Chapter 15 lecture 3

CHAPTER 15: CHEMICAL KINETICS

Temperature and Reaction Rates: The Arrhenius Equation

Chemical kinetics explores the speed at which chemical reactions occur and the factors that influence these rates. One of the most important relationships in kinetics is the Arrhenius equation, which connects the rate constant of a reaction to its activation energy and temperature.

• Arrhenius Equation: Describes how the rate constant (k) depends on temperature (T) and activation energy (Ea).

• Formula: $k = A e^{-\frac{E_a}{RT}}$ where: - k: rate constant - A: frequency factor (product of collision frequency and orientation probability, A = pZ) - Ea: activation energy (J/mol) - R: universal gas constant (8.314 J/K·mol) - T: absolute temperature (K)

• Exponential Factor: $e^{-\frac{E_a}{RT}}$ increases with higher temperature and decreases with higher activation energy.

• Logarithmic Form: $\ln k = -\frac{E_a}{R}\frac{1}{T} + \ln A$ This linearizes the relationship for graphical analysis.

Activation Energy and Reaction Progress

For a reaction to occur, molecules must collide with sufficient energy and proper orientation. The minimum energy required is called the activation energy (Ea). The transition state or activated complex is formed when reactants reach this energy barrier.

• Activation Energy (Ea): Always positive; represents the energy barrier for the reaction.

• Activated Complex: A transient species with partially broken and formed bonds.

• Effect of Ea: Higher Ea means slower reaction rate at a given temperature.

• Potential Energy Profile: Shows the energy changes as reactants convert to products.

Comparison of Potential Energy Profiles: Exothermic vs. Endothermic Reactions

Potential energy diagrams illustrate the energy changes during a reaction. Exothermic reactions release energy, while endothermic reactions absorb energy.

• Exothermic Reaction: Energy of products is lower than reactants; ΔE is negative.

• Endothermic Reaction: Energy of products is higher than reactants; ΔE is positive.

Effect of Temperature on Reaction Rate

Increasing temperature raises the average kinetic energy of molecules, increasing the fraction of molecules with energy greater than Ea. This results in a higher reaction rate.

• Molecular Kinetic Theory: Higher temperature means more energetic collisions.

• Fraction of Energetic Molecules: Represented by the shaded area under the energy distribution curve above Ea.

• Arrhenius Equation: Shows that k increases with temperature.

Arrhenius Equation: Frequency Factor and Exponential Factor

The Arrhenius equation consists of two main components: the frequency factor (A) and the exponential factor ($e^{-\frac{E_a}{RT}}$).

• Frequency Factor (A): Related to collision frequency and orientation; usually treated as constant over a moderate temperature range.

• Exponential Factor: Depends on Ea and T; values range from 0 to 1.

• Low Ea and High T: Exponential factor approaches 1, reaction rate increases.

• High Ea and Low T: Exponential factor approaches 0, reaction rate decreases.

Graphical and Two-Point Forms of the Arrhenius Equation

The logarithmic form of the Arrhenius equation allows determination of Ea from experimental data. A plot of ln k vs. 1/T yields a straight line with slope (–Ea/R).

• Graphical Method: Requires multiple temperature measurements of k.

• Two-Point Form: Used when k is known at two temperatures. $\ln \frac{k_2}{k_1} = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)$

• Applications: Calculate Ea or k at a given temperature.

Example Calculations

Example 1: Calculating k at a higher temperature given Ea and k1. Example 2: Calculating Ea given k at two temperatures.

• Example 1: $\ln \frac{k_2}{k_1} = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)$ $k_2 = k_1 \times e^{2.039} = 1.644 \times 10^{-3} \text{ M}^{-1}\text{s}^{-1}$

• Example 2: $\ln \frac{k_2}{k_1} = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)$ $E_a = 1.83 \times 10^2 \text{ kJ/mol}$

Tabular Data: Effect of Temperature on Rate Constant

The following table summarizes experimental data showing how temperature affects the rate constant for a reaction:

T (ºC)	T (K)	1/T (1/K)	k (M-1s-1)	ln k
5.0	278.2	3.595E-3	4.93E-06	-12.22
15.0	288.2	3.470E-3	1.40E-05	-11.18
21.0	294.2	3.400E-3	4.54E-05	-10.00
31.0	304.2	3.288E-3	8.96E-05	-9.32
39.0	312.2	3.204E-3	2.14E-4	-8.45

Purpose: Demonstrates the increase in rate constant with temperature, supporting the Arrhenius equation.

Summary

• The Arrhenius equation quantitatively relates reaction rate to temperature and activation energy.

• Activation energy is the energy barrier for a reaction; higher Ea means slower rates.

• Temperature increases the fraction of molecules able to react, thus increasing the rate constant.

• Graphical and two-point forms of the Arrhenius equation allow determination of Ea and prediction of rate constants.