5.2 Reaction Kinetics, Rate Laws & Catalysis

5.2 Reaction Kinetics, Rate Laws & Catalysis

Chemical kinetics is the quantitative study of reaction rates, the factors that influence these rates, and the microscopic mechanisms by which chemical transformations take place. While chemical thermodynamics determines whether a reaction is spontaneous ($\Delta G < 0$), kinetics dictates how rapidly the reaction proceeds toward equilibrium. Understanding rate expressions, reaction order, collision theory, and catalysis is paramount for the AMC examination.

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Rate of Reaction: Instantaneous vs. Average

The rate of reaction represents the speed at which reactants are consumed or products are formed per unit volume per unit time. For a general chemical reaction: $aA + bB \rightarrow cC + dD$

The rate of reaction is expressed differentially as: $\text{Rate} = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = +\frac{1}{c}\frac{d[C]}{dt} = +\frac{1}{d}\frac{d[D]}{dt}$

• Negative Sign: Indicates a decrease in reactant concentration over time.
• Units of Rate: $\text{mol dm}^{-3} \text{s}^{-1}$ or $\text{M s}^{-1}$ (or $\text{atm s}^{-1}$ for gas-phase reactions).
• Average Rate: The change in concentration over a finite time interval: $\text{Rate}_{avg} = -\frac{\Delta [A]}{\Delta t}$.
• Instantaneous Rate: The reaction rate at a specific single instant in time, given by the slope of the concentration-time curve at that point: $\text{Rate}{inst} = \lim{\Delta t \to 0} -\frac{\Delta [A]}{\Delta t} = -\frac{d[A]}{dt}$.

Experimental Methods for Rate Determination

1. Physical Methods (Non-destructive):
   • Refractometry: Used when refractive index changes during liquid-phase reactions.
   • Polarimetry: Used for reactions involving optically active substances (e.g., inversion of cane sugar).
   • Spectrophotometry: Used when reactants or products absorb specific wavelengths of light.
   • Electrical Conductance: Used when the total concentration of ions changes (e.g., hydrolysis of alkyl halides).
   • Dilatometry: Measures small volume changes in liquid systems.
2. Chemical Methods: Periodic withdrawal of aliquots followed by rapid quenching (dilution or cooling) and chemical titration.

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Rate Law, Order of Reaction & Specific Rate Constant

The empirical relationship connecting reaction rate to reactant concentrations is called the rate law or rate equation: $\text{Rate} = k [A]^m [B]^n$ where:

• $k = \text{Specific Rate Constant}$ (or velocity constant). When $[A] = [B] = 1.0\text{ M}$, $\text{Rate} = k$.
• $m$ and $n$ are the partial orders of reaction with respect to reactants $A$ and $B$, respectively.
• Overall Order of Reaction ($n_{total}$): The sum of exponents in the rate law: $n_{total} = m + n$.

Characteristics of Order of Reaction

• Experimental Quantity: Cannot be deduced merely from stoichiometric balanced chemical equations.
• Values: Can be zero, positive integers ($1, 2, 3$), or fractions.
• Zero-Order Reactions: Reaction rate is completely independent of reactant concentration (e.g., photochemical combination of $H_2$ and $Cl_2$ over water, thermal decomposition of $NH_3$ on a hot platinum surface). $\text{Rate} = k [A]^0 = k$.
• Pseudo-First-Order Reactions: Second-order reactions made to follow first-order kinetics by keeping one reactant in overwhelming excess (e.g., acid hydrolysis of ethyl acetate: $CH_3COOCH_2CH_3 + H_2O \xrightarrow{H^+} CH_3COOH + CH_3CH_2OH$, where $[H_2O]$ remains essentially constant).

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Molecularity vs. Order of Reaction

It is essential to distinguish between theoretical molecularity and empirical reaction order:

• Molecularity: The number of reacting species (atoms, ions, or molecules) that must collide simultaneously in an elementary step to bring about chemical reaction.
  • Unimolecular ($O_3 \rightarrow O_2 + O$)
  • Bimolecular ($2HI \rightarrow H_2 + I_2$)
  • Trimolecular ($2NO + O_2 \rightarrow 2NO_2$)
• Key Distinctions:
  1. Molecularity is a theoretical concept derived from elementary mechanisms; Order is an experimentally measured property.
  2. Molecularity is always a positive non-zero integer ($1, 2, 3$); Order can be zero, fractional, or negative.
  3. Molecularity cannot exceed 3 because simultaneous collisions of four or more species are statistically impossible. Order can be zero or fractional.
  4. In complex multi-step reactions, molecularity applies only to individual elementary steps, whereas overall order is governed by the slowest elementary step, known as the Rate-Determining Step (RDS).

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Factors Affecting Reaction Rates & Collision Theory

Reaction rates are influenced by reactant concentration, physical state, temperature, light, and catalysts.

Collision Theory of Reaction Rates

According to collision theory (proposed by Arrhenius, Trautz, and Lewis):

1. Reacting molecules must physically collide with one another.
2. Collisions must possess a minimum threshold energy known as the Activation Energy ($E_a$).
3. Collisions must occur with proper steric orientation (proper spatial alignment) at the instant of impact.

$\text{Rate} = Z \cdot P \cdot e^{-E_a / RT}$ where $Z$ is collision frequency, $P$ is the steric/orientation factor, and $e^{-E_a/RT}$ is the Boltzmann fraction of molecules possessing energy $\ge E_a$.

Temperature Dependence: The Arrhenius Equation

For most chemical reactions, the rate roughly doubles or triples for every $10^\circ\text{C}$ rise in temperature. This temperature coefficient ($\gamma$) is defined as: $\gamma = \frac{k_{T+10}}{k_T} \approx 2 \text{ to } 3$

Quantitatively, temperature dependence is described by the Arrhenius Equation: $k = A e^{-\frac{E_a}{RT}}$

Taking the natural logarithm yields the linear form: $\ln k = \ln A - \frac{E_a}{R}\left(\frac{1}{T}\right) \quad \implies \quad \log_{10} k = \log_{10} A - \frac{E_a}{2.303 R T}$

Plotting $\log_{10} k$ against $\frac{1}{T}$ gives a straight line with a negative slope equal to $-\frac{E_a}{2.303 R}$. For two temperatures $T_1$ and $T_2$: $\log_{10}\left(\frac{k_2}{k_1}\right) = \frac{E_a}{2.303 R} \left( \frac{T_2 - T_1}{T_1 T_2} \right)$

   Energy ▲
          │             Activated Complex [‡]
          │                 ┌───┐
          │                ┌┘   └┐
          │               ┌┘     └┐  ▲
          │  Reactants   ┌┘       └┐ │ Ea (Uncatalyzed)
          │  ┌───────────┘         └┐│
          │  │                      └───────┐ Products
          │  │                              │
          │  └──────────────────────────────┘ ▲ ΔH (Overall Enthalpy)
          └───────────────────────────────────┼────────────────────────►
                                             Reaction Coordinate

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Catalysis: Types & Mechanisms

A catalyst is a substance that alters the rate of a chemical reaction without undergoing any net permanent chemical change itself.

Key Principles of Catalytic Action:

1. Provides an alternative reaction mechanism with a lower activation energy barrier ($E_a$).
2. Increases both the forward and reverse reaction rates by equal factors.
3. Does NOT alter the enthalpy change ($\Delta H$), free energy change ($\Delta G$), or chemical equilibrium constant ($K_{eq}$) of a reversible reaction.

Classification of Catalytic Systems

• Homogeneous Catalysis: Catalyst and reactants exist in the same physical phase (e.g., oxidation of $SO_2$ to $SO_3$ catalyzed by gaseous $NO$ in the Chamber process).
• Heterogeneous Catalysis: Catalyst exists in a different physical phase than reactants, acting via surface adsorption (e.g., synthesis of $NH_3$ by Haber's process using solid iron catalyst; hydrogenation of oils using nickel powder).
• Enzyme Catalysis (Biocatalysts): Complex protein molecules produced by living cells that catalyze biochemical reactions with absolute specificity.
  • Characteristics: Highly specific, operate at optimum temperature ($37^\circ\text{C}$ or $310\text{ K}$) and optimum pH ($6 - 8$ for most cellular enzymes, $1.5 - 2.0$ for pepsin in stomach).
  • Lock and Key Model (Fisher): The active site of the enzyme ($E$) possesses a complementary geometric shape that binds the substrate ($S$) to form an enzyme-substrate complex ($ES$), yielding product ($P$): $E + S \rightleftharpoons ES \rightarrow E + P$