Collision Theory and Activation Energy

The previous chapter defined reaction rate and showed how to measure it; this chapter explains, at the level of individual particles, why reactions have the rates they do, and why those rates respond so strongly to temperature, concentration and catalysts. Collision theory is the molecular-level model that answers these questions: it treats a chemical reaction as the outcome of particles colliding, and asks what must be true of a collision for it to actually produce new substances. From this single idea follow activation energy, the Maxwell-Boltzmann distribution, and — in the next chapter — a quantitative account of how rate responds to the conditions under which a reaction is run.

You should already be comfortable with the definition of reaction rate and the idea of a concentration-time graph from the previous chapter; this chapter builds the conceptual model that chapter 3 onward will quantify.

1. The Central Claim of Collision Theory

Collision theory states that for a reaction to occur between two particles, those particles must first collide. This sounds almost too obvious to state, but it immediately explains a wealth of qualitative observations: reactions in the gas phase or in solution, where particles move freely and collide often, can proceed; a reaction between two solids with no contact beyond their touching surfaces proceeds, if at all, only at that limited interface, which is why powdering a solid (increasing its surface area) so reliably speeds up heterogeneous reactions.

Definition. Collision theory models a chemical reaction as a two-stage requirement: particles must collide, and that collision must satisfy two further conditions (Sections 2 and 3) before it leads to a chemical change. A collision failing either condition simply bounces the particles apart unchanged.

However, collision alone is not sufficient. The vast majority of collisions between reacting particles do not result in a reaction — if they did, essentially every reaction would be instantaneous, which observation plainly contradicts. Two further requirements separate an effective collision, one that leads to a reaction, from the much larger number of ineffective (elastic) collisions, which leave the colliding particles chemically unchanged.

2. The Energy Requirement: Activation Energy

The first requirement is energetic. As particles approach each other closely enough to react, existing bonds must begin to stretch and break before new bonds can begin to form — and this bond-reorganisation passes through an unstable, high-energy arrangement of atoms called the transition state (or activated complex). Reaching the transition state costs energy, and that energy must come from somewhere: specifically, from the kinetic energy of the colliding particles, converted into potential energy as the bonds distort.

Definition. The activation energy, $E_a$, is the minimum kinetic energy, summed over the colliding particles, that a collision must possess for the particles to reach the transition state and react. A collision with kinetic energy below $E_a$ cannot overcome this energy barrier, however favourably the particles happen to be oriented.

This is most clearly shown on an energy profile diagram, which plots the potential energy of the reacting system against the reaction coordinate (a measure of progress from reactants to products, not a physical distance). The reactants sit at one energy level; the products sit at another (lower, for an exothermic reaction; higher, for an endothermic one); and between them rises a single hump, the transition state, whose height above the reactants is $E_a$.

Worked example 1. A reaction has reactants at a potential energy of $0~\text{kJ}\,\text{mol}^{-1}$ (taken as the reference), a transition state at $75~\text{kJ}\,\text{mol}^{-1}$, and products at $-40~\text{kJ}\,\text{mol}^{-1}$. Reading the diagram directly:

The reaction is exothermic ($\Delta H < 0$), yet it still requires $75~\text{kJ}\,\text{mol}^{-1}$ of activation energy to proceed — the size and sign of $\Delta H$ say nothing about how large $E_a$ is, a point returned to in Section 6. The activation energy of the reverse reaction, read from the same diagram, is the height of the hump above the products rather than the reactants:

Key relationship. For any reaction, $\Delta H = E_{a,\text{forward}} - E_{a,\text{reverse}}$. An exothermic forward reaction always has a smaller forward activation energy than reverse activation energy; an endothermic forward reaction always has a larger one.

3. The Orientation Requirement

Energy alone is not sufficient either. Many molecules are not simple spheres, and a bond can only form between the specific atoms positioned to interact — the reacting groups must meet with roughly the correct orientation relative to each other, not simply with enough combined energy.

Worked example 2. Consider the gas-phase reaction $\text{NO}_2(g) + \text{CO}(g) \rightarrow \text{NO}(g) + \text{CO}_2(g)$, in which an oxygen atom is transferred from $\text{NO}_2$ to the carbon atom of $\text{CO}$. A collision in which the carbon atom of $\text{CO}$ strikes an oxygen atom of $\text{NO}_2$ can, with enough energy, lead to oxygen transfer. A collision in which the carbon atom instead strikes the nitrogen atom of $\text{NO}_2$ cannot lead to this particular reaction however energetic it is, because the carbon is simply not positioned to bond with an oxygen atom. The fraction of collisions with a favourable orientation is a geometric property of the reacting species, essentially independent of temperature, and it is one reason why measured rates are often far smaller than the total collision frequency alone would suggest.

Definition. An effective collision is one that satisfies both requirements simultaneously: the combined kinetic energy along the line of approach is at least $E_a$, and the particles are correctly oriented for the necessary bonds to form and break.

4. The Maxwell-Boltzmann Distribution

In any sample of gas or solution at a fixed temperature, individual particles do not all move at the same speed; they possess a continuous range of kinetic energies, constantly redistributed by further collisions between the particles themselves. The Maxwell-Boltzmann distribution describes this spread as a continuous curve of the number of particles (vertical axis) against kinetic energy (horizontal axis).

The curve has several essential features worth committing to memory:

• It starts at the origin: no particle has exactly zero kinetic energy.
• It rises to a single peak (the most probable energy, not the mean energy) and then falls away, with a long tail extending to high energies, never quite reaching zero.
• The total area under the curve equals the total number of particles in the sample, a quantity that does not change as temperature changes.

Key distinction. The peak of the Maxwell-Boltzmann curve marks the most probable kinetic energy, not the mean. The curve's asymmetric shape — a sharper rise and a long high-energy tail — means the mean kinetic energy always lies slightly to the right of the peak.

Marking the activation energy $E_a$ as a vertical line on this curve divides the particles into two groups: those with $E < E_a$ (to the left, unable to react even in a head-on, perfectly oriented collision) and those with $E \ge E_a$ (to the right, under the tail, able to react if also correctly oriented). The area under the curve to the right of $E_a$, as a fraction of the total area, is the fraction of collisions energetic enough to react.

Worked example 3. At a fixed temperature, suppose $4\%$ of collisions have combined kinetic energy at or above $E_a$, and separately, one collision in five has a favourable orientation. The fraction of all collisions that are effective is the product of these independent probabilities:

If the total collision frequency in the sample is $2.0\times 10^{28}~\text{collisions}\,\text{dm}^{-3}\,\text{s}^{-1}$, the rate of effective (reaction-producing) collisions is

a number still astronomically large by everyday standards, which is exactly why even "slow" chemical reactions involve enormous numbers of particles reacting every second.

5. How Temperature Shifts the Distribution

Raising the temperature of a sample increases the average kinetic energy of its particles, which reshapes the Maxwell-Boltzmann curve in a precise way: the peak shifts to the right (toward higher energy) and becomes lower and broader, since the same total number of particles (fixed area) is now spread over a wider range of energies.

Crucially, the area under the curve beyond $E_a$ — the fraction of particles with enough energy to react — increases disproportionately for even a modest temperature rise, because that region sits under the distribution's long, sensitive tail. This disproportionate effect on the high-energy tail, not any change in the collision frequency itself, is what Chapter 3 identifies as the dominant reason reaction rate responds so strongly to temperature; collision frequency itself rises only modestly with temperature (roughly with $\sqrt{T}$), far too little to explain the large rate increases actually observed.

Warning. It is not that "more collisions happen" at higher temperature that chiefly explains the increased rate — collision frequency barely changes. It is that a much larger fraction of those collisions now exceed $E_a$, because of how the distribution's tail reshapes with temperature.

6. Activation Energy Is Independent of $\Delta H$

A recurring point of confusion is the relationship between activation energy and the enthalpy change of reaction. These are entirely independent quantities, fixed by different physical facts: $\Delta H$ depends only on the relative energies of reactants and products (a thermodynamic quantity, covered in Block B), while $E_a$ depends on the height of the energy barrier that must be crossed between them (a kinetic quantity, specific to the mechanism by which the reaction proceeds).

A strongly exothermic reaction can still have a very large activation energy and therefore be extremely slow at room temperature — the combustion of paper is hugely exothermic, yet paper does not spontaneously combust on a desk, precisely because its activation energy is far too large for a significant fraction of collisions to overcome at room temperature. Conversely, some endothermic reactions proceed readily because their activation energy happens to be small. A catalyst, developed fully in the next chapter, works by providing an alternative reaction pathway of lower $E_a$, without changing $\Delta H$ at all.

Common Exam Pitfalls

• Treating "collision" and "effective collision" as synonyms. The vast majority of collisions between reacting particles are ineffective; only those satisfying both the energy and the orientation requirement lead to a reaction.
• Reading the Maxwell-Boltzmann peak as the mean energy. The distribution is asymmetric; the mean lies to the right of the peak, under the longer high-energy tail.
• Claiming that higher temperature "causes more collisions," as the main explanation of a faster rate. Collision frequency rises only modestly with temperature; the dominant effect is the disproportionate growth of the high-energy tail beyond $E_a$.
• Confusing $E_a$ with $\Delta H$, or assuming an exothermic reaction must have a small activation energy. The two are set by unrelated features of the energy profile — the height of the barrier versus the difference between the two energy levels either side of it.
• Forgetting the orientation requirement for a molecule that is not simple and symmetric, especially in mechanism questions (developed further in a later chapter), where only specific atoms of a larger molecule can undergo the necessary bond change.
• Omitting units, or using the wrong sign, when reading $E_a$ or $\Delta H$ off an energy profile diagram. Both are always measured as a vertical energy difference on the diagram, in $\text{kJ}\,\text{mol}^{-1}$, read consistently from the same reference level.

Summary

• Collision theory: a reaction occurs only when particles collide with both sufficient energy and the correct orientation; most collisions are ineffective.
• Activation energy, $E_a$, is the minimum combined kinetic energy a collision needs to reach the transition state and react; it is read as the height of the energy-profile hump above the reactants.
• The orientation requirement reflects the geometric need for the reacting atoms to meet correctly; it is largely independent of temperature.
• The Maxwell-Boltzmann distribution describes the spread of particle kinetic energies at a given temperature: it starts at the origin, peaks at the most probable (not mean) energy, and has a long high-energy tail whose area beyond $E_a$ gives the fraction of energetically capable collisions.
• Raising temperature shifts the distribution's peak right and flattens it, disproportionately increasing the fraction of particles with $E \ge E_a$ — far more than the modest accompanying rise in collision frequency, which is why rate is so sensitive to temperature.
• $E_a$ and $\Delta H$ are independent quantities: $\Delta H = E_{a,\text{forward}} - E_{a,\text{reverse}}$, and a reaction's exothermicity says nothing by itself about how large or small its activation energy is.