 K, then:
Eₐ = -(-5000) × 8.314 = 41570 J/mol = 41.57 kJ/mol
Temperature Effects
Rule of Thumb: Rate doubles for every 10°C rise (for Eₐ ~ 50 kJ/mol)
Quantitative: Fractional increase depends on Eₐ
Example: For Eₐ = 50 kJ/mol:
k(310K)/k(300K) = e^(Eₐ/R × (1/300 – 1/310))
= e^(50000/8.314 × 0.0001075) ≈ 1.8
High Eₐ Reactions: More sensitive to T changes
Low Eₐ Reactions: Less sensitive to T changes
Catalysts: Lower Eₐ, making reaction faster at same T
Arrhenius Equation Practice
Find Eₐ: ln(k₂/k₁) = (Eₐ/R)(1/T₁ – 1/T₂)
ln(0.06/0.0023) = (Eₐ/8.314)(1/298 – 1/323)
Eₐ = 86.5 kJ/mol
Factors Affecting Reaction Rates
Concentration Effect
Rate Law: Rate ∝ [Reactant]ⁿ
For n > 0: Increase [Reactant] increases rate
Zero Order: Rate independent of [Reactant] at high concentrations
Molecularity: Related to number of molecules colliding
Elementary Reactions:
- Unimolecular: Rate ∝ [A]
- Bimolecular: Rate ∝ [A][B] or [A]²
- Termolecular: Rate ∝ [A][B][C] (rare)
Complex Reactions: Overall order from rate-determining step
Temperature Effect
Arrhenius Explanation: More molecules have E ≥ Eₐ
Fraction with E ≥ Eₐ: f = e^(-Eₐ/RT)
At 300K: f ≈ e^(-Eₐ/2500) where Eₐ in J/mol
Example: For Eₐ = 50 kJ/mol:
f = e^(-50000/(8.314×300)) ≈ e^(-20) ≈ 2 × 10⁻⁹
At 310K: f ≈ e^(-19.3) ≈ 4 × 10⁻⁹ (doubled!)
Collision Frequency: Also increases with √T
Overall: Rate increase is exponential with T
Catalysts
Definition: Substance that increases rate without being consumed
Mechanism: Provides alternative pathway with lower Eₐ
Homogeneous: Same phase as reactants (e.g., acid catalysis)
Heterogeneous: Different phase (e.g., Pt in contact with gases)
Enzymes: Biological catalysts (Michaelis-Menten kinetics)
Characteristics:
- Doesn’t affect equilibrium position
- Speeds up both forward and reverse reactions equally
- Appears in reaction mechanism but not overall equation
Other Factors
Surface Area: For heterogeneous reactions, rate ∝ surface area
Pressure: For gases, rate ∝ pressure (increases concentration)
Light: Photochemical reactions (quantum yield concept)
Solvent: Polarity affects ionic reactions
Radiation: High-energy radiation initiates chain reactions
Orientation: Steric effects in organic reactions
Medium: Solid state vs solution vs gas phase
Ionic Strength: Salt effect in ionic reactions
Common Catalysts and Their Actions
| Reaction | Catalyst | Type | Mechanism |
|---|---|---|---|
| 2H₂O₂ → 2H₂O + O₂ | MnO₂, I⁻ | Heterogeneous/Homogeneous | Provides lower energy pathway |
| N₂ + 3H₂ → 2NH₃ | Fe (Haber process) | Heterogeneous | Adsorption and weakening of bonds |
| 2SO₂ + O₂ → 2SO₃ | V₂O₅ (Contact process) | Heterogeneous | Forms intermediate complexes |
| Hydrolysis of esters | H⁺ or OH⁻ | Homogeneous | Acid/base catalysis |
| Biological reactions | Enzymes | Homogeneous | Lock-and-key model, active sites |
Interactive Reaction Rate Simulator
Explore Reaction Kinetics in Real-Time
Observations:
- Increase Temperature: Exponential increase in rate constant
- Increase Eₐ: Exponential decrease in rate constant
- Increase Concentration: Linear/quadratic increase in rate (depends on order)
- Higher Order: More sensitive to concentration changes
- High Eₐ Reactions: More temperature sensitive
Memory Aids & Problem-Solving Tips
Kinetics Mnemonics
Zero Order: “Flat rate” (rate constant, independent of [A])
First Order: “Log linear” (ln[A] vs t is straight)
Second Order: “Inverse plot” (1/[A] vs t is straight)
Half-life: “0th: depends, 1st: constant, 2nd: inverse”
Arrhenius: “k = A e^(-Eₐ/RT)” → “Constant A times e to the minus Ea over RT”
Units of k: “M¹⁻ⁿ time⁻¹” where n = order
Problem Solving Steps
1. Identify: What’s given? What’s asked? Rate constant? Half-life? Order?
2. Choose Equation: Based on order (0, 1, 2, or Arrhenius)
3. Convert Units: T to Kelvin, Eₐ to J/mol if using R=8.314
4. Substitute: Plug values into appropriate equation
5. Solve: Rearrange algebraically, use ln/exp as needed
6. Check: Units correct? Sign reasonable? Magnitude plausible?
7. Graph Interpretation: Which plot is linear? Slope gives what?
Common Mistakes to Avoid
❌ Using Celsius instead of Kelvin in Arrhenius equation
❌ Confusing rate constant (k) with rate (k[A]ⁿ)
❌ Forgetting that half-life for 1st order is constant
❌ Mixing up differential and integrated rate laws
❌ Assuming stoichiometry gives reaction order
❌ Using wrong R value (8.314 for J, 0.0821 for L·atm)
❌ Not converting Eₐ to J/mol when using R=8.314
❌ Confusing activation energy with enthalpy change
Graph Interpretation Tips
Zero Order: [A] vs t → straight line, slope = -k
First Order: ln[A] vs t → straight line, slope = -k
Second Order: 1/[A] vs t → straight line, slope = k
Arrhenius Plot: ln k vs 1/T → straight line, slope = -Eₐ/R
Rate vs [A]:
• 0 order: horizontal line
• 1st order: straight through origin
• 2nd order: parabola through origin
Experimental Connections
Clock Reactions: Iodine clock, persulfate-iodide
Gas Evolution: Measure volume vs time
Colorimetry: Measure absorbance vs time for colored species
Conductometry: For ionic reactions
Pressure Measurements: For gas phase reactions
Quenching Methods: Stop reaction at specific times
Fast Reactions: Stopped-flow, flash photolysis
Quick Calculations
Approximation: Rate doubles for 10°C rise when Eₐ ~ 50 kJ/mol
Fraction with E ≥ Eₐ: f ≈ e^(-Eₐ/RT)
At 300K: RT ≈ 2.5 kJ/mol
Half-life Estimates:
• 1st order: t₁/₂ = 0.693/k
• 2nd order: t₁/₂ = 1/(k[A]₀)
Time for x% completion:
1st order: t = (1/k) ln(100/(100-x))