Chemical Kinetics: Rates, Mechanisms, and Catalysis

Chemical Kinetics

Introduction to Kinetics

Chemical kinetics is the study of the speed (rate) 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 proceed.

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

• Applications: Industrial synthesis, biological processes, environmental chemistry.

• Key Questions: How fast do reactions occur? What factors influence the rate?

Variation in Reaction Rates

• Reaction rates can vary by many orders of magnitude.

• Examples:

  • Decomposition of H2O2 (with catalyst): seconds

  • Formation of rust: years

  • Precipitation of AgCl: instantaneous

Factors Affecting Reaction Rates

• Concentration: Higher concentrations generally increase rate.

• Temperature: Higher temperatures increase rate.

• Catalysts: Lower activation energy, increasing rate.

• Surface Area: Greater surface area (for solids) increases rate.

Defining and Measuring Reaction Rates

Average and Instantaneous Rate

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

• Instantaneous Rate: Rate at a specific moment; mathematically, the slope of the tangent to the concentration vs. time curve.

Example: For the reaction 2 NO2(g) → 2 NO(g) + O2(g):

• Average rate: $\text{Rate} = -\frac{1}{2} \frac{\Delta [\text{NO}_2]}{\Delta t}$

• Instantaneous rate: $\text{Rate} = -\frac{1}{2} \frac{d[\text{NO}_2]}{dt}$

Monitoring Reaction Progress

• Rates can be measured by monitoring the concentration of any reactant or product over time.

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

• Rate can be expressed as:

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

Differential Rate Laws

Definition and Form

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)

Determining Reaction Order

• Reaction order is not necessarily related to stoichiometry; it must be determined experimentally.

• Common orders: zero, first, second.

• Units of k depend on overall order.

Experimental Determination: Initial Rates Method

• Vary the concentration of one reactant while keeping others constant; observe the effect on rate.

• Use data from multiple experiments to deduce the order with respect to each reactant.

Exp	[A] (M)	[B] (M)	Rate (M/s)
1	0.010	0.010	1.15×10-4
2	0.020	0.010	2.30×10-4
3	0.010	0.020	2.30×10-4

Additional info: Table shows how doubling [A] or [B] affects the rate, allowing deduction of reaction order.

Integrated Rate Laws

Definition

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

• Zero Order: $[A]_t = [A]_0 - kt$

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

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

Half-Life

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

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

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

Graphical Determination

• Plotting $[A]$ vs. $t$ (zero order), $\ln [A]$ vs. $t$ (first order), or $1/[A]$ vs. $t$ (second order) yields a straight line if the reaction is of that order.

Pseudo-First Order Reactions

When one reactant is in large excess, its concentration remains nearly constant, simplifying the rate law to first order with respect to the limiting reactant.

• Example: Hydrolysis of esters in water, where [H2O] is constant.

The Arrhenius Equation

Temperature Dependence of Rate Constant

• The rate constant, k, increases with temperature.

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

• A: Frequency factor (related to collision frequency and orientation)

• Ea: Activation energy

• R: Gas constant (8.314 J/mol·K)

• T: Temperature in Kelvin

Graphical Determination of Ea

• Taking the natural log: $\ln k = \ln A - \frac{E_a}{R} \frac{1}{T}$

• Plotting $\ln k$ vs. $1/T$ yields a straight line with slope $-E_a/R$.

Reaction Mechanisms

Elementary Steps and Rate Laws

• Overall reactions may occur via a series of elementary steps.

• Elementary Step: A single molecular event.

• Intermediate: Species produced and consumed during the mechanism.

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

Rate-Determining Step (RDS)

• The slowest step in a mechanism determines the overall rate law.

• Rate law for the overall reaction is based on the RDS.

Proposing and Testing Mechanisms

• Propose a mechanism, write the rate law for the RDS, and compare to the experimentally determined rate law.

• If they match, the mechanism is supported (but not proven).

Catalysis

Definition and Types

• Catalyst: A substance that increases the rate of a reaction without being consumed.

• Homogeneous Catalysis: Catalyst in the same phase as reactants.

• Heterogeneous Catalysis: Catalyst in a different phase (e.g., solid catalyst with gaseous reactants).

• Enzymes: Biological catalysts, highly specific and efficient.

Effect on Reaction Profile

• Catalysts lower the activation energy, providing an alternative pathway for the reaction.

• Do not affect the overall thermodynamics (ΔG, ΔH) of the reaction.

Summary Table: Rate Laws and Integrated Forms

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

Key Takeaways

• Reaction rates depend on concentration, temperature, catalysts, and surface area.

• Rate laws must be determined experimentally.

• Integrated rate laws allow calculation of concentrations at any time.

• Arrhenius equation relates rate constant to temperature and activation energy.

• Mechanisms explain the stepwise process of reactions; the slowest step controls the rate.

• Catalysts increase reaction rates by lowering activation energy.