Chapter 14: Chemical Kinetics – Study Notes

Chemical Kinetics

Introduction to Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed, or rate, at which chemical reactions occur. Understanding reaction rates is essential for controlling industrial processes, biological systems, and laboratory experiments.

• Reaction rates measure the change in concentration of reactants or products over time.

• Reactants typically decrease in concentration as the reaction proceeds.

• Products increase in concentration as the reaction proceeds.

Molecular Collisions

Collision Theory

For a chemical reaction to occur, molecules must collide with each other. The probability and effectiveness of these collisions are influenced by several factors:

• Physical state of reactants (solid, liquid, gas)

• Reactant concentration (higher concentration increases collision frequency)

• Reactant temperature (higher temperature increases kinetic energy and collision frequency)

• Presence of a catalyst (catalysts lower activation energy and increase reaction rate)

Rates of a Reaction

Defining Reaction Rate

The rate of a reaction can be expressed for a general reaction:

$A \rightarrow B$

The rate is defined as:

$\text{Rate} = -\frac{d[A]}{dt} = \frac{d[B]}{dt}$

• The rate of decrease of reactants equals the rate of increase of products.

Average Rate Example

To calculate the average rate at which a reactant disappears over a time interval, use:

$\text{Average Rate} = -\frac{\Delta [A]}{\Delta t}$

Example: If the concentration of A changes from 1.00 mol to 0.30 mol over 40 seconds, the average rate is:

$\text{Average Rate} = -\frac{0.30 - 1.00}{40 - 0} = 0.0175 \text{ mol/s}$

Chemical Reaction Rates

Instantaneous vs. Average Rates

Reaction rates are not constant throughout the course of a reaction. Typically, rates decrease as reactant concentration decreases.

• Instantaneous rate is the rate at a specific moment, determined by the slope of the concentration vs. time curve.

• Average rate is calculated over a time interval.

• Lower concentration reduces the chances of collision between reactant molecules.

Stoichiometric Rates

Stoichiometry and Rate Expressions

Stoichiometric ratios affect how rates are expressed for reactions involving multiple reactants and products.

For a general reaction:

$aA + bB \rightarrow cC + dD$

The rate is:

$\text{Rate} = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}$

• This rate expression accounts for stoichiometric coefficients.

Example: Ozone Decomposition

Given the rate at which $O_2$ appears is $6.0 \times 10^{-5}$ mol/L·s, use stoichiometry to find the rate at which $O_3$ disappears.

Example: For $2O_3 \rightarrow 3O_2$, the rate of disappearance of $O_3$ is:

$\text{Rate}_{O_3} = \frac{2}{3} \times \text{Rate}_{O_2}$

Rate Laws

General Rate Law Expression

Rate laws (or rate equations) mathematically describe the rate of a chemical reaction based on reactant concentrations.

For a general reaction:

$aA + bB \rightarrow \text{products}$

$\text{Rate} = k[A]^m[B]^n$

• $k$ = rate constant (units depend on reaction order)

• $m$ and $n$ = reaction orders (experimentally determined)

Reaction Orders

Types of Reaction Orders

Reaction orders indicate how the rate depends on the concentration of each reactant.

• Zero order (0): Rate is independent of reactant concentration.

• First order (1): Rate is directly proportional to reactant concentration.

• Second order (2): Rate is proportional to the square of reactant concentration.

• Overall reaction order is the sum of the individual orders.

Examples of Rate Laws

• $\text{Rate} = k[H_2O_2]$ (First order in $H_2O_2$)

• $\text{Rate} = k[C_4H_6]^2$ (Second order in $C_4H_6$)

• $\text{Rate} = k[H^+][OH^-]$ (First order in each reactant, overall second order)

Experimental Determination

Reaction orders must be determined experimentally by observing how concentration changes over time in different scenarios.

Rate Constant (k)

Definition and Units

The rate constant (k) determines the speed of a particular reaction. Its value and units depend on the overall reaction order.

• If $k > 10^9$, the reaction is considered fast.

• If $k < 10^0$, the reaction is considered slow.

• Units for rate: always M/s (molarity per second).

• Units for $k$ vary:

  • First order: $k$ in s$^{-1}$

  • Second order: $k$ in M$^{-1}$s$^{-1}$

Example: $\text{Rate} = k[H_2O_2]$, $k$ = s$^{-1}$

Example: $\text{Rate} = k[H^+][OH^-]$, $k$ = M$^{-1}$s$^{-1}$

Determining Rate Law: Method of Initial Rates

Experimental Approach

The method of initial rates uses data from various experiments to determine the rate law of a reaction.

• Reaction order and rate constant are determined from experimental data.

• Compare trials where only one reactant concentration changes to find its order.

• Plug values into the rate law to solve for $k$.

Example Table:

[NO] (mol/L)	[Cl2] (mol/L)	Initial Rate (M·s-1)
0.10	0.10	0.00300
0.10	0.15	0.00450
0.15	0.10	0.00675

Use the initial rates method and the experimental data to determine the rate law and the value of $k$ for the reaction.

Integrated Rate Laws

Zero Order Reactions

For zero order reactions:

$\text{Rate} = k$

Integrated rate law:

$[A]_t = -kt + [A]_0$

A plot of $[A]_t$ vs. $t$ yields a straight line.

First Order Reactions

For first order reactions:

$\text{Rate} = k[A]$

Integrated rate law:

$\ln[A]_t = -kt + \ln[A]_0$

A plot of $\ln[A]_t$ vs. $t$ yields a straight line.

Second Order Reactions

For second order reactions:

$\text{Rate} = k[A]^2$

Integrated rate law:

$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$

A plot of $1/[A]_t$ vs. $t$ yields a straight line.

Half-Life of a Reaction

Definition

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

• Zero order: $t_{1/2} = \frac{[A]_0}{2k}$

• First order: $t_{1/2} = \frac{0.693}{k}$

• Second order: $t_{1/2} = \frac{1}{k[A]_0}$

Collision Model

Fundamentals

The collision model explains reaction rates by considering the frequency and energy of molecular collisions.

• Reactants must collide with proper orientation and sufficient energy to react.

• The minimum energy required is called activation energy ($E_a$).

• Collisions with energy greater than $E_a$ can form an unstable activated complex (transition state).

Arrhenius Equation

Temperature Dependence of Rate Constant

The Arrhenius equation relates the rate constant to activation energy and temperature:

$k = Ae^{-E_a/(RT)}$

• $A$ = frequency factor (related to collision frequency and orientation)

• $E_a$ = activation energy

• $R$ = gas constant ($8.314$ J/mol·K)

• $T$ = temperature (K)

Linear form:

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

A plot of $\ln k$ vs. $1/T$ yields a straight line with slope $-E_a/R$.

Reaction Mechanisms

Elementary Steps and Molecularity

Many reactions occur through a series of elementary steps rather than a single event. The molecularity of a step is the number of molecules involved:

• Unimolecular: One reactant molecule

• Bimolecular: Two reactant molecules

• Termolecular: Three reactant molecules (rare)

Intermediates and Mechanism Representation

Reaction mechanisms show the stepwise conversion of reactants to products, often involving intermediates (species produced and consumed during the mechanism).

• Each elementary step has its own rate law, determined by its molecularity.

• The rate-determining step is the slowest step, which limits the overall reaction rate.

Catalysts

Role of Catalysts

A catalyst is a substance that increases the rate of a chemical reaction without being consumed. Catalysts provide an alternative pathway with lower activation energy.

• Homogeneous catalysts: Same phase as reactants

• Heterogeneous catalysts: Different phase than reactants

Example: In the decomposition of hydrogen peroxide, iodide ion acts as a catalyst.

Summary Table: Reaction Orders and Integrated Rate Laws

Order	Rate Law	Integrated Rate Law	Half-Life Expression
Zero	$\text{Rate} = k$	$[A]_t = -kt + [A]_0$	$t_{1/2} = \frac{[A]_0}{2k}$
First	$\text{Rate} = k[A]$	$\ln[A]_t = -kt + \ln[A]_0$	$t_{1/2} = \frac{0.693}{k}$
Second	$\text{Rate} = k[A]^2$	$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$	$t_{1/2} = \frac{1}{k[A]_0}$

Additional info: Some examples and context were inferred to ensure completeness and clarity for exam preparation.