Chemical Kinetics: Reaction Rates, Mechanisms, and Catalysis

Chapter 15: Chemical Kinetics

Introduction to Kinetics

Chemical kinetics is the study of reaction rates and the factors that affect how quickly chemical reactions occur. Understanding kinetics allows chemists to control and optimize reactions in laboratory and industrial settings.

• Reaction Rate: The change in concentration of a reactant or product per unit time.

• Key Questions: How fast does a reaction proceed? What factors influence the speed?

• Example: Combustion of methane: CH4(g) + 2 O2(g) → CO2(g) + 2 H2O(g)

Kinetics vs. Thermodynamics

Thermodynamics tells us if a reaction is spontaneous (favorable), but kinetics determines how fast it occurs. Some spontaneous reactions require an input of energy (e.g., a spark) to proceed due to slow kinetics.

Reaction Rates and Rate Laws

Defining Reaction Rate

The rate of a reaction is measured by the change in concentration of reactants or products over time. For a general reaction A → B:

• Rate = $-\frac{\Delta [A]}{\Delta t} = \frac{\Delta [B]}{\Delta t}$

• For instantaneous rate: $-\frac{d[A]}{dt}$

Stoichiometry and Rate

For reactions with coefficients, the rate must account for stoichiometry. For example, for 2A → B + 3C:

• If $\frac{\Delta [B]}{\Delta t} = 6.0$ M/s, then $\frac{\Delta [A]}{\Delta t} = -12.0$ M/s (since 2 moles of A are consumed for every 1 mole of B produced).

Rate Laws

The rate law expresses the rate as a function of reactant concentrations:

• General form: Rate = $k[A]^n$

• k: Rate constant (depends on temperature)

• n: Order of reaction (determined experimentally; can be integer, fraction, or negative)

Determining Rate Laws from Data

Experimental data is used to determine the order of reaction with respect to each reactant and the value of the rate constant.

• Example: For 2 HI → H2 + I2, doubling [HI] doubles the rate, so the reaction is first order in HI.

Overall Reaction Order

The overall order is the sum of the exponents in the rate law. For Rate = $k[NO]^2[Cl_2]^2$, the overall order is 4.

Units of the Rate Constant

The units of k depend on the overall order of the reaction. For Rate = $k[A]^{1/2}[B]$, k has units of M-3/2 s-1.

Integrated Rate Laws

First Order Reactions

For a first order reaction (A → Products):

• Integrated rate law: $[A] = [A_0]e^{-kt}$

• Linear form: $\ln[A] = -kt + \ln[A_0]$

• A plot of $\ln[A]$ vs. t yields a straight line with slope -k.

First Order Half-Life

The half-life ($t_{1/2}$) is the time required for half of the reactant to be consumed:

• $t_{1/2} = \frac{\ln 2}{k}$ (independent of concentration)

Second Order Reactions

For a second order reaction (A → Products):

• Integrated rate law: $\frac{1}{[A]} = kt + \frac{1}{[A_0]}$

• A plot of $1/[A]$ vs. t yields a straight line with slope k.

• Half-life: $t_{1/2} = \frac{1}{k[A_0]}$ (depends on initial concentration)

Zero Order Reactions

For a zero order reaction (A → Products):

• Integrated rate law: $[A] = -kt + [A_0]$

• A plot of [A] vs. t yields a straight line with slope -k.

• Half-life: $t_{1/2} = \frac{[A_0]}{2k}$

Summary Table: Integrated Rate Laws

Order	Integrated Rate Law	Straight Line Plot	Slope	Half-life ($t_{1/2}$)
0	$[A] = -kt + [A_0]$	[A] vs t	-k	$\frac{[A_0]}{2k}$
1	$\ln[A] = -kt + \ln[A_0]$	ln[A] vs t	-k	$\frac{\ln 2}{k}$
2	$\frac{1}{[A]} = kt + \frac{1}{[A_0]}$	1/[A] vs t	k	$\frac{1}{k[A_0]}$

Collision Theory and Activation Energy

Collision Theory

For a reaction to occur, molecules must collide with sufficient energy (greater than or equal to the activation energy, $E_a$) and proper orientation.

• Activation Energy ($E_a$): The minimum energy required for a reaction to proceed.

• Activated Complex: A high-energy, unstable arrangement of atoms at the peak of the energy barrier.

Effect of Temperature and the Arrhenius Equation

Increasing temperature increases the average kinetic energy of molecules, leading to more collisions with energy greater than $E_a$, thus increasing the rate constant k.

• Arrhenius equation: $k = Ae^{-E_a/RT}$

• Linear form: $\ln k = -\frac{E_a}{R}\left(\frac{1}{T}\right) + \ln A$

Determining Activation Energy Experimentally

By measuring the rate constant at two temperatures, $E_a$ can be calculated:

• $\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$

Reaction Mechanisms

Elementary and Complex Reactions

Most reactions occur in a series of steps called a mechanism. Each step may be elementary (single collision) or complex (multiple steps).

• Elementary Step: Rate law can be written directly from stoichiometry.

• Overall Reaction: The sum of all elementary steps.

• Intermediate: Produced in one step and consumed in another; does not appear in the overall reaction.

Rate-Determining Step

The slowest step in a mechanism determines the overall reaction rate and the observed rate law.

Example: NO2 + CO → NO + CO2

• Experimental rate law: Rate = $k[NO_2]^2$

• Mechanism must be consistent with both the overall reaction and the observed rate law.

Catalysts and Reaction Rate

Role of Catalysts

A catalyst increases the rate of a reaction by providing an alternative pathway with a lower activation energy. Catalysts are not consumed in the reaction and appear unchanged at the end.

• Example: Decomposition of H2O2 is sped up by KI, FeCl3, or catalase.

• Catalysts appear as reactants in one step and are regenerated as products in another.

Effect of a Catalyst on Energy Diagram

The catalyzed pathway has a lower activation energy, resulting in a faster reaction rate.

Summary of Key Concepts

• Difference between spontaneity (thermodynamics) and reaction speed (kinetics)

• How to draw and interpret potential energy diagrams

• How to write and determine rate laws and reaction order from experimental data

• Use of integrated rate laws for zero, first, and second order reactions

• Understanding collision theory and the effect of temperature and catalysts on reaction rates

• How to analyze reaction mechanisms, identify intermediates and catalysts, and derive rate laws for multi-step reactions