# Collision Theory

The collision theory explains that gas-phase chemical reactions occur when two gas molecules collide with sufficient kinetic energy. The minimum amount of energy required for a successful collision, which results in a successful reaction, is called the activation energy. Thus, only a fraction of collisions lead to successful reactions.

### Overview

The collision theory is based on the kinetic theory of gases; therefore we are only dealing with gas-phase chemical reactions. Ideal gas assumptions are applied. Furthermore, we also are assuming:

1. All molecules are traveling through space in a straight line.
2. All molecules are rigid spheres.
3. The reactions concerned are between only two molecules.
4. The molecuels need to collide.

Ultimately the collision theory of gases gives us the rate constant for bimolecular gas-phase reactions; it is equal to the rate of successful collisions. The rate of successful collisions is proportional to the fraction of successful collisions multiplied by the overall collision frequency.

### Collision Frequency

The rate at which molecules collide which is the frequency of collisions is called the collision frequency, Z with units of (collisions/ unit of time). Given a box of molecules A and B, the collision frequency between molecules A and B is given by:

$Z=N_{A}N_{B}\sigma_{AB}\sqrt{\dfrac{8k_{B}T}{\pi\mu_{AB}}}$

where

• $$N_A$$ and $$N_B$$ are the number of molecules $$A$$ and $$B$$, and is directly related to the concentrations of $$A$$ and $$B$$.
• $$\sqrt{\dfrac{8k_{B}T}{\pi\mu_{AB}}}$$ is the mean speed of molecules.
• $$\sigma_{AB}$$ is the averaged sum of the collision cross sections of molecules A and B. The collision cross section represents the collision region presented by one molecules to another.

### Successful Collisions

In order for a successful collision to occur, the reactant molecules have to collide with enough kinetic energy to break original bonds and form new bonds to become the product molecules. Thus, it is called the activation energy for the reaction; it is also often referred to as the energy barrier.

The fraction of collisions that have enough energy to go over the activation barrier is given by:

$$f = e^{\frac{-E_a}{RT}}$$

where

• $$f$$ is the fraction of collisions with enough energy
• $$E_a$$ is the activation energy

The fraction of successful collisions is directly proportional to the temperature and inversely proportional to the activation energy.

### Putting it all Together

The rate constant of the gas-phase reaction is proportional to the product of the collision frequency and the fraction of successful reactions. As stated earlier, sufficient kinetic energy is required for a successful reaction; however they also have to collide properly(see more below). Compare the following equation to the Arrhenius equation:

$$k = Z\rho{e^{\frac{-E_a}{RT}}}$$

where

• $$k$$ is the rate constant for the reaction
• $$(Missing file: File:/\\rho)\rho$$ is the steric factor. It is the probability of the reactant molecules colliding with the right orientation and positioning to achieve a product with the desirable geometry and stereospecificity. The steric factor is very difficult to assess on paper so it is determined experimentally.
• $$Z\rho$$ is the pre-exponential factor, A, of the Arrhenius equation. In theory, it is the frequency of total collisions that collide with the right orientation. In practice, it is the pre-exponential factor that is directly determined by experiment and then used to calculate the steric factor.
• Ea is activation energy, T is temperature, and R is gas constant.

### Collision Frequency Equation

Z= NAρAB (8KBT/ πμAB)1/2

• NA is number of molecules per unit volume
• KB is Boltzmann's constant

Examples of gas-phase reactions and their steric factors

Reactions with more complex reactants and greater needs for collision orientation specificity will have a smaller steric factor (lower success):

H2 + C2H4 → C2H6

The opposite holds true for simpler reactions; they will have a relatively larger steric factor:

H. + .H ?→ H2

### Applications

As you might already know, the collision theory is used to predict the reaction rates for gas-phase reactions. It is a rough approximation due to the complications of the steric factor; furthermore, some of the assumptions fall apart. For example, in real life molecules are not perfect spheres. Finally, the concepts of collision frequency can be applied in laboratory:

• We know that the temperature of the environment affects the average speed of molecules. Thus, when we heat our reactions to increase the reaction rate.
• The initial concentration of reactants is directly proportional to the collision frequency; increasing the initial concentration will speed up the reaction.

Although the collision theory deals with gas-phase reactions, its concepts can also be applied to reactions that take place in solvents; however you must keep in mind that the properties of the solvents (for example: solvent cage) will affect the rate of reactions. Ultimately, the collision theory tells us how reactions occur; it can be used to approximate the rate constants of reactions, and its concepts can be directly applied in lab.

### References

1. Atkins, Peter and Julio de Paula. Physical Chemistry for the Life Sciences. 2006. New York, NY: W.H. Freeman and Company. p.281-3, 290
2. Petrucci, R. H., Harwood, W. S., & Herring, F. G. (2002). General Chemistry: Principles and Modern Applications. Upper Saddle River, NJ: Prentice-Hall, Inc. p.597-9

### Contributor

• Lu Zhao

17:40, 9 Jan 2014

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