Chemical Kinetics: Rates, Mechanisms, and Reaction Dynamics

Chemical Kinetics

Introduction to Kinetics

Chemical kinetics is the study of the rate (or speed) at which chemical processes occur. It also provides insight into the reaction mechanism, which describes the step-by-step pathway by which reactants are converted into products.

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

• Reaction mechanism: The detailed sequence of elementary steps that make up the overall reaction.

Factors that Affect Reaction Rate

Physical State of Reactants

The physical state of reactants influences how readily molecules can interact and react.

• Reactants must come into contact to react.

• Homogeneous mixtures (all reactants in the same phase) generally react faster than heterogeneous mixtures.

Other Factors Influencing Rate

• Concentration of Reactants: Higher concentrations typically increase reaction rate.

• Temperature: Raising temperature usually increases rate by providing more energy for collisions.

• Presence of a Catalyst: Catalysts lower the activation energy, increasing rate without being consumed.

• Nature of Reactants: Some substances are inherently more reactive.

• Surface Area of Solids: Greater surface area allows more collisions.

• Pressure (for gases): Increasing pressure (decreasing volume) increases concentration and rate.

Measuring Reaction Rate

Definition and Calculation

Reaction rates are determined by monitoring the change in concentration of reactants or products over time.

• For a reaction: A → B, rate can be measured as the decrease in [A] or increase in [B] per unit time.

• Average rate over an interval: $\text{Average rate} = \frac{\Delta [\text{Reactant}]}{\Delta t}$

• Instantaneous rate: The rate at a specific moment, found as the slope of the tangent to the concentration vs. time curve.

Example Table: Concentration vs. Time

Time (s)	[C4H9Cl] (M)
0.0	0.1000
50.0	0.0905
100.0	0.0820
200.0	0.0671
300.0	0.0549
500.0	0.0448
1000.0	0.0200
10,000.0	0

Note: The average rate decreases as the reaction proceeds due to decreasing reactant concentration.

Stoichiometry and Rate

When the stoichiometric coefficients are not 1:1, use them to relate rates of disappearance and appearance:

• For $2\text{HI}(g) \rightarrow \text{H}_2(g) + \text{I}_2(g)$:

$\text{Rate} = -\frac{1}{2}\frac{\Delta [\text{HI}]}{\Delta t} = \frac{\Delta [\text{H}_2]}{\Delta t} = \frac{\Delta [\text{I}_2]}{\Delta t}$

Method of Initial Rates

Determining Rate Laws Experimentally

The method of initial rates involves measuring the initial rate of reaction for different initial concentrations of reactants.

• By comparing how the rate changes as concentrations change, the order of reaction with respect to each reactant can be determined.

Example Table: Initial Rates

Experiment	[NH4+] (M)	[NO2-] (M)	Initial Rate (M/s)
1	0.0100	0.0200	5.4 × 10-7
2	0.0200	0.0200	10.8 × 10-7
3	0.0400	0.0200	21.5 × 10-7
4	0.100	0.0200	53.2 × 10-7
5	0.200	0.0200	108 × 10-7
6	0.200	0.0400	216 × 10-7
7	0.200	0.0800	324 × 10-7
8	0.200	0.160	433 × 10-7

Rate Laws and Reaction Order

Rate Law

A rate law expresses 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 is the rate constant; m and n are the reaction orders with respect to A and B.

• The rate law must be determined experimentally.

Orders of Reaction

• Zeroth Order: Rate is independent of concentration ($\text{Rate} = k$).

• First Order: Rate is directly proportional to concentration ($\text{Rate} = k[A]$).

• Second Order: Rate is proportional to the square of concentration ($\text{Rate} = k[A]^2$).

Overall Reaction Order

• The sum of the exponents in the rate law gives the overall order.

• Example: $\text{Rate} = k [\text{NH}_4^+][\text{NO}_2^-]$ is second order overall (first order in each reactant).

Units of the Rate Constant (k)

• Units of k depend on the overall reaction order:

Order	Rate Law	Units of k
Zeroth	rate = k	M/s
First	rate = k[A]	s-1
Second	rate = k[A]2	M-1s-1

Integrated Rate Laws

First Order Reactions

• Differential rate law: $\text{Rate} = k[A] = -\frac{d[A]}{dt}$

• Integrated rate law: $\ln \frac{[A]_t}{[A]_0} = -kt$

• Linear form: $\ln [A]_t = -kt + \ln [A]_0$ (y = mx + b)

• Graph of $\ln [A]$ vs. time is a straight line with slope -k.

Second Order Reactions

• Integrated rate law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$

• Graph of $1/[A]$ vs. time is a straight line with slope k.

Zero Order Reactions

• Integrated rate law: $[A]_t = [A]_0 - kt$

• Graph of [A] vs. time is a straight line with slope -k.

Summary Table: Integrated Rate Laws and Half-Lives

	Zeroth Order	First Order	Second Order
Integrated Rate Law	$[A]_t = [A]_0 - kt$	$\ln [A]_t = \ln [A]_0 - kt$	$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
Half-life ($t_{1/2}$)	$\frac{[A]_0}{2k}$	$\frac{0.693}{k}$	$\frac{1}{k[A]_0}$

Collision Theory and Mechanisms

Collision Theory

For a reaction to occur, reactant molecules must collide with sufficient energy and proper orientation.

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

• Only collisions with energy ≥ Ea and correct orientation lead to product formation.

Reaction Mechanisms

• Reactions may occur in a single step (elementary reaction) or multiple steps (multistep mechanism).

• The rate-determining step is the slowest step in a multistep mechanism and controls the overall rate.

• Molecularity refers to the number of molecules involved in an elementary step (unimolecular, bimolecular, termolecular).

Activation Energy and Temperature

• Increasing temperature increases the fraction of molecules with energy ≥ Ea, thus increasing rate.

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

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

• Where A is the frequency factor, R is the gas constant, and T is temperature in Kelvin.

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

Sample Problems and Applications

Sample Rate Law Problem

• Given data for the reaction $2\text{NO}(g) + \text{O}_2(g) \rightarrow 2\text{NO}_2(g)$, determine:

  • (A) Rate law: $\text{rate} = k[\text{NO}]^2[\text{O}_2]$

  • (B) Order: 2nd order in NO, 1st order in O2, 3rd order overall

  • (C) $k = 1822$ M-2s-1

  • (D) Rate for given concentrations: 0.0559 M/s

Half-Life Calculations

• First order: $t_{1/2} = \frac{0.693}{k}$ (independent of initial concentration)

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

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

Graphical Determination of Order

• Zero order: [A] vs. time is linear

• First order: ln[A] vs. time is linear

• Second order: 1/[A] vs. time is linear

Key Terms

• Rate law: Mathematical relationship between rate and concentrations

• Order of reaction: Exponent of concentration in rate law

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

• Activation energy (Ea): Minimum energy for reaction

• Half-life (t1/2): Time for half of reactant to be consumed

• Elementary step: Single step in a reaction mechanism

• Rate-determining step: Slowest step in a mechanism