# Lecture 28: Arrhenius Equation, Thermal Energy Distribution, and Collision Theory

### The Effect of Temperature on Rate

#### Arrhenius Equation

﻿$K=A(e\frac{-Ea}{RT})$﻿

T = Temperature, R = Gas constant ﻿$\cdot$﻿ 8.314 J/mol ﻿$\cdot$﻿ K

A = Frequency factor

Ea = Activation energy, the extra energy needed to start the molecules reaction

• Rate of a reaction increases with Temperature
• For any reaction, Activation Energy is needed. (Ea)

Activation Energy - The amount of energy needed to convert reactants into the activated complex

Activation Complex - Chemical species with partially broken and partially formed bonds (always very high in energy because of its partial bonds)

#### Isomerization of Methyl Isonitrile

﻿$CH_{3}-N\equiv C\rightarrow CH_{3}-C\equiv N$﻿

• Methyl Isonitrile rearranges to acetonitrile
• For this reaction to occur the H﻿$_{3}$﻿C-N bond must break and a new H﻿$_{3}$﻿C-C bond forms
• As the reaction begins the C-N bond weakens enough for the C﻿$\equiv$﻿ N group to start to rotate

#### The Exponential Factor (number between 0 and 1)

• Fraction of reactant molecules with sufficient energy to cross over the energy barrier. The higher the energy barrier - fewer molecules overcome it.
• Extra energy comes from Kinetic energy of motion
• Increasing the temperature increases the average kinetic energy.
• Increasing the temperature increases the number of molecules with sufficient energy to overcome the energy barrier
• Increasing the temperature increases the reaction rate

Thermal Energy Distribution

• When temperature increases, the fraction of molecules with enough energy to surmount the activation energy barrier also increases.

#### The Arrhenius Equation

﻿$ln(K)=\frac{-Ea}{R}\left ( \frac{1}{T} \right )+ln(A)$﻿

(In the form of a line y=mx + b)

﻿$y = ln(K)$﻿

﻿$x =$﻿ ﻿$(\frac {1} {T} )$﻿

Determine the activation energy and frequency factor of the reaction ﻿$O_{3(g)}\rightarrow O_{2(g)}+O_{(g)}$﻿ given the following data.

﻿$y = -1.12 ﻿ ⋅﻿ 10﻿^{4} ﻿x + 26.8$﻿

Slope = ﻿$1.12 ﻿ ⋅ 10﻿^{4} ﻿K$﻿ y intercept = 26.8

Ea = (1.12 ﻿$\cdot$﻿ 10﻿$^{4}$﻿K) (8.314 J/mol ﻿$\cdot$﻿ K) = ﻿$9.31 ﻿ ⋅﻿ 10﻿^{4} ﻿ J/mol$﻿

ln (K) = ﻿$\frac {-Ea}{R}$﻿ ﻿$(\frac {1}{T})$﻿ + ﻿$ln(A)$﻿

Ea = ﻿$93.1 kJ/mol$﻿

A = e﻿$^{26.8}$﻿ = ﻿$4.36 ﻿ ⋅﻿ 10﻿^ {11} ﻿$﻿

A = ﻿$4.36 \cdot 10^{11} M^{-1} \cdot S^{-1}$﻿

A = e﻿$^{y-intercept}$﻿

### Collision Theory of Kinetics

• Reaction to take place - reacting molecules must collide with each other
• Once molecules collide they may react together or they may not, depending on 2 factors
• Collision should have enough energy to "break the bonds holding reactant molecules together"
• Reacting molecules collide in the proper orientation for new bonds to form

For a collision to lead to overcoming the energy barrier the reacting molecules much have sufficient kinetic energy so that when they collide the activated complex can form

#### Energetic Collision leads to product

Effective Collision - meeting the conditions of energy and orientation

The higher the frequency of effective collision, the faster the reaction rate.

Activated Complex - 2 molecules - Effective collision, a temporary, high energy (unstable) chemical species formed.