Result

Collision theory

T-041Home CU-202Threads kinetics

Statement

Rate from collision frequency, energy and orientation.

Why it matters

rate-law-order and integrated-rate-laws describe how concentration affects rate purely empirically, as observed proportionalities, without explaining the underlying physical mechanism that produces them. Collision theory supplies exactly that missing physical picture at the molecular level: a reaction happens because molecules collide, and only a fraction of those collisions are both energetic and correctly oriented enough to actually react, a picture that directly explains the empirical Arrhenius equation's exponential form and pre-exponential factor rather than leaving them as purely fitted parameters.

Collision theory is also the natural companion, and to some extent the precursor, to transition-state-theory, which develops a more rigorous, thermodynamically framed treatment of the same activated-crossing idea; both, together with steady-state-approximation's treatment of multi-step mechanisms, form the mechanistic backbone of the kinetics unit.

Hypotheses

Molecules are treated as hard spheres of a definite collision cross-section \(\sigma\), colliding according to the kinetic theory of gases.This is a deliberate simplification of real molecular shape and the true, softer intermolecular potential, but it allows the collision frequency to be computed from well-established kinetic theory results (mean relative speed, number density) without needing a detailed intermolecular potential. Only collisions with kinetic energy along the line of centres exceeding the activation energy \(E_a\) can lead to reaction.This links directly back to the Boltzmann distribution used in deriving the Arrhenius equation: the fraction of collisions energetic enough to react is exactly the same exponential Boltzmann fraction, \(e^{-E_a/RT}\), applied here specifically to the relative kinetic energy of a colliding pair. Colliding molecules must also be correctly oriented relative to one another for reaction to occur, captured by an empirical steric factor \(p\) (typically \(p\le1\)).For simple atom-transfer reactions between small, nearly spherical species, \(p\) is often reasonably close to \(1\); for reactions requiring a specific, geometrically demanding approach (e.g. attack at a particular carbon in a larger organic molecule), \(p\) can be several orders of magnitude smaller than \(1\), reflecting how narrow the range of reactive orientations actually is.

Proof

\[ Z = \sigma\,\bar{v}_{\text{rel}}\,[\text{A}][\text{B}]\,N_A \]

The total collision frequency between molecules of A and B per unit volume per unit time, a standard result from the kinetic theory of gases, is proportional to the collision cross-section \(\sigma\), the mean relative speed \(\bar v_{\text{rel}}\) (itself proportional to \(\sqrt{T}\)), and the number densities (concentrations) of both reacting species. B

\[ f_{\text{energy}} = e^{-E_a/RT} \]

Of all the collisions occurring at frequency \(Z\), only the fraction with sufficient kinetic energy along the line of centres to overcome the activation barrier actually leads to reaction; this fraction follows the same Boltzmann exponential already used to derive the empirical Arrhenius equation directly. A

\[ \text{rate} = p\cdot Z\cdot f_{\text{energy}} = p\,\sigma\,\bar{v}_{\text{rel}}\,N_A\,e^{-E_a/RT}\,[\text{A}][\text{B}] \]

Multiplying the total collision frequency by both the energetic fraction (Step 2) and the steric (orientation) factor \(p\) gives the overall reaction rate, expressed explicitly as a second-order rate law in the two reactant concentrations, with a rate constant built entirely from molecular-level quantities. A

\[ k = p\,\sigma\,\bar{v}_{\text{rel}}\,N_A\,e^{-E_a/RT} \equiv A\,e^{-E_a/RT} \]

Comparing this explicit, molecular-level rate constant directly with the empirical Arrhenius equation identifies the pre-exponential factor \(A\) with the product of the steric factor, collision cross-section, and mean relative speed — collision theory's central achievement, giving Arrhenius's previously purely empirical \(A\) a genuine physical interpretation. A

Result

\[ k = p\,\sigma\,\bar{v}_{\text{rel}}\,N_A\,e^{-E_a/RT} \]

Reading. A reaction's rate constant is built from three physically distinct factors: how often molecules collide at all, what fraction of those collisions are energetic enough to react, and what fraction are additionally oriented correctly — giving the empirically fitted Arrhenius parameters \(A\) and \(E_a\) a concrete molecular interpretation.

Scope. Works best for simple, small-molecule gas-phase reactions with roughly hard-sphere-like collision behaviour (Hypotheses); the steric factor \(p\), while conceptually clear, is rarely calculable from first principles for complex molecules and is usually treated as an empirically fitted correction instead.

Corollaries & converses

Because \(\bar v_{\text{rel}}\propto\sqrt T\), collision theory predicts a mild, additional \(\sqrt T\) temperature dependence in \(A\) beyond the dominant exponential term, consistent with the modified Arrhenius form \(k=AT^{1/2}e^{-E_a/RT}\) sometimes used for a more precise fit over a wide temperature range.
transition-state-theory refines this picture further, replacing the empirical steric factor \(p\) with a more rigorous statistical-thermodynamic treatment of the activated complex's entropy, giving a more complete, first-principles account of the pre-exponential factor.
Converse: an experimentally determined pre-exponential factor \(A\) that is dramatically smaller than the simple collision-frequency estimate (\(\sigma\bar v_{\text{rel}}N_A\) alone, with \(p=1\)) is itself evidence of a small steric factor, i.e. a reaction with unusually stringent orientational requirements.

Fails without

Assume every collision with sufficient energy also leads to reaction, ignoring orientation: without accounting for the steric factor (Hypotheses, third point), the predicted rate constant overestimates the true value, sometimes by several orders of magnitude for geometrically demanding reactions.
Apply simple hard-sphere gas-kinetic collision theory to a solution-phase reaction: solvent caging and mediated encounters mean reacting molecules no longer behave as freely colliding hard spheres (Hypotheses, first point), so the simple gas-phase collision-frequency formula of Step 1 no longer applies unmodified.

Common errors

Assuming every molecular collision leads to reaction; only the small fraction that is simultaneously energetic enough and correctly oriented actually reacts (Steps 2-3).
Treating the steric factor \(p\) as always close to \(1\); for many real reactions, especially those involving larger or more geometrically complex molecules, \(p\) can be very much smaller.
Forgetting that collision frequency \(Z\) itself already carries a weak temperature dependence (through \(\bar v_{\text{rel}}\propto\sqrt T\)), separate from and much weaker than the dominant exponential temperature dependence carried by the energetic fraction.
Applying simple hard-sphere collision theory uncritically to complex, multi-step, or condensed-phase (solution) reactions, where the underlying hard-sphere gas-kinetic assumptions (Hypotheses) are much less directly applicable.

Discussion

Collision theory was developed in the early twentieth century, following soon after Arrhenius's 1889 empirical rate-temperature relationship, as chemists sought a genuine physical mechanism to explain why reaction rates depended on temperature in exactly that exponential way; it represented the first serious attempt to connect macroscopic reaction kinetics directly to the microscopic kinetic theory of gases already well established for other properties (pressure, diffusion, viscosity).

Common misconception: that the steric factor \(p\) is simply a fudge factor introduced to make the theory fit poorly-behaved reactions, with no independent physical meaning. In fact \(p\) has a genuine geometric interpretation, related directly to the fraction of the full range of relative molecular orientations at collision that are actually capable of leading to reaction — a real physical quantity that transition-state-theory later recasts more rigorously in terms of the activated complex's rotational and configurational entropy.

Worked examples

\[ \text{Estimated collision-theory } A \approx 10^{11}\,\text{L mol}^{-1}\text{s}^{-1}\text{ (with } p=1\text{)}; \qquad \text{measured } A \approx 4\times10^{8}\,\text{L mol}^{-1}\text{s}^{-1} \]

A bimolecular reaction's pre-exponential factor is measured experimentally and compared against the simple hard-sphere collision-frequency estimate (Step 1 of the Proof, evaluated with a typical molecular cross-section and relative speed, assuming every correctly energetic collision reacts). A

\[ p = \frac{A_{\text{measured}}}{A_{\text{collision estimate}}} = \frac{4\times10^{8}}{10^{11}} \approx 4\times10^{-3} \]

The measured pre-exponential factor is roughly \(250\) times smaller than the simple collision-frequency estimate, implying a steric factor of only about \(0.004\); this reaction evidently requires a fairly narrow, specific relative orientation between the colliding molecules for reaction to succeed, consistent with a moderately geometrically demanding bimolecular mechanism. A

\[ p \approx 4\times10^{-3} \]

Reading. Comparing a measured pre-exponential factor against the simple collision-frequency estimate gives a direct, quantitative measure of how orientationally demanding a given reaction actually is.

Scope. Reactions between simple, nearly spherical atoms or small radicals typically show \(p\) much closer to \(1\) than this example, while reactions demanding a specific approach geometry to a larger molecule can show \(p\) many orders of magnitude smaller still.

Further example. For the gas-phase bimolecular decomposition of nitrogen dioxide, \(2\,\text{NO}_2 \rightarrow 2\,\text{NO} + \text{O}_2\), the experimentally determined pre-exponential factor is \(A_{\text{exp}} = 2.5\times10^{9}\,\text{L mol}^{-1}\text{s}^{-1}\). Using the hard-sphere collision theory model, the estimated collision frequency factor is \(Z_{AB} = 4.0\times10^{11}\,\text{L mol}^{-1}\text{s}^{-1}\). Calculate the steric factor \(p\) and discuss its physical implication for this specific reaction mechanism.

\[ p = \frac{A_{\text{exp}}}{Z_{AB}} \]

The steric factor \(p\) is defined as the ratio of the experimentally observed pre-exponential factor to the theoretical collision frequency factor. This ratio quantifies the fraction of collisions that possess the correct spatial orientation for reaction, assuming the energy criterion is met. A

\[ p = \frac{2.5\times10^{9}}{4.0\times10^{11}} \]

Substituting the given values for the experimental and theoretical pre-exponential factors into the definition. A

\[ p = \frac{2.5}{4.0} \times 10^{9-11} = 0.625 \times 10^{-2} = 6.25\times10^{-3} \]

Performing the arithmetic: dividing the coefficients \(2.5\) by \(4.0\) gives \(0.625\), and subtracting the exponents \(9\) and \(11\) gives \(10^{-2}\). Adjusting to standard scientific notation yields \(6.25\times10^{-3}\). A

\[ p \approx 0.006 \]

Rounding to two significant figures, consistent with the precision of the input data. This value indicates that only about \(0.6\%\) of correctly energetic collisions lead to reaction. A

\[ p \approx 6.3\times10^{-3} \]

Reading. The steric factor is significantly less than 1, indicating that the reaction is highly orientation-dependent. For this reaction of \(\text{NO}_2\), the molecules must approach in a specific geometry to allow for the breaking of one \(\text{N}-\text{O}\) bond and the formation of the \(\text{O}-\text{O}\) bond in the transition state; random orientations do not facilitate this specific bond rearrangement.

Scope. This analysis assumes the transition state theory framework where the pre-exponential factor is dominated by collision frequency and steric effects. It holds for gas-phase reactions where intermolecular forces are negligible compared to kinetic energy, and the hard-sphere approximation is a reasonable first-order model for molecular size.

Problems

A reaction has a collision-theory-estimated \(A\approx5\times10^{10}\,\text{L mol}^{-1}\text{s}^{-1}\) and a measured \(A\approx5\times10^{9}\,\text{L mol}^{-1}\text{s}^{-1}\). Estimate the steric factor \(p\).
Solution
\(p=A_{\text{measured}}/A_{\text{estimate}} = 5\times10^{9}/5\times10^{10} = 0.1\), suggesting roughly \(1\) in \(10\) correctly energetic collisions is also correctly oriented for reaction.
Explain, using collision theory, why increasing the temperature of a reaction mixture increases the rate constant far more through the energetic fraction (Step 2) than through the collision frequency itself (Step 1).
Solution
The collision frequency \(Z\) depends on temperature only mildly, through \(\bar v_{\text{rel}}\propto\sqrt T\) (Corollaries); the energetic fraction, by contrast, depends on temperature exponentially, \(e^{-E_a/RT}\). For any activation energy that is not extremely small compared with \(RT\), the exponential term's sensitivity to a given fractional change in \(T\) vastly exceeds the much weaker square-root dependence of \(Z\), so essentially all of the observed rate increase with temperature comes from the growing energetic fraction, not from molecules simply colliding more often.
Why might collision theory, in its simple hard-sphere form, be expected to work less well for a reaction occurring in solution than for the same reaction occurring in the gas phase?
Solution
The Hypotheses assume molecules behave as free, hard spheres colliding according to gas-kinetic theory; in solution, reacting molecules are instead surrounded by solvent molecules, which can cage reactants together for extended periods (increasing the effective number of encounters between a given pair without necessarily increasing true collision frequency in the gas-kinetic sense), impede their approach, or otherwise mediate the encounter in ways the simple gas-phase collision-frequency formula (Step 1) does not account for at all.
For a bimolecular reaction in the gas phase, the collision frequency factor is estimated to be \(Z = 1.2\times10^{11}\,\text{L mol}^{-1}\text{s}^{-1}\). If the experimentally determined pre-exponential factor is \(A = 3.6\times10^{10}\,\text{L mol}^{-1}\text{s}^{-1}\), calculate the steric factor \(p\). Interpret the result in terms of the probability of a successful collision.
Solution
The steric factor is calculated as the ratio of the experimental pre-exponential factor to the collision frequency factor: \[ p = \frac{A}{Z} = \frac{3.6\times10^{10}}{1.2\times10^{11}} \] \[ p = \frac{3.6}{1.2} \times 10^{10-11} = 3.0 \times 10^{-1} = 0.30 \] The steric factor is \(0.30\). This implies that approximately \(30\%\) of collisions with sufficient energy are also correctly oriented to lead to a reaction. This is a relatively high steric factor, suggesting that the reaction is not extremely sensitive to the specific orientation of the colliding molecules, which is typical for reactions involving small, symmetric molecules or those with loose transition states.
For the gas-phase reaction \(\text{K} + \text{Br}_2 \rightarrow \text{KBr} + \text{Br}\), the measured pre-exponential factor is about five times the hard-sphere collision value, i.e. the steric factor \(p \approx 5\). How can \(p\) exceed 1, and what does the “harpoon” mechanism propose?
Solution
\(p\) compares the measured rate with collisions at the hard-sphere (contact) distance, so \(p \gt 1\) means the reaction happens at distances larger than the molecules’ size. In the harpoon mechanism the K atom transfers its electron to Br\(_2\) at a large separation \(R^*\), where the energy cost (ionisation energy of K minus electron affinity of Br\(_2\)) is repaid by the Coulomb attraction, \(e^2/4\pi\varepsilon_0R^* = I - E_{ea}\), giving \(R^*\) of several hundred pm. The resulting ion pair K\(^+\)Br\(_2^-\) is pulled together and reacts, so the effective cross-section \(\pi R^{*2}\) is several times the hard-sphere value.