Chemical Kinetics: Reaction Rates, Rate Laws, and Temperature Dependence

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. Understanding kinetics allows chemists to control reactions and optimize conditions for desired outcomes.

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

• Rate law: An equation that relates the reaction rate to the concentrations of reactants, each raised to a power (the order with respect to that reactant).

Reaction Order and Rate Laws

Units of the Rate Constant

The units of the rate constant (k) depend on the overall order of the reaction:

• Zero-order: M/s or mol/(L·s)

• First-order: 1/s or s-1

• Second-order: 1/(M·s) or L/(mol·s)

If you know the units of the rate constant, you can deduce the overall order of the reaction.

Summary Table: Rate Laws and Integrated Rate Laws

Order	Rate Law	Integrated Rate Law
0	Rate = k	$[A]_t = -kt + [A]_0$
1	Rate = k[A]	$\ln\left(\frac{[A]_t}{[A]_0}\right) = -kt$
2	Rate = k[A]^2	$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$

Half-Life and Straight-Line Plots

Order	Integrated Rate Law	Half-Life	Straight-Line Plot
0	$[A]_t = -kt + [A]_0$	$\frac{[A]_0}{2k}$	$[A]$ vs $t$
1	$\ln\left(\frac{[A]_t}{[A]_0}\right) = -kt$	$\frac{0.693}{k}$	$\ln[A]$ vs $t$
2	$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$	$\frac{1}{k[A]_0}$	$1/[A]$ vs $t$

Graphical Determination of Reaction Order

Plots of concentration versus time can be used to determine the order of a reaction:

• Zero-order: $[A]$ vs $t$ is linear (slope = –k, intercept = $[A]_0$)

• First-order: $\ln[A]$ vs $t$ is linear (slope = –k, intercept = $\ln[A]_0$)

• Second-order: $1/[A]$ vs $t$ is linear (slope = k, intercept = $1/[A]_0$)

Example: For the decomposition of NO2, a plot of $\ln[NO_2]$ vs $t$ is not linear, but $1/[NO_2]$ vs $t$ is linear, indicating a second-order reaction in NO2.

Worked Examples in Kinetics

Example 1: First-Order Reaction Calculation

The isomerization of cyclopropane to propene is a first-order reaction with $k = 9.2\ \text{s}^{-1}$. If $[\text{cyclopropane}]_0 = 6.00\ \text{M}$, what is $[\text{cyclopropane}]$ after 1.00 s?

• Use the first-order integrated rate law: $\ln\left(\frac{[A]_t}{[A]_0}\right) = -kt$

Temperature Dependence of Rate Constants

Collision Theory and Activation Energy

Collision theory states that molecules must collide with sufficient energy and proper orientation to react. The minimum energy required is the activation energy ($E_a$).

• The rate constant can be expressed as $k = Zfp$

• $Z$ = collision frequency

• $f$ = fraction of collisions with energy ≥ $E_a$

• $p$ = orientation factor

The fraction of collisions with enough energy is:

$f = e^{-E_a/RT}$

As temperature increases, $f$ increases exponentially, making reactions faster.

Importance of Molecular Orientation

For some reactions, only certain orientations of colliding molecules lead to product formation. For example, in the reaction of NO and Cl2, only collisions where NO approaches Cl2 with its N atom lead to NOCI formation.

Transition-State Theory and Energy Diagrams

Transition-state theory describes the formation of an activated complex (transition state), a high-energy, unstable arrangement of atoms. The potential energy diagram shows the energy changes during a reaction.

• Endothermic reaction: Products have higher energy than reactants; $\Delta H > 0$

• Exothermic reaction: Products have lower energy than reactants; $\Delta H < 0$

Example: For $N_2O(g) \rightarrow N_2(g) + O(g)$, $E_a$(forward) = 251 kJ/mol, $\Delta H = +167$ kJ/mol, $E_a$(reverse) = 84 kJ/mol.

The Arrhenius Equation

Arrhenius Equation Forms

The temperature dependence of the rate constant is given by the Arrhenius equation:

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

Where:

• $A$ = frequency factor

• $E_a$ = activation energy

• $R$ = gas constant

• $T$ = temperature (K)

A linearized, two-point form is:

$\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$

Example Calculations

• Example 3: Given $k_1$ and $k_2$ at two temperatures, use the Arrhenius equation to find $E_a$ and predict $k$ at a third temperature.

• Example 4: For a first-order reaction, the half-life is $t_{1/2} = 0.693/k$.

• Example 5: If the half-life increases as the reaction proceeds, the reaction is likely second order.

Determining Rate Laws from Experimental Data

Example 6: Rate Law Analysis

Given a rate law $\text{Rate} = k[Q][R]^2$, changing the concentration of Q or R affects the rate according to their exponents. Zero-order reactants do not affect the rate when their concentration changes.

Example 7: Initial Rate Method

Given experimental data for $2NO(g) + O_2(g) \rightarrow 2NO_2(g)$:

Exp.	[NO]0, M	[O2]0, M	Initial Rate of NO, M/s
1	0.0125	0.0253	0.0281
2	0.0250	0.0253	0.112
3	0.0125	0.0506	0.0561

Use the method of initial rates to determine the reaction order with respect to each reactant and the rate constant.

Example 8: Multicomponent Rate Law

For $2Fe^{2+}(aq) + Cl_2(aq) \rightarrow 2Fe^{3+}(aq) + 2Cl^-(aq)$, experimental data is used to determine the order with respect to Fe2+, Cl2, and H+:

Exp.	[Fe2+], M	[Cl2], M	[H+], M	Rate, M/s
1	0.0020	0.0020	1.0	1.0 × 10-5
2	0.0040	0.0020	1.0	2.0 × 10-5
3	0.0020	0.0040	1.0	2.0 × 10-5
4	0.0040	0.0040	1.0	4.0 × 10-5
5	0.0020	0.0020	0.5	2.0 × 10-5
6	0.0020	0.0020	0.1	1.0 × 10-5

By comparing experiments, deduce the order with respect to each reactant.

Applications and Extensions of Rate Laws

Rate Laws Beyond Chemistry

Rate laws can be used to describe processes outside chemistry, such as biological growth or aging. The exponents in such rate laws must be determined experimentally.

• Example: Rate of growth = (soil type)a(temperature)b(light)c(fertilizer)d

• To determine exponents, vary one factor at a time and measure the effect on the rate.

Example: If doubling a factor quadruples the rate, the exponent for that factor is 2.

Overall Reaction Order

The overall order of a reaction is the sum of the exponents in the rate law. For example, for $\text{Rate} = k[NO_2][F_2]$, the reaction is first order in NO2, first order in F2, and second order overall.

Additional info: Where the original slides or notes were incomplete, standard textbook explanations and equations were provided for clarity and completeness.