Chemical Kinetics: Rates, Mechanisms, and Catalysis

Chemical Kinetics

Introduction to Kinetics

Chemical kinetics is the study of the rates at which chemical reactions occur and the factors that affect these rates. Unlike thermodynamics, which tells us if a reaction is possible, kinetics tells us how fast it will happen.

• Reaction Rate: Measures how quickly reactants are converted to products.

• Examples: Some reactions are instantaneous (e.g., precipitation of AgCl), while others (e.g., cement hardening) can take years.

Factors Affecting Reaction Rate

Several factors influence how fast a reaction proceeds:

• Concentration of reactants

• Temperature

• Presence of a catalyst

• Physical state of reactants

Defining Reaction Rate

Average and Instantaneous Rate

The rate of a reaction can be measured as the change in concentration of a reactant or product over time.

• Average Rate: Change in concentration over a time interval.

• Instantaneous Rate: Rate at a specific moment, found by the tangent to the concentration vs. time curve or by the first derivative.

Example: For the reaction $2NO_2(g) \rightarrow 4NO(g) + O_2(g)$, the average rate is:

• $\text{Rate} = -\frac{1}{2} \frac{\Delta[NO_2]}{\Delta t}$

• $\text{Rate} = \frac{1}{4} \frac{\Delta[NO]}{\Delta t}$

Monitoring Reaction Progress

Rates can be monitored by measuring the concentration of any reactant or product and relating it to others using stoichiometry.

• For a general reaction: $aA + bB \rightarrow cC + dD$

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

Differential Rate Laws

Formulation of Rate Laws

The differential rate law expresses the rate as a function of reactant concentrations and a rate constant.

• General form: $\text{Rate} = k[A]^x[B]^y$

• k: Rate constant (depends on temperature)

• x, y: Reaction orders (determined experimentally)

Units of k: Depend on the overall order of the reaction.

• First order: $\text{Rate} = k[A]$

• Second order: $\text{Rate} = k[A]^2$

Determining Reaction Orders Experimentally

Reaction orders are found by varying the concentration of one reactant at a time (Isolation or Initial Rates Method) and observing the effect on rate.

Exp	[CH3COOH] (M)	[I2] (M)	[H+] (M)	Rate (M/s)
1	0.010	0.010	0.00050	1.15e-5
2	0.020	0.010	0.00050	2.30e-5
3	0.010	0.020	0.00050	2.30e-5
4	0.010	0.010	0.00100	1.15e-5

Additional info: Orders are deduced by comparing how rate changes with concentration changes.

Integrated Rate Laws

Time Dependence of Concentration

Integrated rate laws relate reactant concentration to time, allowing prediction of concentrations at any time.

• First Order: $[A]_t = [A]_0 e^{-kt}$

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

• Zero Order: $[A]_t = [A]_0 - kt$

Half-life ($t_{1/2}$): Time for half the reactant to be consumed.

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

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

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

Graphical Determination of Reaction Orders and Rate Constants

Linear Plots for Integrated Rate Laws

Reaction order can be determined by plotting concentration data:

• First order: Plot $\ln[A]$ vs. time (straight line)

• Second order: Plot $1/[A]$ vs. time (straight line)

• Zero order: Plot $[A]$ vs. time (straight line)

Pseudo First Order Reactions

Manipulating Reaction Conditions

If one reactant is in large excess, its concentration remains nearly constant, simplifying the rate law to first order in the other reactant.

• Example: $NO(g) + O_2(g) \rightarrow NO_2(g)$, with $[O_2]$ in excess, rate depends only on $[NO]$.

The Arrhenius Equation

Temperature Dependence of Rate Constant

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

• $k = A e^{-E_a/(RT)}$

• A: Frequency factor

• $E_a$: Activation energy

• R: Gas constant

Graphical determination: Plot $\ln k$ vs. $1/T$ yields a straight line with slope $-E_a/R$.

Reaction Mechanisms

Elementary Steps and Rate Laws

Complex reactions occur via a series of elementary steps. The overall rate law is determined by the slowest (rate-determining) step.

• Intermediate: Species formed and consumed during the mechanism.

• Molecularity: Number of reactant particles in an elementary step (unimolecular, bimolecular, termolecular).

Rate Law from Mechanism

For a mechanism, the rate law can be written directly from the molecularity of the slow step.

• Example: If the slow step is $A + B \rightarrow C$, then $\text{Rate} = k[A][B]$.

Catalysis

Role of Catalysts

A catalyst increases the reaction rate by providing an alternative pathway with lower activation energy. It is not consumed in the reaction.

• Homogeneous catalyst: Same phase as reactants.

• Heterogeneous catalyst: Different phase from reactants.

• Enzymes: Biological catalysts that increase rates by stabilizing transition states.

Summary Table: Rate Laws and Integrated Forms

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

Key Terms

• Rate constant (k): Proportionality constant in the rate law.

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

• Intermediate: Species formed and consumed during a reaction mechanism.

• Catalyst: Substance that increases reaction rate without being consumed.

Additional info: These notes expand on the original slides and fill in missing definitions, equations, and context for clarity and completeness.