Comprehensive Notes on Chemical Kinetics and Reaction Rates

Course Curriculum & Academic Progression

• The sequence of advanced general chemistry topics increases progressively in complexity:
  • Foundational and challenging topics include net ionic equations and chemical thermodynamics.
  • Current and upcoming core units cover chemical kinetics, chemical equilibrium, and acid-base chemistry.
  • Exam 4 represents the most challenging examination in the curriculum; missed material from Exam 3 compounds the difficulty, making catching up exceptionally demanding without persistent study effort.

Thermodynamics Versus Chemical Kinetics

• Thermodynamics:
  • Focuses strictly on whether a chemical transformation is spontaneous or non-spontaneous under given conditions.
  • Determines the reaction enthalpy ($\Delta H$), designating whether a reaction is endothermic (requires/absorbs energy) or exothermic (releases energy).
  • Determines the reaction entropy ($\Delta S$), quantifying the increase or decrease in molecular disorder/randomness.
  • Operates independent of time or reaction pathways; it provides no information regarding the duration required for a system to reach equilibrium.
• Chemical Kinetics:
  • Focuses exclusively on the rate at which a reaction occurs—how fast or slow a chemical transformation proceeds.
  • Spontaneous reactions vary drastically in timescale:
  • Some spontaneous processes, such as the oxidative rusting of iron, take years or millions of years to occur.
  • Other spontaneous processes proceed rapidly, reaching completion within fractions of a second.
  • Chemical kinetics investigates the mechanistic steps and experimental conditions that dictate these reaction velocities.

Potential Energy Diagrams & Activation Energy

• Enthalpy Calculation:
  • The enthalpy change of a reaction ($\Delta H$) is mathematically defined as the difference between the potential energy of the products and the reactants:     $\Delta H = \text{Energy}_{\text{products}} - \text{Energy}_{\text{reactants}}$
  • For exothermic reactions, the potential energy of the products is lower than that of the reactants ($\text{lower} - \text{higher} = \text{negative}$), resulting in a negative value ($\Delta H < 0$) and the liberation of net energy.
  • For endothermic reactions, the potential energy of the products is higher than that of the reactants, resulting in a positive value ($\Delta H > 0$) and the absorption of net energy.
• The Energy Barrier and Transition State:
  • Potential energy diagrams exhibit an initial peak before descending to product energies.
  • The summit of this energy barrier is defined as the transition state (activated complex).
  • In order to react, molecules must absorb sufficient energy to sever existing chemical bonds and separate into free atoms; these atoms subsequently rearrange and form new chemical bonds, which releases energy.
• Activation Energy ($E_a$):
  • The minimum energy input required to activate and initiate a chemical reaction.
  • Required for all chemical processes regardless of whether the overall reaction is exothermic or endothermic.
  • Kinetic significance:
  • A high activation energy ($E_a$) corresponds to a slow reaction rate because fewer reactant molecules possess the necessary kinetic energy to surpass the barrier.
  • A low activation energy ($E_a$) corresponds to a fast reaction rate because a larger fraction of reactant molecules can readily overcome the barrier.

Collision Theory of Reaction Rates

• Chemical reactions occur as a consequence of collisions between reacting molecules or free atoms, followed by the reorganization and formation of new bonds.
• Collision theory dictates that three primary factors govern whether a collision leads to a reaction:
  • 1. Collision Rate:
  • The total number of collisions occurring per unit of time (e.g., collisions per second or per hour).
  • The reaction rate is directly proportional to the collision frequency: a higher collision rate produces a faster reaction rate, whereas a lower collision rate produces a slower reaction rate.
  • 2. Molecular Orientation:
  • Reacting species must collide with a specific geometric alignment to facilitate the redistribution of electrons and formation of new bonds.
  • Example: In the reaction of carbon monoxide ($CO$) with oxygen ($O_2$) to form carbon dioxide ($CO_2$):
    • A collision between an oxygen atom of $CO$ and an oxygen atom of $O_2$ will not yield a reaction.
    • A collision oriented such that the carbon atom of $CO$ impacts an oxygen atom of $O_2$ provides the proper geometry to successfully synthesize $CO_2$.
  • 3. Activation Energy ($E_a$):
  • Colliding species must possess kinetic energy equal to or exceeding the activation energy threshold to overcome electrostatic repulsions and break existing bonds.

Factors Affecting Reaction Rates

• 1. Characteristics and Nature of Reactants:
  • Molecular Size: Smaller molecules react more rapidly than larger molecules because smaller species experience greater mobility and collide more readily.
  • Experimental Modifiability: The intrinsic size of a given reactant molecule is fixed and cannot be artificially decreased for that specific compound.
  • Physical State (Phase): Reaction rates proceed in the general order of:     $\text{Gas} > \text{Liquid} > \text{Solid}$
  • Gas molecules are widely separated, travel rapidly, and collide frequently.
  • Solid particles are locked in fixed positions within a lattice, restricting motion and limiting collision opportunity.
  • Experimental Modifiability: The physical state can often be manipulated experimentally (e.g., vaporizing a liquid into the gas phase to accelerate a reaction).
  • Surface Area: For heterogeneous reactions involving solids, powdered reagents react significantly faster than solid blocks or chunks.
  • Increasing the surface area exposes a larger number of reactive sites to surrounding reactants.
  • Experimental Modifiability: Surface area is easily modified experimentally by pulverizing, grinding, or milling solid reagents into fine powders.
  • Bonding and Chemical Reactivity: Ionic compounds dissolved in aqueous solution react much faster than covalent molecular substances.
  • Dissolved ionic compounds dissociate into free, mobile ions that combine directly without prior bond breakage.
  • Molecular substances contain rigid covalent bonds connecting atoms that must first be cleaved, requiring significant bond dissociation energy.
  • Experimental Modifiability: Inherent chemical reactivity and bond type cannot be altered for a chosen chemical species.
• 2. Reactant Concentration (and Partial Pressure):
  • An increase in reactant concentration directly increases the reaction rate by packing more particles into a given volume, thereby increasing collision frequency.
  • For gas-phase reactions, partial pressure is used in place of molar concentration (dependent upon the mole fraction). Elevated partial pressure increases gas-phase reaction rates analogously to increasing liquid-phase concentration.
• 3. Temperature:
  • Higher temperatures produce faster reaction rates.
  • Thermal energy elevates the average kinetic energy of reactant molecules, causing them to move faster, collide more frequently, and collide with energy exceeding the activation energy ($E_a$).
• 4. Catalysts:
  • A catalyst is a chemical substance added to a reaction mixture that accelerates the reaction rate without being consumed in the overall process.
  • Mechanism: A catalyst alters the reaction route/mechanism by providing an alternative pathway characterized by a lower activation energy ($E_a$).
  • Intermediates: The catalyst participates by reacting with a starting material to generate an unstable reaction intermediate. This intermediate subsequently reacts in a secondary step to yield the target products and regenerate the catalyst in its original form.
  • Reaction Profile Features: Catalyzed pathways frequently display multiple transition state peaks (e.g., a two-peak profile corresponding to a two-step mechanism separated by an intermediate energy trough), with all peak energies remaining lower than the uncatalyzed activation barrier.
  • Hypothetical Mechanism: For an uncatalyzed reaction $A + B \rightarrow C + D$, introducing catalyst $M$ yields:     $\text{Step 1: } B + M \rightarrow BM$$\text{Step 2: } BM + A \rightarrow C + D + M$
  • Net Reaction: $A + B \rightarrow C + D$.
  • Substance $BM$ functions as an unstable chemical intermediate.
  • Catalyst $M$ enters the reaction initially and emerges chemically identical at the conclusion.

Quantitative Potential Energy Diagram Analysis

• Given Reaction Parameters:
  • Reaction Enthalpy: $\Delta H = +40\,kJ$
  • Uncatalyzed Forward Activation Energy: $E_{a,\text{fwd, uncatalyzed}} = 120\,kJ$
  • Catalyzed Forward Activation Energy: $E_{a,\text{fwd, catalyzed}} = 80\,kJ$
• Analysis of the Forward Reaction:
  • Since $\Delta H = +40\,kJ$, the reaction is endothermic; the potential energy of the products is $40\,kJ$ higher than that of the reactants.
  • The uncatalyzed forward barrier rises $120\,kJ$ above the reactant baseline to reach the primary transition state.
  • The catalyzed pathway lowers the barrier to a peak $80\,kJ$ above the reactant baseline.
• Analysis of the Reverse Reaction:
  • In a reverse reaction ($C + D \rightarrow A + B$), the former products become the reactants, and the former reactants become the products.
  • The activation energy of the reverse reaction is measured relative to the product energy level:     $E_{a,\text{rev, uncatalyzed}} = E_{a,\text{fwd, uncatalyzed}} - \Delta H = 120\,kJ - 40\,kJ = 80\,kJ$$E_{a,\text{rev, catalyzed}} = E_{a,\text{fwd, catalyzed}} - \Delta H = 80\,kJ - 40\,kJ = 40\,kJ$
  • Reversing an endothermic reaction lowers the activation energy by exactly the magnitude of $\Delta H$ ($40\,kJ$) in both the catalyzed and uncatalyzed systems.

Mathematical Definition and Formulation of Reaction Rate

• Definition of Rate:
  • The reaction rate measures the speed of a chemical process, defined quantitatively as the change in the concentration of a reactant or product over an interval of time:     $\text{Rate} = \frac{\Delta [\text{Concentration}]}{\Delta t}$
• Sign Conventions:
  • Reaction rates are defined as strictly positive quantities ($\text{Rate} > 0$).
  • Reactants are consumed during a reaction, meaning final concentration is less than initial concentration ($\Delta [\text{Reactant}] = [\text{Reactant}]_{\text{final}} - [\text{Reactant}]_{\text{initial}} < 0$). A negative sign must be applied to reactant terms to ensure the calculated rate is positive.
  • Products are formed over time ($\Delta [\text{Product}] = [\text{Product}]_{\text{final}} - [\text{Product}]_{\text{initial}} > 0$), requiring a positive sign.
• Stoichiometric Coefficients:
  • Reaction rate describes the velocity of the overall chemical reaction, not merely individual chemical species.
  • In a reaction such as $3H_2 + N_2 \rightarrow 2NH_3$, three moles of $H_2$ are consumed for every one mole of $N_2$ consumed and every two moles of $NH_3$ produced.
  • To ensure the overall reaction rate remains a single, uniform numerical value regardless of which species is monitored, each term is divided by its stoichiometric coefficient.
• General Rate Expression:
  • For the general balanced reaction $aA + bB \rightarrow cC + dD$:     $\text{Rate} = -\frac{1}{a}\frac{\Delta [A]}{\Delta t} = -\frac{1}{b}\frac{\Delta [B]}{\Delta t} = \frac{1}{c}\frac{\Delta [C]}{\Delta t} = \frac{1}{d}\frac{\Delta [D]}{\Delta t}$
  • Rate relationships are set as strictly equal across species; individual components are never added or multiplied together to compute overall reaction rate.

Step-by-Step Kinetics Calculations: Methane Combustion

• Model Reaction:$CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(g)$
• Comprehensive Rate Expression:$\text{Rate} = -\frac{1}{1}\frac{\Delta [CH_4]}{\Delta t} = -\frac{1}{2}\frac{\Delta [O_2]}{\Delta t} = +\frac{1}{1}\frac{\Delta [CO_2]}{\Delta t} = +\frac{1}{2}\frac{\Delta [H_2O]}{\Delta t}$
• Calculation 1: Concentration Change of $CH_4$ over 10 Minutes:
  • Evaluated as final concentration minus initial concentration:     $\Delta [CH_4] = [CH_4]_{10} - [CH_4]_0$
  • The resulting value is negative, reflecting the consumption of methane reactant over time.
• Calculation 2: Calculating Produced $[CO_2]$ at 20 Minutes from $[CH_4]$ Data:
  • Experimental time interval: $t = 0\,\text{min}$ to $t = 20\,\text{min}$ ($\Delta t = 20\,\text{min}$).
  • Change in methane concentration: $\Delta [CH_4] = -0.03\,M$.
  • Applying the rate equality:     $-\frac{1}{1}\frac{\Delta [CH_4]}{\Delta t} = +\frac{1}{1}\frac{\Delta [CO_2]}{\Delta t}$$-(-0.03\,M) = \Delta [CO_2]$$\Delta [CO_2] = 0.03\,M$
  • Given an initial concentration $[CO_2]_0 = 0\,M$:     $[CO_2]_{20} = [CO_2]_0 + \Delta [CO_2] = 0\,M + 0.03\,M = 0.03\,M$
• Calculation 3: Rate of Reaction for the Formation of $CO_2$ between 10 and 20 Minutes:
  • Time interval: $\Delta t = 20\,\text{min} - 10\,\text{min} = 10\,\text{min}$.
  • Concentration values: $[CO_2]_{20} = 0.03\,M$; $[CO_2]_{10} = 0.02\,M$.
  • Change in concentration: $\Delta [CO_2] = 0.03\,M - 0.02\,M = 0.01\,M$.
  • Calculation:     $\text{Rate} = \frac{\Delta [CO_2]}{\Delta t} = \frac{0.01\,M}{10\,\text{min}} = 0.001\,M\,\text{min}^{-1}$
• Calculation 4: Average Reaction Rate between 0 and 30 Minutes Using $CH_4$ Data:
  • Experimental concentrations: Initial $[CH_4]_0 = 0.05\,M$; final $[CH_4]_{30} = 0.015\,M$.
  • Time elapsed: $\Delta t = 30\,\text{min} - 0\,\text{min} = 30\,\text{min}$.
  • Concentration change: $\Delta [CH_4] = 0.015\,M - 0.05\,M = -0.035\,M$.
  • Calculation:     $\text{Rate} = -\frac{\Delta [CH_4]}{\Delta t} = -\frac{-0.035\,M}{30\,\text{min}} = 0.0011667\,M\,\text{min}^{-1} \approx 0.0012\,M\,\text{min}^{-1}$
  • Dimensional units ($M\,\text{min}^{-1}$ or $mol\,dm^{-3}\,min^{-1}$) must always be explicitly stated, as numerical values without units lack physical meaning.
• Calculation 5: Overall Reaction Rate vs. Rate of $O_2$ Consumption:
  • The expression for overall reaction rate with respect to oxygen is:     $\text{Rate} = -\frac{1}{2}\frac{\Delta [O_2]}{\Delta t}$
  • The rate of oxygen consumption refers strictly to the rate at which $O_2$ itself disappears:     $\text{Rate of } O_2 \text{ consumption} = -\frac{\Delta [O_2]}{\Delta t} = 2 \times \text{Rate}$
  • The individual consumption/formation rate of a substance does not incorporate the reciprocal of its stoichiometric coefficient.

Questions & Discussion

• Controlling Collision Orientation in Industry:
  • Question: Can molecular orientation be actively controlled during collisions, or is it an uncontrolled factor?
  • Response: In industrial chemical manufacturing, specialized physical or catalytic techniques might exist to influence orientation, but ordinary reaction vessels rely on statistical molecular orientations.