Collision Theory and Activation Energy

Collision theory states that a reaction occurs only when reactant particles collide with at least the activation energy (E_a) and with the correct orientation, so that the rate depends on the collision frequency, the fraction of energetic collisions and the orientation factor. In brief: for E_a = 50 kJ/mol at 298 K, only about 1.7 × 10⁻⁹ of the collisions have enough energy, and a 10 °C increase in temperature roughly doubles that fraction.

Collision theory gives a molecular picture of reaction rates and explains the effects of concentration and temperature. It also introduces the concept of activation energy, the energy barrier that has to be overcome in every reaction. This article describes the three requirements for an effective collision, explains the energy profile and the transition state, and calculates the fraction of molecules that can react.

What makes a collision effective?

In a gas at room temperature and atmospheric pressure, each molecule undergoes roughly 10⁹ to 10¹⁰ collisions per second, yet most reactions are far slower than that. The reason is that only a tiny fraction of the collisions lead to a reaction. For a collision to be effective, three conditions must be met:

1. The particles must collide.
2. They must have a combined kinetic energy of at least the activation energy.
3. They must be in the correct orientation so that the atoms that form the new bonds meet.

The rate constant can be written as k = Z × p × e^−E_a/RT, where Z is the collision frequency, p is the steric (orientation) factor and the exponential is the fraction of collisions with energy at least E_a.

What is activation energy?

The activation energy is the minimum energy that the colliding particles must have in order to react. It is needed to break or weaken bonds in the reactants before new bonds can form. On an energy profile diagram, the reactants are at one energy level, the products at another and between them there is a hill. The height of the hill above the reactants is E_a. The top of the hill is the transition state, in which the old bonds are partly broken and the new bonds are partly formed. The species at the top is called the activated complex, and it cannot be isolated, as it has a lifetime of only about 10⁻¹³ s.

The activation energy of the reverse reaction is E_a(reverse) = E_a(forward) − ΔH. A catalyst lowers E_a for both directions, but does not change ΔH.

How many collisions have enough energy?

The energies of molecules in a gas have a spread described by the Maxwell–Boltzmann distribution. The fraction of molecules or collisions with energy of at least E_a is approximately e^−E_a/RT. At 298 K, RT = 8.314 × 298 = 2478 J/mol = 2.48 kJ/mol, so for E_a = 50 kJ/mol the exponent is −50 / 2.48 = −20.2 and the fraction is 1.7 × 10⁻⁹. This number is extremely sensitive to E_a and to T.

Fraction of molecules above E_a

A rise of 20 kJ/mol in activation energy cuts the fraction by a factor of about 3000. This is the reason why activation energy is the most important factor in the rate of a reaction, and why catalysts that reduce it have such a large effect.

Why does orientation matter?

Even a collision with sufficient energy will not necessarily lead to a reaction if the particles hit the wrong way. In the reaction CH₃Br + OH⁻ → CH₃OH + Br⁻, the hydroxide ion must approach the carbon atom from the side opposite to the bromine atom. A collision with the hydroxide ion hitting the hydrogen atoms or the bromine side is not productive. For simple reactions between atoms the orientation factor p is close to 1, but for reactions between large molecules it can be 10⁻³ or smaller.

Worked examples

Example 1: the effect of temperature

For E_a = 50 kJ/mol, compare the fraction at 298 K and at 308 K.

1. At 298 K: RT = 2478 J/mol, E_a/RT = 20.18, fraction = e^−20.18 = 1.7 × 10⁻⁹.
2. At 308 K: RT = 2561 J/mol, E_a/RT = 19.52, fraction = e^−19.52 = 3.3 × 10⁻⁹.
3. The ratio is 3.3 / 1.7 = about 1.9.

A rise in temperature of only 10 °C nearly doubles the fraction of effective collisions, which explains the rule of thumb for the effect of temperature. The collision frequency rises by only about 1.6 % over the same interval.

Example 2: forward and reverse activation energies

A reaction has E_a = 85 kJ/mol and ΔH = −40 kJ/mol. For the reverse reaction, E_a = 85 − (−40) = 125 kJ/mol. The reverse reaction has to climb a higher hill, since the products lie lower in energy.

Example 3: effect of a catalyst

A catalyst reduces E_a from 75 to 50 kJ/mol at 298 K. The rate increases by a factor of e^25,000/2478 = e^10.09 = 2.4 × 10⁴.

What are the limits of the theory?

Simple collision theory treats molecules as hard spheres, and it overestimates the rates of reactions between complex molecules, for which the orientation factor must be fitted. It also does not explain how the energy is distributed within the molecules. More advanced approaches, such as transition state theory, calculate the rate constant from the properties of the activated complex.

How did the theory develop?

Svante Arrhenius suggested in 1889 that molecules need a minimum energy to react, and that an equilibrium exists between normal and “active” molecules. Collision theory was developed in 1916–1918 by Max Trautz and William Lewis, who applied the kinetic theory of gases to chemical rates. Henry Eyring, Michael Polanyi and Meredith Evans developed transition state theory in 1935. Dudley Herschbach, Yuan Lee and John Polanyi received the Nobel Prize in Chemistry in 1986 for the study of the dynamics of individual reactive collisions.

Which mistakes are common?

• Thinking that all collisions lead to a reaction: only a very small fraction do.
• Confusing activation energy and ΔH: E_a is the barrier, and ΔH is the overall energy difference.
• Saying that a catalyst changes ΔH: it changes E_a only.
• Believing that exothermic reactions have no activation energy: most still need to overcome a barrier.
• Treating the transition state as an intermediate: an intermediate is a real species with a finite lifetime.