Chemical Kinetics (Pt. 8)

The Arrhenius Equation, Activation Energy, and Catalysts

By Shawn P. Shields, Ph.D.

This work is licensed by Shawn P. Shields-Maxwell under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Reaction Rates and Temperature

The rate of a reaction generally goes up with increasing temperature.

The rate constant k is temperature dependent.

Why?

Factors Affecting Reaction Rates

The overall reaction rate depends on:

The frequency of collisions

The fraction of collisions with enough energy to react

The orientation of colliding molecules (steric factor)

The Arrhenius Equation

Molecules only react if they collide with each other in the correct orientation.

Increasing the temperature increases the energy and frequency of molecular collisions.

The Arrhenius equation relates the rate constant k to all of these factors.

The Arrhenius Equation

𝒌 = 𝐀𝐞−𝐄𝐚𝐑𝐓

k is the rate constant

Ea is the activation energy (to be discussed)

“A” is the pre-exponential factor representing the likelihood that collisions with the proper orientation occur.

R is the gas constant (8.314 J/mol K)

T is the temperature in Kelvin

The Arrhenius Equation and the Rate Law

For the elementary reaction:

D + B C

The rate law is: Rate = k[D][B]

Plug in the Arrhenius equation to visualize its effect on the rate of reaction:

𝐑𝐚𝐭𝐞 = 𝐀𝐞−𝐄𝐚𝐑𝐓 𝐃 𝐁𝒌 = 𝐀𝐞

−𝐄𝐚𝐑𝐓

Activation Energy, Ea

The macroscopic reaction rate is determined by the frequency of collisions having sufficient energy to overcome the activation energy barrier (subtracting those that do not have the correct orientation to react (steric factor)).

Molecules need to have enough kinetic energy to overcome bonding and repulsions of the reactants.

This required minimum amount of energy is called the activation energy Ea

Temperature and Activation EnergyOnly a small percentage of collisions have enough energy to react.

This proportion increases with increasing temperature.

Fraction of molecules with energy E

Energy Activation Energy Ea

Fraction of molecules greater than Ea(enough E to react)

Low T

High T

Finding Ea by Graphing

𝒌 = 𝐀𝐞−𝐄𝐚𝐑𝐓

Take the natural log (ln) of both sides

𝐥𝐧 𝒌 =−𝐄𝐚𝐑𝐓

+ 𝐥𝐧 𝐀

Plot ln k vs 1/T to determine the activation energy.

Finding Ea by Graphing

𝐥𝐧 𝒌 =−𝐄𝐚𝐑𝐓

+ 𝐥𝐧 𝐀

Finding Ea by Graphing

𝐥𝐧 𝒌 =−𝐄𝐚𝐑𝐓

+ 𝐥𝐧 𝐀

Reactions with a larger Ea show a higher sensitivity of the rate constant k to the temperature.

Arrhenius Equation

Another useful form of the equation relates the rate constant k at two temperatures

ln𝑘2𝑘1

= −EaR

1

T2−1

T1

Where k1 is the rate constant at T1, and k2is the rate constant at T2

Reaction Progress and Activation Energy

Exothermic Reaction Endothermic Reaction

Increasing the Rate Constant k

There are two ways to increase the rate constant for a given reaction:

raise the temperature

use a catalyst to lower the activation energy Ea

Catalysts change the reaction mechanism in such a way as to decrease the activation energy Ea.

Catalysts Lower Ea

𝐥𝐧 𝒌 =−𝐄𝐚𝐑𝐓

+ 𝐥𝐧 𝐀

Catalysts essentially increase the number of molecules with the required activation energy to react by lowering the requirement.

Another View: Catalysts Lower Ea

Instead of increasing the temperature, use a catalyst to lower the energy barrier to reaction.

Catalysts change the reaction mechanism.

Catalysts

Catalysts do not change the thermodynamics of the reaction (i.e., they do not change the position of equilibrium). More on that later…

Catalysts are not used up or produced in a reaction.

They can be used in less than “stoichiometric quantities” (or very small amounts).

They can be homogeneous (same phase) or heterogeneous (alternate phase) catalysts.

Example Problems

will be posted separately.