Unit 6: Chemical Kinetics: Rate Laws, Collision Theory & Catalysis

While chemical thermodynamics establishes the thermodynamic spontaneity ($\Delta G < 0$) and ultimate equilibrium composition of a system, it is blind to time. A reaction may possess a colossal thermodynamic driving force ($\Delta G^\circ \ll 0$) yet remain kinetically dormant at room temperature for millennia (such as the oxidation of diamond to graphite, or the reaction of hydrogen and oxygen gas to form water). Chemical kinetics is the science of reaction velocities, transition states, and microscopic reaction mechanisms.

The Stoichiometric Definition of Reaction Rate

Consider a general chemical reaction:

where $\nu_i$ are stoichiometric coefficients (negative for reactants, positive for products).

The rate of disappearance of a reactant $A$ is $-\frac{d[A]}{dt}$, and the rate of appearance of product $C$ is $+\frac{d[C]}{dt}$. To obtain a unique, single value for the reaction rate that is independent of which chemical species is being monitored, IUPAC defines the Rate of Reaction ($r$ or $v$) as the rate of change of extent of reaction $\xi$ per unit volume $V$:

For our general reaction:

The dimensions of reaction rate are always $[\text{concentration}]\cdot[\text{time}]^{-1}$, with coherent SI units of $\text{mol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$ (or $\text{M}\cdot\text{s}^{-1}$).

Experimental Measurement Techniques

Determining reaction rates requires measuring concentration $[A_i](t)$ as a continuous function of time:

1. Spectrophotometry / UV-Vis Absorption: Based on the Beer-Lambert Law ($A = \epsilon b c$). Particularly suited when reactants or products possess characteristic chromophores (e.g., transition metal complexes, iodine, conjugated polyenes).

2. Conductometry: Measures the electrical conductance of ionic solutions over time (e.g., ester saponification releasing acetate ions and consuming hydroxide ions with distinct limiting ionic mobilities $\lambda^\circ$).

3. Polarimetry: Measures the optical rotation angle $\alpha(t)$ of chiral solutions over time (e.g., the historical inversion of sucrose catalyzed by acid into glucose and fructose).

4. Gas Manometry: Measures total pressure $P(t)$ inside a constant-volume reactor for gas-phase reactions where stoichiometric moles of gas change ($\Delta n_g \neq 0$).

The Differential Rate Law

For most homogeneous chemical reactions, the reaction rate at a given temperature is experimentally found to be proportional to powers of the concentrations of the participating species:

• $k$: The Rate Constant (Specific Reaction Rate). It is independent of concentrations, but depends strongly on temperature, pressure, and the presence of catalysts.
• $m$: The Partial Reaction Order with respect to reactant $A$.
• $n$: The Partial Reaction Order with respect to reactant $B$.
• $m + n$: The Overall Reaction Order.

Crucial Conceptual Distinction: Reaction orders ($m, n$) are strictly empirical experimental quantities. They cannot be deduced from the stoichiometric coefficients ($a, b$) of an overall chemical equation! A reaction such as $2\text{N}_2\text{O}_5 \rightarrow 4\text{NO}_2 + \text{O}_2$ is first-order ($r = k[\text{N}_2\text{O}_5]$), not second-order. Reaction orders can be positive integers ($0, 1, 2$), fractional ($1/2, 3/2$ in radical chain reactions), or even negative (when a product inhibits the rate).

Dimensional Analysis of the Rate Constant ($k$):

From $r = k [C]^n$, the dimensions of $k$ depend directly upon the overall reaction order $n$:

• For $n = 0$ (Zero-Order): $[k] = \text{mol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$
• For $n = 1$ (First-Order): $[k] = \text{s}^{-1}$
• For $n = 2$ (Second-Order): $[k] = \text{L}\cdot\text{mol}^{-1}\cdot\text{s}^{-1}$
• For $n = 3$ (Third-Order): $[k] = \text{L}^2\cdot\text{mol}^{-2}\cdot\text{s}^{-1}$

Differential rate laws relate reaction rate to instantaneous concentrations. To predict concentration $[A](t)$ as an explicit function of elapsed time $t$, we must integrate the differential equations.

1. Zero-Order Reactions

A reaction is zero-order when its rate is completely independent of reactant concentration:

Separating variables and integrating from $t = 0$ ($[A] = [A]_0$) to time $t$ ($[A] = [A]_t$):

• Linear Plot: A plot of $[A]_t$ versus time $t$ yields a straight line with slope $-k$ and y-intercept $[A]_0$.
• Half-Life ($t_{1/2}$): The time required for $[A]$ to reach half its initial value ($[A]_{t_{1/2}} = \frac{1}{2}[A]_0$):

For zero-order reactions, half-life is directly proportional to initial concentration.

• Examples: Heterogeneous surface catalysis under saturated adsorbate coverage (e.g., decomposition of $\text{NH}_3$ on tungsten or platinum filaments at high pressure) and biological alcohol metabolism by hepatic alcohol dehydrogenase when enzyme binding sites are saturated.

2. First-Order Reactions

A reaction is first-order when its rate is directly proportional to the concentration of a single reactant:

Separating variables:

Expressed in exponential decay form:

• Linear Plot: A plot of $\ln[A]_t$ versus $t$ yields a straight line with slope $-k$ and y-intercept $\ln[A]_0$.
• Half-Life ($t_{1/2}$):

Fundamental First-Order Principle: The half-life of a first-order process is completely independent of initial concentration $[A]_0$. Exactly $50\%$ of the reactant decays every $t_{1/2}$ interval, irrespective of whether the sample begins at $10\text{ M}$ or $10^{-6}\text{ M}$. All nuclear radioactive decays obey first-order kinetics identically.

3. Second-Order Reactions

Type 1: Single Reactant ($2A \rightarrow \text{Products}$)

Separating variables:

• Linear Plot: A plot of $\frac{1}{[A]_t}$ versus $t$ yields a straight line with positive slope $+k$ and y-intercept $\frac{1}{[A]_0}$.
• Half-Life ($t_{1/2}$):

For second-order reactions, half-life is inversely proportional to initial concentration ($t_{1/2} \propto [A]_0^{-1}$). Each successive half-life is twice as long as the previous one ($t_{1/2}^{(2)} = 2 t_{1/2}^{(1)}$).

Type 2: Two Reactants with Non-Equal Initial Concentrations ($A + B \rightarrow \text{Products}$)

Let $[A]_0 = a, [B]_0 = b$. At time $t$, let $x$ moles per liter have reacted: $[A]_t = a - x, [B]_t = b - x$:

Integrating by partial fractions:

4. Pseudo-First-Order Reactions (The Flooding Method)

When a reaction involves multiple reactants, isolating individual orders is achieved experimentally via the Isolation (Flooding) Method. For $r = k [A][B]$: If reactant $B$ is introduced in colossal stoichiometric excess relative to $A$ ($[B]_0 \gg [A]_0$, e.g., $[B]_0 = 1.0\text{ M}$ and $[A]_0 = 0.001\text{ M}$): Throughout the entire reaction, $[B]_t \approx [B]_0 = \text{const}$. The rate law simplifies to:

where $k' \equiv k [B]_0$ is the pseudo-first-order rate constant. By measuring $k'$ across multiple runs with varying $[B]_0$, the true second-order constant $k$ is deduced from the slope of $k'$ vs $[B]_0$.

Virtually all macroscopic chemical reactions proceed through a sequence of discrete, sub-picosecond microscopic events known collectively as the Reaction Mechanism.

Elementary Reactions vs Complex Reactions

• Elementary Reaction: A reaction that occurs in a single microscopic event or collision, traversing a single transition state without forming any isolable chemical intermediates.
• Molecularity: The exact number of reactant molecules, atoms, or ions that collide and participate in an elementary step:

1. Unimolecular ($\text{Molecularity} = 1$): A single energized molecule dissociates or isomerizes:

2. Bimolecular ($\text{Molecularity} = 2$): Two species collide and exchange bonds:

3. Termolecular ($\text{Molecularity} = 3$): Simultaneous three-body collision ($A + B + C \rightarrow P$). Extremely rare in gas-phase chemistry because the statistical probability of three independent particles colliding within a window of $\sim 10^{-13}\text{ s}$ is minute. Reactions with molecularity $> 3$ are physically non-existent.

The Fundamental Law of Elementary Reactions: For an elementary step only, the reaction orders are identically equal to its molecularity and stoichiometric coefficients! For complex multi-step reactions, this correspondence fails completely.

The Rate-Determining Step (RDS) Approximation

In a multi-step reaction mechanism composed of sequential elementary steps, if one step has an activation barrier significantly higher than all other steps, that step proceeds much slower than the rest. It acts as a kinetic bottleneck, termed the Rate-Determining Step (RDS). The rate of the overall reaction is dictated entirely by the rate of this slowest elementary step.

Example: Oxidation of Nitric Oxide

This reaction exhibits experimental third-order kinetics: $r = k[\text{NO}]^2[\text{O}_2]$. A three-body collision is physically implausible. The accepted mechanism involves a rapid pre-equilibrium followed by a slow bimolecular RDS:

• Step 1 (Fast, Reversible): $\text{NO} + \text{NO} \xrightleftharpoons[k_{-1}]{k_1} \text{N}_2\text{O}_2$ (intermediate dimer)
• Step 2 (Slow, RDS): $\text{N}_2\text{O}_2 + \text{O}_2 \xrightarrow{k_2} 2\text{NO}_2$

From Step 2:

Since $\text{N}_2\text{O}_2$ is a transient intermediate, it cannot appear in the final rate law. Using the Pre-Equilibrium Approximation on Step 1 ($r_1 = r_{-1}$):

Substituting into the rate equation:

where $k_{obs} = \frac{k_1 k_2}{k_{-1}}$. This mechanism perfectly reproduces the experimental third-order rate law!

The Steady-State Approximation (SSA)

When a reaction mechanism involves highly reactive, unstable intermediates $I$ that do not achieve pre-equilibrium, the Steady-State Approximation (Max Bodenstein, 1913) assumes that after an initial induction period, the concentration of the reactive intermediate remains constant and small:

The Lindemann-Hinshelwood Mechanism of Unimolecular Reactions

Frederick Lindemann (1922) and Cyril Hinshelwood resolved a major paradox of physical chemistry: How can a unimolecular gas-phase reaction ($A \rightarrow P$) exhibit first-order kinetics when thermal excitation requires bimolecular collisions ($A + A \rightarrow A^* + A$)?

The proposed mechanism:

1. Collisional Activation:

(where $M$ is any collision partner, including other $A$ molecules).

2. Collisional Deactivation (Quenching):

3. Unimolecular Chemical Reaction:

Applying the Steady-State Approximation to the energized intermediate $A^*$:

The overall rate of product formation is:

Limiting Pressure Regimes:

• High-Pressure Limit ($P \rightarrow \infty$, $k_{-1}[M] \gg k_2$):

Deactivation dominates over reaction. The denominator simplifies to $k_{-1}[M]$:

• Low-Pressure Limit ($P \rightarrow 0$, $k_2 \gg k_{-1}[M]$):

Every energized molecule reacts before collision can quench it. The denominator simplifies to $k_2$:

The Lindemann-Hinshelwood mechanism proves that gas-phase unimolecular reactions transition from second-order at low pressures to first-order at high pressures!

Reaction rates exhibit an extraordinarily steep dependence on temperature. For most chemical reactions near room temperature, a modest increase of just $10\text{ K}$ doubles or triples the reaction velocity.

The Empirical Arrhenius Equation (1889)

Svante Arrhenius unified empirical observations into the master equation of chemical kinetics:

• $E_a$: The Activation Energy ($\text{J}\cdot\text{mol}^{-1}$ or $\text{kJ}\cdot\text{mol}^{-1}$)—the minimum kinetic energy that colliding reactant molecules must possess to overcome electrostatic repulsion and initiate bond breaking.
• $A$: The Pre-Exponential Factor (Frequency Factor)—measures the total frequency of collisions occurring with appropriate molecular orientation.
• $\exp(-E_a / RT)$: The Boltzmann Factor—the statistical fraction of collisions possessing kinetic energy equal to or exceeding $E_a$.

Taking the natural logarithm:

A plot of $\ln k$ versus $1/T$ (an Arrhenius Plot) yields a straight line with:

Evaluating Eq. (6.17) at two distinct temperatures $T_1$ and $T_2$:

Hard-Sphere Collision Theory

For a bimolecular gas-phase reaction $A + B \rightarrow P$, collision theory models molecules as hard, rigid spheres of collision diameters $d_A$ and $d_B$:

• Mean collision cross-section: $\sigma_{AB} = \pi d_{AB}^2 = \pi \left(\frac{d_A + d_B}{2}\right)^2$.
• Total collision frequency per unit volume between $A$ and $B$:

where $\mu = \frac{m_A m_B}{m_A + m_B}$ is the reduced mass of the colliding pair.

If every collision possessing kinetic energy along the line of centers exceeding $E_a$ led to reaction, the rate constant would be:

The Steric Factor ($P$):

For complex polyatomic molecules, experimental rate constants are frequently orders of magnitude smaller than predicted by simple collision frequency ($k_{exp} \ll k_{theor}$). Molecules must collide not only with sufficient energy, but with the correct spatial orientation (e.g., in an $S_N2$ reaction, the nucleophile must attack the back-side of the carbon-leaving group bond). Collision theory introduces the Steric Factor ($P \le 1$):

For reactions between spherical atoms, $P \approx 1$. For reactions involving large enzymes or sterically hindered organic molecules, $P$ can drop to $10^{-4} - 10^{-6}$.

Transition State Theory (Activated Complex Theory)

Developed by Henry Eyring, Michael Polanyi, and Eugene Wigner (1935), Transition State Theory (TST) abandons rigid hard spheres and models the continuous evolution of molecular geometry along a multi-dimensional Born-Oppenheimer potential energy surface.

Reactants pass through a transient, high-energy molecular configuration called the Activated Complex (Transition State, denoted $\ddagger$) located at the saddle point of the potential energy surface:

The activated complex is assumed to be in quasi-thermodynamic equilibrium with the reactants ($K^\ddagger = \frac{[\ddagger]}{[A][B]}$), decomposing into products at a universal frequency governed by quantum mechanics: $\nu = \frac{k_B T}{h}$.

This yields the iconic Eyring-Polanyi Equation:

Since $\Delta G^\ddagger = \Delta H^\ddagger - T \Delta S^\ddagger$:

• $\Delta H^\ddagger$: Enthalpy of Activation ($\Delta H^\ddagger = E_a - RT$ for liquid reactions).
• $\Delta S^\ddagger$: Entropy of Activation.
• $\Delta S^\ddagger > 0$: Transition state is more disordered or dissociated than reactants (e.g., unimolecular bond cleavage).
• $\Delta S^\ddagger < 0$: Transition state requires highly constrained, ordered orientation of two or more colliding partners, explaining the physical origin of the collision theory steric factor:

A catalyst is a chemical substance that increases the velocity of a chemical reaction without being consumed in the net chemical transformation.

Energetic Principles of Catalytic Action

The Fundamental Thermodynamic Rule of Catalysis:

1. A catalyst provides an alternative reaction pathway possessing a significantly lower activation energy barrier ($E_a^{cat} < E_a^{uncat}$ or $\Delta G^{\ddagger, cat} < \Delta G^{\ddagger, uncat}$).
2. Because both forward and reverse reaction rates are accelerated by identically the same factor:

1. A catalyst does NOT alter the equilibrium constant $K_{eq}$, does NOT alter $\Delta G^\circ$, does NOT alter $\Delta H^\circ$, and does NOT alter the maximum theoretical equilibrium yield of products. It merely dramatically accelerates the speed with which the system approaches equilibrium.

Homogeneous vs Heterogeneous Catalysis

1. Homogeneous Catalysis: The catalyst exists in the same physical phase as the reactants (typically liquid solution). Examples:

• Acid-base catalysis: Protonation of carbonyl oxygens in ester hydrolysis accelerating nucleophilic attack by water.

2. Heterogeneous Catalysis: The catalyst exists in a distinct physical phase, most commonly a solid transition metal or metal oxide accelerating gas-phase or liquid-phase reactions.

• Mechanism of Surface Catalysis:

1. Diffusion of reactants to the solid surface.
2. Adsorption: Reactants bind to active surface sites.

• Physisorption: Weak van der Waals binding ($\Delta H_{ads} \sim -10\text{ to } -40\text{ kJ/mol}$), reversible.
• Chemisorption: Chemical bond formation with surface metal d-orbitals ($\Delta H_{ads} \sim -80\text{ to } -400\text{ kJ/mol}$), accompanied by bond weakening and dissociation of reactant molecules (e.g., $\text{H}_2 \rightarrow 2\text{H}_{ads}$ on Pt).

1. Surface reaction between adsorbed species:

• Langmuir-Hinshelwood Mechanism: Both reacting species adsorb on adjacent surface sites and react.
• Eley-Rideal Mechanism: An adsorbed molecule reacts directly with an incoming gas-phase molecule.

4. Desorption: Product molecules detach from the surface into the bulk fluid.

• Industrial Applications:
• Haber-Bosch ammonia synthesis ($\text{Fe}$ promoted with $\text{K}_2\text{O}$ and $\text{Al}_2\text{O}_3$ at $450^\circ\text{C}, 200\text{ bar}$).
• Ostwald nitric acid synthesis ($\text{Pt-Rh}$ gauze wire nets).
• Automotive catalytic converters ($\text{Pt/Pd/Rh}$ honeycomb monoliths oxidizing $\text{CO}$ and hydrocarbons while reducing $\text{NO}_x$).

Enzyme Kinetics: The Michaelis-Menten Model

Enzymes are biological catalysts (globular proteins) exhibiting extraordinary catalytic power (accelerating rates by $10^6 - 10^{17}$ fold) and stereochemical specificity.

In 1913, Leonor Michaelis and Maud Menten proposed the foundational kinetic mechanism:

where $E$ is free enzyme, $S$ is substrate, $ES$ is the enzyme-substrate complex, and $P$ is product.

Applying the Briggs-Haldane Steady-State Approximation to $[ES]$:

Total enzyme concentration is conserved: $[E]_0 = [E] + [ES] \implies [E] = [E]_0 - [ES]$. Substitute into the steady-state equation:

where $K_m \equiv \frac{k_{-1} + k_2}{k_1}$ is the Michaelis Constant ($\text{M}$).

The initial reaction rate is $v_0 = k_2 [ES]$:

where $V_{max} \equiv k_2 [E]_0$ is the Maximum Catalytic Velocity (achieved when all enzyme active sites are fully saturated with substrate).

• $k_2 = k_{cat}$ is the Turnover Number (catalytic constant, $\text{s}^{-1}$)—the number of substrate molecules converted to product per active site per second.
• $k_{cat} / K_m$ is the Catalytic Efficiency ($\text{M}^{-1}\cdot\text{s}^{-1}$). Upper physiological limit is bounded by the diffusion-controlled limit ($10^8 - 10^9\text{ M}^{-1}\cdot\text{s}^{-1}$).

The Lineweaver-Burk Double-Reciprocal Plot:

Taking the reciprocal of Eq. (6.24):

A plot of $1/v_0$ versus $1/[S]$ yields a straight line with: