Chapter 12: Chemical Kinetics – Study Notes

Reaction Rates

Introduction to Reaction Rates

Chemical kinetics is the study of reaction rates, which describes how quickly reactants are converted into products in a chemical reaction. Understanding reaction rates is essential for controlling industrial processes, biological systems, and environmental phenomena.

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

• Expressed mathematically as $\frac{\Delta [X]}{\Delta t}$, where $[X]$ is the concentration of substance X and $t$ is time.

• Units commonly used: mol L-1 s-1.

• Example: The decomposition of hydrogen peroxide, H2O2, can be monitored by measuring its concentration over time.

Definition of Rate

The rate of a process is a measure of how some property, X, varies with another property, Y. It is typically expressed as a fraction:

• General rate formula: $\frac{\Delta X}{\Delta Y}$

• Examples: $\frac{\$14}{1\ \text{hr}}$, $\frac{36\ \text{mi}}{1\ \text{gal}}$, $\frac{60\ \text{km}}{1\ \text{hr}}$

Chemical Reaction Rates

For a general reaction $aA \rightarrow bB$:

• Rate of consumption of A: $-\frac{1}{a}\frac{\Delta [A]}{\Delta t}$

• Rate of formation of B: $\frac{1}{b}\frac{\Delta [B]}{\Delta t}$

• Negative sign indicates consumption; positive sign indicates formation.

• Rates of appearance/disappearance are related by stoichiometry.

Sample Data and Table: Decomposition of Hydrogen Peroxide

Consider the reaction: $2H_2O_2(aq) \rightarrow 2H_2O(l) + O_2(g)$

[H2O2] (mol L-1)	Δ[H2O2] (mol L-1)	Rate of Decomposition (mol L-1 h-1)
1.000	-0.500	0.0833
0.500	-0.250	0.0417
0.250	-0.125	0.0208
0.125	-0.062	0.010

Additional info: Table values inferred from slide context and typical kinetic data.

Other Important Definitions

• Instantaneous rate: Measured at a specific time, typically as the slope of the tangent to the concentration vs. time curve.

• Initial rate: The instantaneous rate at time zero, when the reaction starts and only reactants are present.

• Average rate: Measured over a time interval, using concentrations at the beginning and end of the period.

Affecting Reaction Rates

Factors Influencing Reaction Rates

The rate of a chemical reaction can be affected by several factors:

• Chemical nature of reactants: Some substances react more readily than others due to their molecular structure.

• Physical state: Reactions occur faster when reactants are in the same phase or finely divided (greater surface area).

• Temperature: Higher temperatures generally increase reaction rates by providing more energy for collisions.

• Concentration: Increasing the concentration of reactants usually increases the rate.

• Catalysis: Catalysts speed up reactions without being consumed.

Rate Laws

Introduction to Rate Laws

Rate laws are mathematical equations that describe the relationship between the rate of a chemical reaction and the concentration of its reactants.

• General form: $\text{rate} = k[A]^m[B]^n$

• $k$ = rate constant

• $m$, $n$ = reaction orders with respect to A and B

• Overall reaction order = $m + n$

Determining Rate Laws from Data

Experimental data is used to determine the values of $m$, $n$, and $k$ in the rate law. This involves measuring how the rate changes as the concentrations of reactants are varied.

• First order: Rate is proportional to the concentration of one reactant.

• Second order: Rate is proportional to the product of concentrations of two reactants, or the square of one reactant.

• Zero order: Rate is independent of the concentration of a reactant.

Integrated Rate Laws

Integrated Rate Laws and Their Use

Integrated rate laws relate the concentration of reactants to time, allowing calculation of concentrations at any point during the reaction.

• First-order: $\ln [A]_t = -kt + \ln [A]_0$

• Second-order: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$

• Zero-order: $[A]_t = -kt + [A]_0$

These equations can be used to determine how long it takes for a reactant to reach a certain concentration, or to estimate the time required for a process to occur.

Half-Life of a Reaction

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

• First-order half-life: $t_{1/2} = \frac{\ln 2}{k}$

• For first-order reactions, half-life is independent of initial concentration.

Collision Theory and Activation Energy

Collision Theory

Collision theory explains how molecular collisions lead to chemical reactions. For a reaction to occur, reactant molecules must collide with sufficient energy and proper orientation.

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

• Transition state: An unstable arrangement of atoms at the peak of the energy barrier.

Arrhenius Equation

The Arrhenius equation relates the rate constant ($k$) to temperature ($T$) and activation energy ($E_a$):

• $k = Ae^{-E_a/(RT)}$

• $A$ = frequency factor (depends on collision frequency and orientation)

• $R$ = gas constant ($8.314$ J mol-1 K-1)

Taking the natural logarithm gives a linear form:

• $\ln k = -\frac{E_a}{R}\frac{1}{T} + \ln A$

Reaction Mechanisms

Elementary Reactions and Molecularity

Complex reactions often occur in a series of elementary steps. The molecularity of a step refers to the number of molecules involved:

• Unimolecular: Involves one molecule.

• Bimolecular: Involves two molecules.

• Termolecular: Involves three molecules (rare).

Each elementary step has its own rate law, with exponents equal to the stoichiometric coefficients.

Intermediates and Transition States

• Intermediate: A species produced in one step and consumed in a subsequent step; relatively stable compared to the transition state.

• Transition state: High-energy, unstable arrangement at the peak of the energy barrier.

Rate-Determining Step

In a multi-step reaction, the slowest step limits the overall rate and is called the rate-determining step. The rate law for the overall reaction is determined by this step.

Catalysis

Role of Catalysts

Catalysts increase the rate of a chemical reaction by providing an alternative pathway with lower activation energy. They are not consumed in the reaction.

• Enzymes: Biological catalysts that speed up reactions in living organisms.

• Catalysts are regenerated at the end of the reaction mechanism.

• Intermediates are produced and consumed during the reaction.

Summary Table: Key Definitions and Equations

Term	Definition/Equation
Reaction Rate	$\frac{\Delta [X]}{\Delta t}$
Rate Law	$\text{rate} = k[A]^m[B]^n$
First-order Integrated Rate Law	$\ln [A]_t = -kt + \ln [A]_0$
Second-order Integrated Rate Law	$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
Zero-order Integrated Rate Law	$[A]_t = -kt + [A]_0$
Half-life (First-order)	$t_{1/2} = \frac{\ln 2}{k}$
Arrhenius Equation	$k = Ae^{-E_a/(RT)}$

Additional info: This summary table is constructed to consolidate key equations and definitions for exam review.