3.8485e-19
m²
2.4797e-26
kg
652.18
m/s
1.5115e+8
m³/(mol·s)
1.9697e-9
2.9772e-2
m³/(mol·s)
3.8485e-19
m²
2.4797e-26
kg
652.18
m/s
1.5115e+8
m³/(mol·s)
1.9697e-9
2.9772e-2
m³/(mol·s)
The Collision Theory Calculator computes the rate constant of a bimolecular gas-phase reaction using the fundamental principles of collision theory. This theory, developed by Max Trautz and William Lewis in the early 1900s, explains reaction rates in terms of molecular collisions. It proposes that for a reaction to occur, molecules must (1) collide, (2) with sufficient energy to overcome the activation energy barrier, and (3) with the correct orientation. The calculator determines the collision frequency from molecular parameters, applies the Boltzmann energy factor, and incorporates the steric factor to predict the rate constant. This provides a first-principles estimate of reaction rates and helps bridge the gap between molecular properties and macroscopic kinetics.
Collision theory expresses the rate constant as:
$$k = p \cdot Z_{AB} \cdot e^{-E_a/RT}$$
where p is the steric factor and Z_AB is the collision frequency per unit concentration:
$$Z_{AB} = N_A \cdot \sigma_{AB}^2 \cdot \pi \cdot \bar{v}_{rel}$$
The average relative speed of two molecules comes from kinetic molecular theory:
$$\bar{v}_{rel} = \sqrt{\frac{8 k_B T}{\pi \mu}}$$
where μ is the reduced mass of the two molecules:
$$\mu = \frac{m_A \cdot m_B}{m_A + m_B}$$
The collision cross-section σ_AB is the average of the two molecular diameters. The Boltzmann factor e^(−Ea/RT) gives the fraction of collisions with energy ≥ Ea. The steric factor p (0 to 1) accounts for the requirement of proper molecular orientation.
The collision frequency Z_AB tells you how often molecules collide per second — typically 10⁹-10¹¹ collisions per mol per second in gas phase. However, only a tiny fraction of these collisions have enough energy (Boltzmann factor) and correct orientation (steric factor) to produce a reaction. The calculated rate constant k can be compared with experimental values: good agreement validates the collision theory model, while significant discrepancies suggest more complex mechanisms (tunneling, long-range interactions, or multi-step pathways).
Inputs
Results
Reduced mass μ = (28×32)/(28+32)/1000/6.022e23 = 2.48×10⁻²⁶ kg. Relative speed ≈ 607 m/s. Z_AB ≈ 1.41×10¹¹. With p = 0.1 and Ea = 50 kJ/mol: k = 0.1 × 1.41×10¹¹ × e⁻²⁰·¹⁷ ≈ 2.69 m³/(mol·s).
Inputs
Results
Light H₂ colliding with heavy I₂ at high temperature gives a very fast relative speed. With low Ea = 10 kJ/mol and p = 0.5, the Boltzmann factor is large (e⁻²·⁴ ≈ 0.09), yielding k ≈ 6.67×10¹¹.
Collision theory explains chemical reaction rates by treating molecules as hard spheres that must collide with sufficient energy and proper orientation to react. It provides a molecular-level explanation for the Arrhenius equation and connects macroscopic kinetics to microscopic molecular behavior.
The collision cross-section (σ) is the effective diameter of a molecule for collision purposes. It is approximately the sum of the radii of the two colliding molecules. Typical values for small molecules are 2-5 Å (2-5 × 10⁻¹⁰ m).
The steric factor (p) accounts for the fact that molecules must collide with the correct relative orientation for reaction to occur. It ranges from nearly 1 (atoms, where orientation doesn't matter) to 10⁻⁸ or less (large complex molecules requiring very specific alignment).
Collision theory treats molecules as featureless hard spheres, ignoring: quantum tunneling, long-range attractive forces, molecular vibrations and rotations, and complex potential energy surfaces. For these reasons, transition state theory often provides better quantitative predictions.
Lower reduced mass means faster relative molecular speeds and more frequent collisions. This is why reactions involving light molecules (H₂, He) tend to have higher collision frequencies than those between heavy molecules.
Not directly. In solution, molecules are constantly surrounded by solvent, and the collision frequency is governed by diffusion rates rather than free molecular motion. The Smoluchowski equation handles diffusion-controlled reactions in solution.
Collision theory uses a simple collision model with hard spheres. Transition state theory (Eyring theory) considers the detailed potential energy surface and the properties of the transition state (activated complex). TST provides more accurate predictions and separates entropic and enthalpic contributions.
Use gas kinetic diameters from viscosity or diffusion measurements. Common values: H₂ = 2.89 Å, N₂ = 3.68 Å, O₂ = 3.46 Å, CO₂ = 3.94 Å, CH₄ = 3.78 Å. For the pair AB, σ_AB ≈ (σ_A + σ_B)/2.
Collision theory predicts k ∝ T^(1/2) × e^(−Ea/RT). The T^(1/2) factor comes from the average molecular speed. This weak pre-exponential temperature dependence is usually dominated by the exponential Arrhenius factor.
At standard conditions (1 atm, 298 K), a single molecule undergoes about 10⁹-10¹⁰ collisions per second. The total collision rate in a mole of gas is about 10³⁴ collisions per liter per second, but only a tiny fraction leads to reaction.
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