12.5 Collision Theory

Collision Theory and Reaction Rates

Chemical reactions happen when particles collide, but not just any collision will do. Collision theory explains why factors like temperature and concentration speed up or slow down reactions: they change how often particles collide and how much energy those collisions carry. This section covers the core ideas of collision theory, activation energy, and the Arrhenius equation.

Collision Theory Basics

Collision theory states that a reaction occurs when reactant particles collide with sufficient energy and proper orientation. Collisions that actually produce a reaction are called effective collisions. If either the energy or the orientation is wrong, the particles just bounce off each other unchanged.

• Higher collision frequency means more chances for effective collisions, which leads to a faster reaction rate.
• Not every collision is effective. Only a small fraction of all collisions have both enough energy and the right orientation to break and form bonds.

How Physical State Affects Collisions

The physical state of the reactants determines how freely particles can move and how often they collide.

• Gases have the highest collision frequency. Particles move rapidly in all directions with large spaces between them, so they collide often.
• Liquids have lower collision frequency than gases. Particles are closer together but move more slowly, so collisions happen less frequently.
• Solids have the lowest collision frequency. Particles are locked in fixed positions, so reactions can only occur at the surface where particles are exposed.

This is why grinding a solid into a powder (increasing surface area) speeds up a reaction: more particles are available to collide.

How Temperature Affects Reaction Rate

Raising the temperature increases the average kinetic energy of particles. This has two effects:

1. Particles move faster, so they collide more frequently.
2. A larger fraction of those collisions carry enough energy to overcome the activation energy barrier.

The second effect is actually the more important one. Even a modest temperature increase shifts the energy distribution so that many more particles clear the energy threshold.

A common rule of thumb: a 10°C rise in temperature roughly doubles the reaction rate. This is an approximation and depends on the specific reaction, but it gives you a sense of how sensitive rates are to temperature.

The Maxwell-Boltzmann distribution describes the spread of molecular speeds (and kinetic energies) at a given temperature. At higher temperatures, the curve flattens and shifts to the right, meaning more molecules occupy the high-energy tail where they can react.

How Concentration Affects Reaction Rate

Higher concentration means more reactant particles packed into the same volume. More particles per unit volume means more frequent collisions, which means more effective collisions per second.

• Doubling the concentration of a reactant roughly doubles the collision frequency (assuming the reaction is first order in that reactant).
• This is why reactions often slow down over time: as reactants are consumed, their concentration drops and collisions become less frequent.

Activation Energy and the Transition State

Activation energy ($E_a$) is the minimum energy that colliding particles must have for the collision to result in a reaction. Think of it as an energy hill the reactants need to climb before they can slide down to become products.

• Reactant particles must possess kinetic energy ≥ $E_a$ for a collision to be effective.
• A reaction with a low $E_a$ proceeds faster because a larger fraction of collisions have enough energy. A high $E_a$ means fewer collisions succeed, so the reaction is slower.

The transition state (also called the activated complex) is the highest-energy arrangement of atoms along the reaction pathway. At this point, old bonds are partially broken and new bonds are partially formed. It's an unstable, fleeting configuration that exists only at the peak of the energy barrier. The rate of reaction depends on how often this transition state forms.

On an energy diagram, the transition state sits at the top of the curve between reactants and products. The height of that peak above the reactants is $E_a$.

Collision Factors

Two additional factors determine whether a collision is effective:

• Collision cross-section is the effective target area within which two particles must approach each other for a collision to occur. Larger molecules generally have larger cross-sections.
• Steric factor accounts for orientation. Even if two particles collide with enough energy, the reactive parts of the molecules must be facing each other. The steric factor is always ≤ 1, reflecting that only certain orientations lead to a reaction.

Arrhenius Equation

The Arrhenius equation connects the rate constant ($k$) to temperature and activation energy in a single expression:

$k = Ae^{-E_a/RT}$

• $k$ = rate constant
• $A$ = pre-exponential factor (also called the frequency factor), which accounts for collision frequency and the steric factor
• $E_a$ = activation energy (in J/mol)
• $R$ = universal gas constant ($8.314 \text{ J mol}^{-1} \text{K}^{-1}$)
• $T$ = temperature in Kelvin

The negative exponent $-E_a/RT$ is the key piece. It represents the fraction of molecules with enough energy to react. As $T$ increases, $E_a/RT$ gets smaller, the exponent becomes less negative, and $k$ increases. That's why higher temperature means a faster reaction.

To calculate $k$ at a specific temperature:

1. Identify the values of $A$ and $E_a$ (these are usually given or determined experimentally).
2. Convert temperature to Kelvin if it isn't already.
3. Plug $A$, $E_a$, $R$, and $T$ into $k = Ae^{-E_a/RT}$.
4. Solve for $k$.

The relationship is exponential, not linear. That means small increases in temperature can produce large increases in the rate constant, especially for reactions with high $E_a$.