Collision Theory of Reaction Rates

Drop two reactive gas molecules into a flask and ask a simple question. How often do they actually meet? How often does meeting lead to reaction? Collision theory answers both halves quantitatively. It starts from nothing more exotic than billiard-ball geometry and the Maxwell-Boltzmann speed distribution. The theory matters because it converts the abstract Arrhenius equation into something concrete. That something is built from measurable molecular properties: sizes, masses, temperature, and an energy threshold. It also exposes where simple geometry fails. This motivates the steric factor and eventually transition-state theory. This article derives the collision frequency between hard spheres. It layers in the Boltzmann energy criterion, and defines the steric factor $P$ . It then shows exactly how the resulting rate constant compares to the empirical Arrhenius $A$ and $E_a$ .

Key takeaways

Collision theory models molecules as hard spheres and counts collisions per unit volume per unit time using kinetic theory.
Only collisions with kinetic energy along the line of centers exceeding $E_a$ can react; this gives a Boltzmann factor $e^{-E_a/RT}$ .
The simple hard-sphere prediction for $A$ is almost always too large compared to experiment.
A steric factor $P$ (typically $10^{-1}$ to $10^{-6}$ ) corrects for orientation requirements neglected by pure geometry.
Collision theory predicts a mild $T^{1/2}$ dependence in the pre-exponential factor, refining the Arrhenius picture.
Key idea: the rate constant is the collision frequency times the fraction of collisions with enough energy times the fraction correctly oriented.

The core insight

A bimolecular rate constant is the product of three factors: how often molecules collide (kinetic theory geometry), what fraction of those collisions carry enough energy along the line of centers to cross the barrier (a Boltzmann factor built on $E_a$ ), and what fraction are also geometrically arranged to react (the steric factor $P$ ); collision theory computes the first two from first principles and absorbs everything else into $P$ .

The physical picture

Picture two spherical molecules, A and B, with radii $r_A$ and $r_B$ , moving through a gas. They react only if their centers approach within a critical distance, the collision diameter $d_{AB} = r_A + r_B$ . Kinetic theory tells us how many such close encounters occur per unit time and volume, given the number densities and the relative speed distribution set by temperature. Not every geometric collision leads to reaction. Chemical bonds must break and form, which costs energy. Collision theory imposes an energy criterion: only the component of kinetic energy directed along the line joining the two centers at impact counts toward overcoming the barrier. A glancing, high-speed collision may still fail if most of the kinetic energy is transverse. Finally, even an energetic head-on collision can fail if the molecules are not oriented so that the reactive atoms meet. A steric or orientation requirement, quantified by the empirical steric factor $P$ , closes the gap between hard-sphere theory and real rate data.

Figure

Line-of-centers geometry for a bimolecular collision

Figure

Predicted vs observed rate constant on an Arrhenius plot

The core relations

z_{AB} = \sigma\, \bar{v}_{rel}\, [A][B]\, N_A^2 \times 10^{3}

where $z_{AB}$ is the collision number per unit volume per unit time (SI units mol $^{-1}$ s $^{-1}$ m $^{-3}$ scaling handled by constants), $\sigma = \pi d_{AB}^2$ is the collision cross-section, and $\bar{v}_{rel}$ is the mean relative speed.

\bar{v}_{rel} = \left(\frac{8k_BT}{\pi\mu}\right)^{1/2}

where $\mu = m_Am_B/(m_A+m_B)$ is the reduced mass and $k_B$ is Boltzmann's constant.

k(T) = N_A\, \sigma\, \bar{v}_{rel}\, e^{-E_a/RT}

where $k(T)$ is the bimolecular rate constant (per mole basis), and $e^{-E_a/RT}$ is the fraction of collisions with line-of-centers energy exceeding $E_a$ .

k(T) = P\, N_A\, \sigma\, \left(\frac{8k_BT}{\pi\mu}\right)^{1/2} e^{-E_a/RT}

where $P$ is the steric factor, empirically defined as $P = A_{exp}/A_{theory}$ .

A_{theory}(T) = N_A\,\sigma\left(\frac{8k_BT}{\pi\mu}\right)^{1/2}

where $A_{theory}$ is the collision-theory pre-exponential factor, showing an explicit $T^{1/2}$ dependence absent from the simplest Arrhenius picture.

Derivation AFrom collision counting to the rate constant

Goal. Build $k(T)$ from kinetic theory of gases plus a threshold-energy cutoff, then identify the Arrhenius parameters it predicts.

Step 1: Relative speed distribution. Treat A and B as structureless hard spheres of mass $m_A$ , $m_B$ , colliding in three dimensions. Kinetic theory shows that the relative motion of the pair behaves like a single particle of reduced mass $\mu = m_Am_B/(m_A+m_B)$ moving with a Maxwell-Boltzmann speed distribution at temperature $T$ . Its mean speed is

\bar{v}_{rel} = \left(\frac{8k_BT}{\pi\mu}\right)^{1/2}.

This result follows from integrating the Maxwell-Boltzmann speed distribution $f(v) \propto v^2 e^{-\mu v^2/2k_BT}$ over all speeds, exactly as for a single-species mean speed but with $m$ replaced by $\mu$ .

Step 2: Collision cross-section. Two spheres collide whenever their centers approach within $d_{AB} = r_A + r_B$ . Viewed from the frame of B, A sweeps out an effective target area (cross-section)

\sigma = \pi d_{AB}^2.

In a time $dt$ , A effectively sweeps a cylindrical volume $\sigma \bar{v}_{rel}\,dt$ ; any B molecule with center inside that volume gets struck.

Step 3: Collision frequency density. Multiplying the swept volume rate by the number density of B, $\mathcal{N}_B$ (molecules per unit volume), gives the collision rate for one A molecule with all B molecules, $\sigma \bar{v}_{rel}\mathcal{N}_B$ . Multiplying by the number density of A, $\mathcal{N}_A$ , gives the total collision density

Z_{AB} = \sigma\, \bar{v}_{rel}\, \mathcal{N}_A \mathcal{N}_B.

Converting number densities to molar concentrations ( $\mathcal{N}_A = N_A[A]$ , etc.) and folding Avogadro's number appropriately yields a rate of collisions expressed per unit volume per unit time in mol L $^{-1}$ s $^{-1}$ , once the theory is expressed on a per-mole basis.

Step 4: Applying the energy criterion. Not all collisions react — only those whose kinetic energy along the line of centers exceeds the threshold $E_a$ . Decomposing the relative kinetic energy into a component along the collision axis and components perpendicular to it, and integrating the Boltzmann distribution over all collision energies above $E_a$ , gives the fraction of collisions that qualify:

f = e^{-E_a/RT}.

This is the same exponential that appears empirically in the Arrhenius equation, but here it emerges from a specific microscopic energy criterion rather than being fitted.

Step 5: Assembling the rate law. If every energetically qualifying collision reacted with perfect efficiency, the rate constant predicted by pure hard-sphere geometry would be

k_{theory}(T) = N_A\, \sigma\, \bar{v}_{rel}\, e^{-E_a/RT} = N_A\,\sigma\left(\frac{8k_BT}{\pi\mu}\right)^{1/2} e^{-E_a/RT}.

Comparing this with the empirical Arrhenius form $k = Ae^{-E_a/RT}$ identifies

A_{theory} = N_A\, \sigma\left(\frac{8k_BT}{\pi\mu}\right)^{1/2}.

Step 6: Introducing the steric factor. Experiment almost always gives a smaller pre-exponential factor than $A_{theory}$ , because molecules must also be correctly oriented, and because internal energy redistribution and non-spherical shapes are ignored by the hard-sphere model. Defining the ratio

P \equiv \frac{A_{exp}}{A_{theory}}

absorbs this discrepancy, giving the corrected, experimentally calibrated collision-theory rate constant

\boxed{k(T) = P\, N_A\, \sigma\left(\frac{8k_BT}{\pi\mu}\right)^{1/2} e^{-E_a/RT}}.

Worked examples

Example 1 — Predicting $A_{theory}$ for a simple gas-phase reaction. Consider the classic bimolecular reaction between two hard-sphere-like species with collision diameter $d_{AB} = 4.0 \times 10^{-10}$ m (400 pm, typical for small organic radicals) at $T = 500$ K. Take both masses equal to $m = 50$ u (a small radical), so

\mu = \frac{m \cdot m}{m+m} = \frac{m}{2} = 25\ \text{u} = 25 \times 1.6605\times10^{-27}\ \text{kg} = 4.151\times10^{-26}\ \text{kg}.

Compute the mean relative speed:

\bar{v}_{rel} = \left(\frac{8k_BT}{\pi\mu}\right)^{1/2} = \left(\frac{8(1.381\times10^{-23}\,\text{J K}^{-1})(500\,\text{K})}{\pi(4.151\times10^{-26}\,\text{kg})}\right)^{1/2}.

Numerator: $8 \times 1.381\times10^{-23} \times 500 = 5.524\times10^{-20}$ J. Denominator: $\pi \times 4.151\times10^{-26} = 1.304\times10^{-25}$ kg. Ratio: $5.524\times10^{-20}/1.304\times10^{-25} = 4.236\times10^{5}\ \text{m}^2\text{s}^{-2}$ . Square root: $\bar{v}_{rel} = 651\ \text{m s}^{-1}$ . Compute the cross-section:

\sigma = \pi d_{AB}^2 = \pi(4.0\times10^{-10}\,\text{m})^2 = 5.03\times10^{-19}\ \text{m}^2.

Assemble $A_{theory}$ (per-molecule basis, then convert to a molar rate constant by multiplying by $N_A$ ):

A_{theory} = N_A\,\sigma\,\bar{v}_{rel} = (6.022\times10^{23}\,\text{mol}^{-1})(5.03\times10^{-19}\,\text{m}^2)(651\,\text{m s}^{-1}).

Multiply: $6.022\times10^{23} \times 5.03\times10^{-19} = 3.03\times10^{5}$ ; then $3.03\times10^{5}\times 651 = 1.97\times10^{8}\ \text{m}^3\,\text{mol}^{-1}\,\text{s}^{-1}$ . Convert to the conventional L mol $^{-1}$ s $^{-1}$ : $1\ \text{m}^3 = 10^3$ L, so

A_{theory} \approx 2.0\times10^{11}\ \text{L mol}^{-1}\text{s}^{-1}.

This is a typical hard-sphere estimate: collision theory routinely predicts $A$ values around $10^{10}$ – $10^{11}$ L mol $^{-1}$ s $^{-1}$ for small-molecule gas reactions, consistent with the classic textbook range.

Example 2 — Extracting the steric factor from experimental Arrhenius data. The gas-phase reaction

\text{NO} + \text{Cl}_2 \rightarrow \text{NOCl} + \text{Cl}

has an experimentally measured Arrhenius pre-exponential factor of $A_{exp} = 4.0\times10^{9}\ \text{L mol}^{-1}\text{s}^{-1}$ near 300 K (representative literature value for this class of reaction), with collision diameter $d_{AB} \approx 4.5\times10^{-10}$ m and reduced mass using $m_{NO}=30.0$ u, $m_{Cl_2}=70.9$ u:

\mu = \frac{(30.0)(70.9)}{30.0+70.9}\ \text{u} = \frac{2127}{100.9}\ \text{u} = 21.1\ \text{u} = 3.50\times10^{-26}\ \text{kg}.

At $T = 300$ K:

\bar{v}_{rel} = \left(\frac{8(1.381\times10^{-23})(300)}{\pi(3.50\times10^{-26})}\right)^{1/2}.

Numerator: $8\times1.381\times10^{-23}\times300 = 3.314\times10^{-20}$ J. Denominator: $\pi\times3.50\times10^{-26}=1.100\times10^{-25}$ kg. Ratio: $3.014\times10^{5}\ \text{m}^2\text{s}^{-2}$ ; square root gives $\bar{v}_{rel} = 549\ \text{m s}^{-1}$ . Cross-section:

\sigma = \pi(4.5\times10^{-10})^2 = 6.36\times10^{-19}\ \text{m}^2.

Theoretical pre-exponential factor:

A_{theory} = N_A\,\sigma\,\bar{v}_{rel} = (6.022\times10^{23})(6.36\times10^{-19})(549).

$6.022\times10^{23}\times6.36\times10^{-19} = 3.83\times10^{5}$ ; then $\times 549 = 2.10\times10^{8}\ \text{m}^3\,\text{mol}^{-1}\text{s}^{-1} = 2.10\times10^{11}\ \text{L mol}^{-1}\text{s}^{-1}$ . Steric factor:

P = \frac{A_{exp}}{A_{theory}} = \frac{4.0\times10^{9}}{2.10\times10^{11}} = 0.019.

So roughly 2% of energetically qualified, geometrically colliding pairs actually react — a modest orientation requirement, consistent with NO and Cl $_2$ needing a specific approach geometry to transfer a chlorine atom. This value sits comfortably in the commonly cited $P = 10^{-1}$ to $10^{-3}$ range for moderately demanding reactions, and it is far from the extreme $P \sim 10^{-6}$ seen in highly stereospecific reactions.

Limiting cases & common traps

Situation	What collision theory predicts	Common pitfall
$P \approx 1$	Simple atom-transfer or radical recombination reactions	Assuming this is universal; it is the exception
$P \ll 1$ (10 $^{-2}$ –10 $^{-6}$ )	Reactions needing specific orientation (large organics, stereospecific steps)	Blaming "steric hindrance" alone; entropy of activation and energy redistribution also matter
$P > 1$ (rare)	Harpoon mechanism in some alkali-metal + halogen reactions, long-range electron transfer	Treating $P>1$ as an error; it signals cross-sections larger than hard-sphere geometry
High $T$	$A_{theory} \propto T^{1/2}$	Ignoring this and treating Arrhenius $A$ as strictly $T$ -independent over wide ranges
Complex, floppy molecules	Hard-sphere model breaks down; internal degrees of freedom matter	Applying collision theory numbers directly without invoking transition-state theory

The chemist behind it

Collision theory in its modern quantitative form is usually credited to Max Trautz (1916) and William Lewis (1918), who independently combined kinetic theory of gases with the Arrhenius activation energy to predict absolute rate constants for gas-phase reactions. Lewis's 1918 paper on the decomposition of hydrogen iodide is often cited as the first clean test of the theory against experimental Arrhenius parameters, revealing both its successes for simple atom-transfer reactions and its systematic overestimate of $A$ for anything more structurally demanding — the gap that the steric factor was invented to patch.

Origins · 1 minWilliam C. McC. Lewis and how this was discovered

William C. McC. Lewis

1885–1956 · British physical chemist who applied kinetic theory of gas collisions to the Arrhenius rate law, producing the first quantitative collision-theory rate constants

Ask the tutor: Want to see how transition-state theory replaces the fitted steric factor with an actual entropy of activation calculated from partition functions?

Why does collision theory sometimes predict

P

greater than one?

A steric factor above unity shows up in reactions like alkali-metal atoms reacting with halogens, where an electron transfers over a distance larger than the hard-sphere contact radius before the atoms actually touch — the so-called harpoon mechanism. The effective reactive cross-section is larger than the geometric hard-sphere cross-section used to compute $A_{theory}$ , so the ratio $A_{exp}/A_{theory}$ exceeds one. This does not violate the theory; it simply reveals that the true reactive cross-section differs from simple geometric contact.

How is the collision theory rate constant related to the Arrhenius equation?

Collision theory derives the same exponential Boltzmann form, $k = Ae^{-E_a/RT}$ , from molecular geometry and kinetic theory rather than fitting it empirically. It identifies the Arrhenius $A$ with $N_A\sigma\bar{v}_{rel}$ (times a steric factor), explaining physically why $A$ has units of a rate constant and predicting its mild $T^{1/2}$ temperature dependence, refining the strict Arrhenius assumption that $A$ is constant.

Does collision theory apply to reactions in solution?

Only with heavy modification. In solution, molecules are caged by solvent and undergo many correlated collisions before diffusing apart, so the simple gas-kinetic collision frequency no longer applies directly. Diffusion-controlled rate theories, which use the Stokes-Einstein relation for diffusion coefficients instead of the Maxwell-Boltzmann mean speed, replace the hard-sphere gas picture for solution-phase bimolecular reactions.

Why does transition-state theory eventually replace collision theory?

Collision theory treats the steric factor $P$ as a fitted number with no independent way to calculate it from first principles beyond crude geometric arguments. Transition-state theory replaces $P$ with a calculable entropy of activation, derived from partition functions of a hypothesized activated complex. This gives a predictive, rather than fitted, account of the same pre-exponential discrepancy, and it generalizes more naturally to condensed-phase and complex reactions.

What role does molecular size play in the predicted rate constant?

Larger collision diameters increase the geometric cross-section $\sigma = \pi d_{AB}^2$ , which increases the raw collision frequency and hence $A_{theory}$ . But larger, more complex molecules usually have far smaller steric factors, because only a small patch of the molecular surface is reactive. The two effects often roughly cancel, which is part of why simple hard-sphere collision counting alone cannot reliably predict rates for complex organic reactions.