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# Collision Theory

For a chemical reaction to occur the reacting molecules have to approach each other and collide with each other . These collisions give rise to the products of the reaction . But if all the collisions of the reacting molecules lead to the formation of the products of the reaction , the rate of a reaction , at any temperature , should depend on the number of collisions taking place in unit time . But it is observed that the actual rate measured is much less than what is expected from the calculated number of collisions per unit time . Therefore Arrhenius kinetic molecular theory of rate of collision theory of reaction rates proposed a simple theory to explain the observed rates of reactions in the gaseous state.

The view of collisions between the reacting molecules in collision theory of a chemical reaction is shown below.

The rate of collision of reactant molecules and the fraction of molecules which have at least the critical energy gives an equation to compare with the experimental Arrhenius equation.

K = A e-Ea/RT

The main features of collision theory are

• Molecules are hard spheres that is there are no inter molecular interactions
• Vibrational and rotational structures of reactants and products are ignored
• The activated complex plays no part in the theory
• Redistribution of energy on reaction is ignored.

## Main Postulates of Collision Theory of Chemical Reactions

The main postulates are

1. Reacting molecules shall have to collide for any reaction to occur.
2. All collisions do not lead to the formation of the products.
3. The colliding molecules shall have to possess a minimum energy to give products . This minimum energy is called Threshold energy. This is higher than that of the molecules in the Normal state.
4. The energy of the molecules at STP is very much less than this Threshold energy.
5. The difference between the Threshold energy and the energy of molecules in the Normal state is Activation energy.
Activation energy = [ Threshold energy - energy of the normal molecules ]
6. The molecules possessing the threshold energy are called Activated energy . These are formed in small numbers during collisions between Normal molecules .
7. Collisions occurring between activated molecules are called Activated collisions . Activated collisions alone lead to the formation of the products reaction.
8. The reaction of the activated collisions amongst the total collisions is very much small.

## Collision Theory and Rate Constant

The collision theory explains how chemical reactions occur and why reaction rates differ. For a reaction to occur the reactants particles must collide. However only a certain fraction of the total collisions are successful to cause a chemical change. The successful or effective collisions have
• Sufficient activation energy at the moment of impact to break the existing bonds and form new bonds which results in the formation of products.
• The correct spatial orientation with respect to each other.

The reaction rate is a function of the number of effective collisions per unit of time. three factors affect collision frequency

• Concentration-higher concentration result in greater collision frequencies.
• Temperature-higher temperature result in higher kinetic energies of the reactant particles and a larger proportion of particles having sufficient activation energies. (Maxwell-Boltzmann distribution).
• The surface area of smaller reactant particles results in larger surface area and high probabilities of effective collisions.

The intermolecular collisions are essential for a reaction but every collision does not result in chemical reaction. Those molecular collisions which result in the chemical reaction are called effective collisions. In order for a collision to be effective, the reactant molecules must be activated that is molecules must possesses a certain minimum amount of energy called threshold energy. Thus threshold energy is a certain minimum energy which the colliding molecules of the reactants must acquire before they are capable of reacting. Rate constant or rate of reaction is, therefore equal to the number of effective collisions

Rate constant (k) = ZAB x f
Where f is the fraction of molecules that are activated. The fraction of molecules having energy equal to or greater than activation energy is found to be equal to $e^\frac{-Ea}{RT}$. This is called Boltzmann factor.

Rate constant (k) = ZAB . $e^\frac{-Ea}{RT}$

The collision theory can be generalized by introducing the steric factor P in the equation.
Rate constant (k) = P $Z_{AB} .e^\frac{-Ea}{RT}$
P is also called probability factor.

Steric factor is supposed to be equal to the fraction of effective collision in which reactant molecules possesses proper relative orientation necessary for the reaction. For example, in the reaction.

A2 + B2 $\to$ 2AB

Thus, reactant molecules even though having energy equal to or greater than threshold energy but colliding with improper orientation simply bounce back and do not form products.

This shows that the number of collisions and consequently the rate of the reaction are proportional to the product of the concentration. Simply stated then the rate of reaction is directly proportional to the concentration.

## Kinetic Molecular Interpretation

The ideal gas law describes the relationship among P, V, T and n for ideal gases. The kinetic molecular theory describes at the level of individual particles. This theory developed largely by Boltzmann, Clausius and Maxwell between 1850 and 1880 is often stated as five postulates.
1. Gases consist of molecules or atoms in continuous random motion.
2. Collisions between these molecules or atoms in a gas are elastic.
3. The volume occupied by the atoms or molecules in a gas is negligibly small.
4. The attractive force between the atoms or molecules in a gas are negligible.
5. The average kinetic energy of a molecule or atom in a gas is directly proportional to the kelvin temperature of the gas.

Each collision has a certain force which is related to the velocity of the gas particle. The total force is the sum of the forces of all the collisions occurring each second per unit area. Thus the pressure is dependent on the velocity of gas particle and the collision frequency. In turn the collision frequency depends on the velocity of the gas particle and the distance to the container walls. The frequency of collision can be changed also by altering the size of the container the force of the collisions is not affected.

Based on this understanding of pressure the molecular meanings of the gas laws can be appreciated as follows.

Boyle's law states that the inverse relationship between pressure and volume.

P ∝ 1 / V

The kinetic molecular theory agrees with this observation. If the volume of a gas is decreased the gas particles will strike the walls of the container more frequently. Increasing the frequency of these collisions increases the observed pressure of the gas.

Charles law predicts a direct relationship between temperature and volume.

V ∝ T

An increase in temperature increases both the force of each collisions and the frequency of collisions. Both of these effects increase the pressure. If the pressure of the gas is to remain constant, the volume must increase to correspondingly decrease the frequency of collisions with the walls of the container.

Gay-Lussac's law finds a direct relationship between temperature and pressure.

P ∝ T

In this case, an increase in temperature increases the kinetic energy of the gas particles. This increase has two effects. First the force of each collision is greater second, since the average velocity of the gas particles increases with temperature, the frequency of collisions with the container walls also increases. Thus the kinetic molecular theory predicts an increase in gas pressure as temperature rises.

## Root Mean Square Velocity

Kinetic molecular theory assumes that the average kinetic energy of the gas particles is directly proportional to the temperature in kelvins. Not all the gas particle will move at the same speed, so we refer to the average kinetic energy. The relationship of the average kinetic energy of the gas particles to the speed of the particle is
KE = $\frac{1}{2}$ mu2
Where KE is the kinetic energy and 'u' is the speed and 'm' is the mass of the particle.

The square root of 'u' is called the root mean square (RMS) velocity labelled as urms and is used to indicate the average speed of the gas. From the mathematical treatment of the kinetic theory of gases, we can determine the relative number of gas particles that have any particular speed.

The average kinetic energy of the gas particles is proportional to the temperature and kinetic molecular theory predicts that the rms speed is related to temperature and molar mass by the equation.
urms = $\frac{\sqrt3RT}{\sqrt M}$
To obtain the rms speed of gas molecules using the above equation we must express R as 8.314 J/mol.K and the molar mass M in kilograms per mole.

## Internal Combustion Effects

Often the reaction to take place rapidly enough to be practical but not so rapidly as to be dangerous. The controlled burning of fuel in an internal combustion engine is an example of such a process. On the other hand we want some undesirable reactions, such as the spoiling of food to take place more slowly.
Four factors have marked effects on the rate of chemical reactions.
1. Nature of the reactants
2. Concentration of the reactants
3. Temperature
4. The presence of a catalyst

### Nature of the Reactant

The nature of reactants includes not only the physical state of each reactant but also the particle size the reaction rate is generally faster between liquid-state reactant than between solid-state reactants and is fastest between gaseous state reactants. Of the three states of matter the gaseous state is the one where there is the most freedom of movement hence collision between reactants are the most frequent in this state.

### Concentration of the Reactants

According to law of mass action, the rate of a chemical reaction is directly proportional to the product of molar concentration of reactants. In a chemical reaction, the concentration of reactants decreases with time and simultaneously the concentration of products increases with time. Hence higher the initial concentration of reactants, higher will be the initial rate of reaction and the rate of the reaction decreases with time with decrease in concentration of reactants.

### Temperature

Since the number of activated molecules increases with the increase in temperature the number of effective collisions also increases with the rise in temperature and this increases the rate of reaction.

### Presence of Catalyst

A catalyst increases the rate of a chemical reaction without undergoing any net chemical change. Some catalysts increase the rate of only one specific chemical reaction without affecting similar reactions. Other catalysts are more general and affect an entire set of similar reactions. Catalysts generally reroute the pathway of a chemical reaction so that this "alternate" path although perhaps more circuitous, has a lower activation energy for reaction than the uncatalyzed reaction.

## Collision Theory Problems

Below you could see problems

### Solved Examples

Question 1: Calculate the root mean square (rms) speed in meters per second of argon atoms at 27o
Solution:

Strategy

Use the rms velocity equation and remember to use the proper values and units for R to convert the molar mass into units of kilograms per mole (Kg/mol)

Solution

The molar mass of argon 39.95 g/mol in the proper units is 0.03995 kg / mol. Son that units can be cancelled out in the calculations, expand the joule into its base units Kg.m2 / s2, when we substitute the value of R into rms equation, we get

Question 2: A 40Kg skater traveling at 4 m/s overtakes a 60Kg skater traveling at 2 m/s in the same direction and collides with her. If the two skaters remain in contact what is their final velocity?
Solution:

Let
v1 = initial velocity of 40kg skater
v2 = initial velocity of 60kg skater
v3 = final velocity of the two skater

then
m1v1 + m2v2 = (m1 + m2) v3
V3 = (m1v1 + m2v2) / (m1 + m2)
= [(40Kg x 4m/s) + (60Kg x 2m/s)] /
(40Kg + 60Kg)
= 2.8 m/s

Question 3: The experimental activation energy for the reaction of iodide ion with methyl bromide at 50oC is 7.63 x 104 J/mol. Calculate the rate constant for the reaction. (A = 1.66 x 1010L mol-1 S-1)
Solution:

Given Data

A = 1.66 x 1010L mol-1 S-1
T = 50oC = (273 + 50) = 323K
Ea = 7.63 x 104 J/mol
R = gas constant = 8.314 J/mol

Solution

K = A $e^\frac{-Ea}{RT}$
substitute all the values in the above equation

K = [1.66 x 1010L mol-1 S-1] [$e^\frac{ {-7.63 x 104 J/mol}{/ (8.314 J/mol x 323K)}]}$

K = 7.70 x 10-3 L mol-1 sec-1

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