Reaction Rates and Chemical Kinetics: A Comprehensive Guide

Chemical kinetics studies the speed of chemical reactions, from slow processes like rusting (\( \ce{4Fe + 3O2 -> 2Fe2O3} \)) to rapid ones like explosions (\( \ce{2H2 + O2 -> 2H2O} \)). The reaction rate is the change in concentration over time:

\[ \text{Rate} = -\frac{\Delta [\ce{A}]}{\Delta t} = \frac{\Delta [\ce{B}]}{\Delta t} \]

Where \( [\ce{A}] \) is reactant concentration (decreasing) and \( [\ce{B}] \) is product concentration. This MathMultiverse guide explores factors, rate laws, activation energy, and applications.

Factors Affecting Rates

Concentration

Higher concentrations increase collisions. For \( \ce{2NO + O2 -> 2NO2} \):

\[ \text{Rate} \propto [\ce{NO}]^2 [\ce{O2}] \]

Doubling \( [\ce{NO}] \) quadruples the rate.

Temperature

Higher temperatures increase collision energy. For \( \ce{H2 + I2 -> 2HI} \):

\[ k \propto e^{-\frac{E_a}{RT}} \]

Catalysts

Lower \( E_a \). For \( \ce{2H2O2 -> 2H2O + O2} \), MnO₂ reduces \( E_a \):

\[ k_{\text{catalyzed}} = A e^{-\frac{58000}{RT}} \]

Surface Area

Increases rate in heterogeneous reactions.

Pressure

For gases, higher pressure boosts rate.

Rate Laws

Rate laws relate rate to concentrations:

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

\( k \): rate constant; \( m, n \): reaction orders.

Zero-Order

Rate is constant:

\[ \text{Rate} = k \]
\[ [\ce{A}]_t = [\ce{A}]_0 - kt \]

Half-life: \( t_{1/2} = \frac{[\ce{A}]_0}{2k} \).

First-Order

Rate ∝ concentration:

\[ \text{Rate} = k [\ce{A}] \]
\[ \ln([\ce{A}]_t) = \ln([\ce{A}]_0) - kt \]

Half-life: \( t_{1/2} = \frac{\ln(2)}{k} \). For \( k = 0.02 \, \text{s}^{-1} \):

\[ t_{1/2} = \frac{0.693}{0.02} \approx 34.65 \, \text{s} \]

Example: \( \ce{2N2O5 -> 4NO2 + O2} \).

Second-Order

Rate ∝ \( [\ce{A}]^2 \):

\[ \text{Rate} = k [\ce{A}]^2 \]
\[ \frac{1}{[\ce{A}]_t} = \frac{1}{[\ce{A}]_0} + kt \]

Half-life: \( t_{1/2} = \frac{1}{k [\ce{A}]_0} \). For \( k = 0.5 \, \text{M}^{-1}\text{s}^{-1} \), \( [\ce{A}]_0 = 0.1 \, \text{M} \):

\[ t_{1/2} = \frac{1}{0.5 \times 0.1} = 20 \, \text{s} \]

For \( \ce{CH3CHO -> CH4 + CO} \), rate = \( 0.01 \, \text{M/s} \) at \( [\ce{CH3CHO}] = 0.2 \, \text{M} \):

\[ k = \frac{0.01}{0.2} = 0.05 \, \text{s}^{-1} \]

First-Order Reaction Decay

Concentration vs. time for a first-order reaction (\( k = 0.02 \, \text{s}^{-1} \), \( [\ce{A}]_0 = 0.2 \, \text{M} \)).

Activation Energy

Activation energy (\( E_a \)) is the energy barrier for reactions. Arrhenius equation:

\[ k = A e^{-\frac{E_a}{RT}} \]

Calculating \( E_a \)

For \( k_1 = 0.01 \, \text{s}^{-1} \) at 298 K, \( k_2 = 0.04 \, \text{s}^{-1} \) at 318 K:

\[ \ln\left(\frac{0.04}{0.01}\right) = -\frac{E_a}{8.314} \left( \frac{1}{318} - \frac{1}{298} \right) \]
\[ E_a \approx 55.6 \, \text{kJ/mol} \]

Catalyst Effect

For \( \ce{CO + NO2 -> CO2 + NO} \), \( E_a \) drops from 134 to 90 kJ/mol:

\[ \frac{k_{\text{cat}}}{k_{\text{uncat}}} \approx 4.6 \times 10^7 \]

Applications

Industry: Haber Process

For \( \ce{N2 + 3H2 -> 2NH3} \):

\[ \text{Rate} = k [\ce{N2}] [\ce{H2}]^3 \]

Medicine: Drug Stability

Aspirin decomposition, \( k = 1.5 \times 10^{-6} \, \text{s}^{-1} \):

\[ t_{1/2} \approx 5.35 \, \text{days} \]

Environment: Ozone Depletion

For \( \ce{Cl + O3 -> ClO + O2} \):

\[ \text{Rate} = 10^{10} [\ce{Cl}] [\ce{O3}] \]