Calculate molecular collisions, reaction rates, and temperature effects
Collision Frequency: Number of molecular collisions per unit volume per unit time
Activation Energy: Minimum energy required for successful reaction
Steric Factor (p): Fraction of collisions with proper orientation (0 < p ≤ 1)
Rate Constant: k = p × Z × e^(-Ea/RT), where Z is collision frequency
Collision theory is a fundamental model in chemical kinetics that explains how chemical reactions occur and why reaction rates vary. According to this theory, for a reaction to occur, reactant molecules must collide with sufficient energy (at least equal to the activation energy) and with proper orientation.
Not all collisions result in reactions. Only a fraction of collisions have enough energy and the correct molecular orientation to break existing bonds and form new ones. The rate constant k depends on collision frequency, activation energy, and the steric factor (orientation requirement).
The number of collisions between molecules per unit volume per unit time.
Z = σ × v̄ × n² / √2
where σ = collision cross-section (πd²)
v̄ = mean molecular speed = √(8kT/πm)
n = number density (molecules/m³)
Collision frequency increases with temperature (higher speeds) and concentration (more molecules). Typical values are on the order of 10²⁸-10³⁴ collisions/(m³·s) for gases at standard conditions.
The minimum kinetic energy that colliding molecules must possess for a reaction to occur.
At room temperature (298 K), only about e^(-Ea/RT) fraction of molecules have energy ≥ Ea. For Ea = 50 kJ/mol, this is approximately 10^(-9) or one in a billion collisions!
The fraction of collisions with proper molecular orientation for reaction to occur.
Example: NO + O₃ → NO₂ + O₂ has p ≈ 0.8 (favorable), while H₂ + I₂ → 2HI has p ≈ 0.16 (unfavorable orientation requirement).
Combines all factors affecting reaction rate into a single temperature-dependent constant.
Collision Theory: k = p × Z × e^(-Ea/RT)
Arrhenius Form: k = A × e^(-Ea/RT)
where A = pre-exponential factor (≈ p × Z)
R = gas constant = 8.314 J/(mol·K)
T = absolute temperature (K)
The exponential term e^(-Ea/RT) is the Boltzmann factor representing the fraction of molecules with sufficient energy for reaction.
Shows exponential increase in rate constant with temperature. A 10 K temperature rise typically doubles or triples reaction rate (rough rule of thumb).
Used to calculate k at new temperature or determine Ea from rate constants at two temperatures.
Higher T → faster molecular speeds → more collisions per second. However, this effect is relatively minor (Z ∝ √T).
Higher T → greater fraction of molecules with E ≥ Ea. This is the dominant effect, as e^(-Ea/RT) changes exponentially with T.
For the reaction H₂(g) + I₂(g) → 2HI(g) at 700 K:
• Activation energy Ea = 165 kJ/mol
• Pre-exponential factor A = 1.0 × 10¹¹ L/(mol·s)
• Steric factor p = 0.16
Calculate the rate constant k and the fraction of molecules with sufficient energy for reaction.
Ea = 165 kJ/mol = 165,000 J/mol
R = 8.314 J/(mol·K)
T = 700 K
-Ea/RT = -165,000 / (8.314 × 700)
-Ea/RT = -165,000 / 5,820
-Ea/RT = -28.35
e^(-Ea/RT) = e^(-28.35) = 4.78 × 10^(-13)
This means only about 4.78 × 10^(-11)% of collisions have sufficient energy!
k = A × e^(-Ea/RT)
k = (1.0 × 10¹¹) × (4.78 × 10^(-13))
k = 4.78 × 10^(-2) L/(mol·s) = 0.0478 L/(mol·s)
The low steric factor (p = 0.16) indicates that only about 16% of energetically sufficient collisions have the correct orientation. Combined with the Boltzmann factor, this makes the overall reaction relatively slow despite high temperature. The reaction requires proper alignment of H₂ and I₂ molecules for H-I bond formation.
Design combustion systems by predicting ignition temperatures and flame speeds. Optimize fuel-air mixtures for engines and furnaces. Calculate minimum ignition energy for safety assessments.
Understand how catalysts work by lowering activation energy. Evaluate catalyst effectiveness by measuring changes in Ea. Design more efficient industrial catalytic processes.
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Determine optimal reaction temperatures balancing rate and selectivity. Design reactor systems with appropriate heating/cooling. Minimize energy costs while maximizing production efficiency.
Determine what's provided: Ea, T, A, p, molecular properties (diameter, mass, concentration). Note units carefully - convert kJ to J, °C to K, etc.
Collision frequency: Z = σ × v̄ × n² / √2. Rate constant: k = A × e^(-Ea/RT). Temperature comparison: ln(k₂/k₁) = (Ea/R)(1/T₁ - 1/T₂).
Calculate intermediate values first (collision cross-section, mean speed, -Ea/RT). Use scientific notation for very large or small numbers. Check that units cancel properly.
Does the answer make sense? Higher T should give higher k. Larger Ea should give smaller k. Collision frequencies should be very large numbers (~10²⁸-10³⁴).
R = 8.314 J/(mol·K), so Ea must be in J/mol. Using kJ/mol gives drastically wrong results.
✓ Always convert: 1 kJ/mol = 1000 J/mol.
Temperature must be in Kelvin for Arrhenius equation. Using °C gives completely wrong exponential.
✓ Convert: T(K) = T(°C) + 273.15.
p is not the same as e^(-Ea/RT). Steric factor accounts for orientation; Boltzmann factor accounts for energy. Both reduce successful collisions.
✓ Overall success rate = p × e^(-Ea/RT).
The collision frequency Z increases only as √T (minor effect), but the Boltzmann factor e^(-Ea/RT) changes exponentially (dominant effect).
✓ Temperature primarily affects k through the exponential term, not collision frequency.