Chemical Kinetics and Reaction Mechanisms Study Notes

Introduction to Chemical Kinetics

• Kinetics and Reaction Mechanisms

  • Chemical kinetics is the study of reaction rates and the understanding of the reaction mechanism.

  • Reaction Mechanism (Reaction Pathway): The sequence of events that describes the actual process by which reactants become products.

  • Example Reaction: $2H_2O_2 \rightarrow O_2 + 2H_2O$

Energy Barriers and Activation Energy

• The Energy Barrier: To become products, reactants must overcome an energy barrier. They must possess sufficient energy to "climb the wall" to reach the other side.

• Activation Energy ($E_a$): The specific amount of energy required to perform a reaction.

• Rate and $E_a$ Relationship: In general, a greater activation energy leads to a slower reaction rate.

Catalysts

• Definition and Function: A catalyst speeds up a reaction without being used up or permanently changed. It functions by changing the reaction pathway or mechanism, thereby lowering the activation energy ($E_a$).

• Representation: In chemical equations, catalysts are usually written over the reaction arrow (e.g., $2H_2O_2 \xrightarrow{KI} O_2 + 2H_2O$).

• Homogeneous Catalysts: The catalyst is in the same phase as the reactant(s).

  • Example: Blood acting as a catalyst for the decomposition of $2H_2O_2$.

• Enzymes (Biological Catalysts):

  • Substrate: The reactant molecule (e.g., lactose).

  • Enzyme-Substrate Complex: A temporary complex formed during the reaction.

  • Example: The enzyme lysozyme.

• Heterogeneous Catalysts: The catalyst is in a different phase than the reactant(s).

  • Example 1: $2H_2O_{2(aq)} \xrightarrow{KI_{(s)}} O_{2(g)} + 2H_2O_{(l)}$

  • Example 2: $C_2H_{4(g)} + H_{2(g)} \xrightarrow{Pt_{(s)}} C_2H_{6(g)}$

  • Surface Catalysts: Platinum ($Pt_{(s)}$) acts as a surface catalyst for gas-phase reactions.

Potential Energy Diagrams

• Components of Energy Diagrams:

  • Reactants: The initial energy level.

  • Products: The final energy level.

  • Transition State (Activated Complex): The peak of the energy barrier. It represents the point where reactants are in the process of rearranging into products. This state is very unstable.

  • Activation Energy ($E_a$): The energy difference between the reactants and the transition state ($T.S.$).

  • Enthalpy Change ($\Delta H$ or Enthalpy): The energy difference between the reactants and products.

• Reaction Types:

  • Exothermic Reactions ($-\Delta H$): The products are at a lower energy level than the reactants.

  • Endothermic Reactions ($+\Delta H$): The products are at a higher energy level than the reactants.

• Transitions and Reversibility:

  • Reverse Activation Energy ($E_a(\text{Reverse})$): The energy required to go from products back to the transition state.

  • Case Study: Rearrangement of methyl isonitrile ($H_3C-N\equiv C:$) to acetonitrile ($H_3C-C\equiv N:$). The diagram shows the Potential Energy relative to the reaction pathway, highlighting the hill representing $E_a$.

Factors Affecting Reaction Rates

1. Physical State of the Reactants:

   • Molecules must come into contact to react.

   • Two aqueous solutions typically react faster than two solids.

   • Crushed powders react faster than large crystals due to increased surface area.

2. Concentration of Reactants:

   • An increase in concentration increases the likelihood of molecular collisions.

3. Temperature:

   • At higher temperatures, molecules have more kinetic energy and move faster.

   • They collide more often and with greater energy.

4. Presence of a Catalyst:

   • Speeds up reactions by offering an alternative mechanism that lowers $E_a$.

   • Catalysts can be recovered at the end of the reaction.

The Collision Model

• Molecular Basis: Takes into account the effects of temperature and concentration. For a reaction to occur, bonds must break and new ones must form; this only happens if molecules collide.

• Principles:

  • Greater number of collisions = Greater reaction rate.

  • Increased concentration leads to more collisions.

  • Increased temperature leads to higher energy collisions.

• Orientation Factor ($A$): Molecules must be oriented correctly during collision to react successfully.

• Fraction of Active Molecules ($f$): Only a small fraction of molecules possess the minimum activation energy ($E_a$) at any given time.

  • Formula: $f = e^{-\frac{E_a}{RT}}$

  • $R = 8.314\,J\,mol^{-1}K^{-1}$

  • $T = \text{Kelvin Temperature}$

Calculating Reaction Rates and Stoichiometry

• Average Reaction Rate: Measured in Molarity per second ($M/s$).

  • Formula: $\text{Rate} = -\frac{\Delta[\text{Reactant}]}{\Delta t} = \frac{\Delta[\text{Product}]}{\Delta t}$

• Relationship Between Species: For a reaction $aA + bB \rightarrow cC + dD$:

  • $\text{Rate} = -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$

• Signs: Product rates are positive; reactant rates are reported as negative when describing disappearance, though overall rates are often expressed as positive values.

• Example Problem: $2Fe_2O_3 + 3C \rightarrow 4Fe + 3CO_2$

  • If the rate of $Fe_2O_3$ disappearance is $-0.59\,M/s$, find the rate of $CO_2$.

  • Calculation: $\text{Rate of } CO_2 = \frac{0.59}{2} \times 3 = 0.885\,M/s$ (rounded to $0.89\,M/s$).

• Ozone Example: $2O_{3(g)} \rightarrow 3O_{2(g)}$. If $O_2$ appears at $6.0 \times 10^{-5}\,M/s$, the rate of $O_3$ disappearance is $-4.0 \times 10^{-5}\,M/s$.

Rate Laws and Reaction Orders

• Rate Law Expression: $\text{Rate} = k[A]^m[B]^n$

  • $k = \text{Rate Constant}$

  • $m, n = \text{Reaction Orders}$ (determined experimentally, usually integers like 0, 1, 2).

• Reaction Orders:

  • Zeroth Order ($0$): Concentration changes do not affect the rate.

  • First Order ($1$): Rate doubles if concentration doubles. Rate triples if concentration triples.

  • Second Order ($2$): Rate quadruples if concentration doubles ($2^2 = 4$). Rate increases nine-fold if concentration triples ($3^2 = 9$).

• Calculating Exponents Logarithmically:

  • $\text{Exponent} = \frac{\log\left(\frac{\text{rate}_a}{\text{rate}_b}\right)}{\log\left(\frac{[A]_a}{[A]_b}\right)}$

• Overall Reaction Order: The sum of all individual exponents in the rate law.

Integrated Rate Laws

• Integrated Rate Law Usage: Used when comparing concentration over time or dealing with percentages reacted/remaining.

• Equations and Graphs:

  • Zeroth Order:

    • Equation: $[A]_t = -kt + [A]_0$

    • Linear Plot: $[A]$ vs $t$ (Slope = $-k$)

  • First Order:

    • Equation: $\ln[A]_t = -kt + \ln[A]_0$

    • Linear Plot: $\ln[A]$ vs $t$ (Slope = $-k$)

    • Note: Radioactive decay always follows 1st order kinetics.

  • Second Order:

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

    • Linear Plot: $\frac{1}{[A]}$ vs $t$ (Slope = $k$)

Half-Life ($t_{1/2}$)

• Definition: The time required for one-half of a reactant to react.

• Formulas:

  • 0th Order: $t_{1/2} = \frac{[A]_0}{2k}$

  • 1st Order: $t_{1/2} = \frac{\ln(2)}{k} = \frac{0.693}{k}$

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

• Example Calculation: Insecticide decomposition with $k = 1.45\,yr^{-1}$.

  • $t_{1/2} = \frac{0.693}{1.45} = 0.478\,yr$.

  • Time to reach 1/4 concentration = $2 \times t_{1/2} = 0.956\,yr$.

Arrhenius Equation

• Mathematical Relationship: Relates the rate constant ($k$) to activation energy ($E_a$).

  • $k = A e^{-\frac{E_a}{RT}}$

  • $A = \text{Frequency factor}$

• Calculating Activation Energy via Two-Point Method:

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

  • $R = 8.314\,J\,mol^{-1}K^{-1}$

  • $T_{Kelvin} = ^\circ C + 273.15$

Multistep Reaction Mechanisms

• Elementary Reactions: Reactions that occur in a single step.

• Molecularity:

  • Unimolecular: One molecule involved.

  • Bimolecular: Two molecules involved.

  • Termolecular: Three molecules involved.

• Multistep Mechanisms: One step is usually slower than the rest.

  • Slow Step / Rate-Determining Step (RDS): The step that limits the overall rate. The rate law of the overall reaction is derived from the stoichiometry of the RDS, NOT the overall equation.

• Intermediates vs. Catalysts:

  • Intermediate: Produced in one step and consumed in a subsequent step (appears first as a product, then as a reactant).

  • Catalyst: Present at the start of the reaction and recovered at the end (appears first as a reactant, then as a product).

• Multi-step Energy Diagrams: Feature multiple peaks (transition states) and valleys (intermediates).