Reaction Kinetics, Collision Theory, and Thermodynamic Principles

Reaction Coordinate Diagrams and Activation Energy

• Reaction coordinate diagrams illustrate the energy changes that occur during the course of a chemical reaction, tracking energy as a function of reaction progress.
• Exothermic vs. Endothermic Reaction Energy Profiles:
  • Exothermic reactions feature reactants at a higher potential energy level than the products ($E_{\text{reactants}} > E_{\text{products}}$), resulting in a negative enthalpy change ($\Delta H < 0$).
  • Endothermic reactions feature products at a higher potential energy level than the reactants ($E_{\text{products}} > E_{\text{reactants}}$), resulting in a positive enthalpy change ($\Delta H > 0$).
  • Regardless of whether a reaction is exothermic or endothermic, the energy of activation ($E_a$) is always positive ($E_a > 0$). Energy must always be inputted to reach the transition state.
• Bond Dissociation Enthalpies and Energy Input:
  • All chemical bonds contain stored chemical energy.
  • Even highly unstable bonds require an initial input of energy to break.
  • Even if breaking and reforming bonds eventually results in lower-energy products (an exothermic process), an initial energy barrier must be overcome.
• Key Parameters of Reaction Coordinates:
  • Activation Energy ($E_a$): The quantitative difference in energy between the reactants and the high-energy transition state ($E_a = E_{\text{transition state}} - E_{\text{reactants}}$).
  • In an endothermic reaction, the activation energy hill extends above the product energy level ($E_a > \Delta H$).
  • A large activation energy corresponds to a tall peak/hill on the coordinate diagram, representing a significant energy barrier.
  • A small activation energy corresponds to a short peak/hill on the coordinate diagram, representing a lower energy barrier.

Chemical Kinetics, Rate Laws, and Temperature Dependence

• General Rate Law Expression:
  • $\text{Rate} = k [A]^x [B]^y$
  • The rate constant $k$ is specific to a given chemical reaction and depends directly on temperature ($T$) and activation energy ($E_a$).
• Effect of Activation Energy on Reaction Rate:
  • Reactions with a lower energy of activation ($E_a$) have a shorter energy hill to cross. Consequently, a greater proportion of molecular collisions possess sufficient energy to react, causing the reaction to proceed more rapidly.
  • Reactions with a high activation energy proceed much more slowly because fewer molecules possess enough energy to clear the tall energy barrier.
• Components of the Arrhenius Framework:
  • $RT$: The product of the ideal gas constant $R$ and the absolute temperature $T$ in Kelvin ($K$).
  • Frequency Factor ($A$): Represents the total number of times the chemical reaction approaches the activation energy barrier per unit time.
  • In gas-phase reactions at low pressure, there are fewer moles of gas per unit volume, causing gas particles to collide much less frequently.
  • Only a specific subset of total particle collisions possess both the required energy and the proper spatial orientation for a chemical reaction to occur.
  • Increasing the overall collision frequency increases the absolute size of this effective subset of collisions.
  • Exponential Factor ($e^{-\frac{E_a}{RT}}$):
  • The exponential factor $e^{-\frac{E_a}{RT}}$ ranges strictly between $0$ and $1$.
  • It represents the exact fraction of molecular approaches/collisions that successfully achieve activation.
  • An exponential factor of $0$ indicates that zero approaches possess sufficient energy to cross the barrier.
  • An exponential factor of $1$ indicates that $100\%$ of approaches possess sufficient energy to cross the barrier.
  • As temperature ($T$) increases, kinetic energy and molecular speeds increase, thereby increasing the exponential factor.
• Rule of Thumb for Temperature Dependence:
  • Near room temperature, increasing the temperature by approximately $10\,^\circ\text{C}$ roughly doubles the rate of a chemical reaction.
  • Practical Examples of Temperature Manipulation:
  • Denaturing/cooking fish proteins using acid (e.g., ceviche with lime juice or vinegar) relies on ambient or refrigerated temperatures and takes a long time.
  • Cooking fish in an oven adds thermal energy, dramatically accelerating the reaction rate so the fish cooks in $10\,\text{min}$ to $20\,\text{min}$.
  • Storing perishable items in a freezer lowers kinetic energy, reducing the exponential factor near $0$ to preserve items long-term.
• Determination of Kinetic Parameters via the Arrhenius Plot:
  • The logarithmic form of the Arrhenius equation is linear:     $\ln(k) = -\frac{E_a}{R}\left(\frac{1}{T}\right) + \ln(A)$
  • Plotting $\ln(k)$ on the y-axis against $\frac{1}{T}$ on the x-axis yields a straight line.
  • The slope of the line equals $-\frac{E_a}{R}$, allowing for direct calculation of the activation energy $E_a$.
  • The y-intercept of the line equals $\ln(A)$, allowing for direct calculation of the frequency factor $A$.
  • Determining these parameters experimentally requires measuring rate constants at a minimum of two different temperatures, though ideally more data points are used.

Collision Model and Reaction Mechanisms

• The Collision Model of Chemical Kinetics:
  • Matter consists of particles undergoing continuous rotational, vibrational, and translational motions.
  • For a chemical reaction to occur upon collision, particles must collide with:
  1. Correct spatial orientation.
  2. Sufficient kinetic energy (sufficient velocity).
  • Example: Reaction between ozone ($O_3$) and nitrogen monoxide ($NO$):     $NO(g) + O_3(g) \rightarrow NO_2(g) + O_2(g)$
  • If particles collide with improper orientation or inadequate speed, they bounce apart unchanged.
  • When orientation and energy criteria are met, an oxygen atom transfers from $O_3$ to $NO$, yielding $NO_2$ and $O_2$
• Experimental Workflow for Deriving Reaction Mechanisms:
  1. Vary the concentrations of individual reactants in the laboratory.
  2. Measure the rate of reaction by tracking either the disappearance of a reactant or the appearance of a product over time.
  3. Construct the empirical rate law and solve for the rate constant $k$.
  4. Measure rates at multiple temperatures and construct an Arrhenius plot to determine $E_a$ and $A$
  5. Propose a multi-step reaction mechanism that is consistent with the experimental rate law.
  • Rate law reaction orders ($x, y$) depend entirely on the individual elementary steps of the reaction mechanism, not on overall stoichiometric coefficients.
• Molecularity of Elementary Steps:
  • Unimolecular Step: Involves a single reactant molecule breaking apart or rearranging ($A \rightarrow \text{products}$). The rate law is first order ($\text{Rate} = k [A]$).
  • Bimolecular Step: Involves two colliding molecules ($A + B \rightarrow \text{products}$ or $2A \rightarrow \text{products}$). The rate law is second order overall ($\text{Rate} = k [A][B]$ or $\text{Rate} = k [A]^2$).
  • Decomposition Reactions: Often follow second-order kinetics because two molecules must collide to provide the necessary activation energy to break a molecule apart.
  • Trimolecular (Termolecular) Step: Involves the simultaneous collision of three reactant molecules ($A + B + C \rightarrow \text{products}$). The rate law is third order overall ($\text{Rate} = k [A][B][C]$). Trimolecular steps are extremely rare because the statistical probability of three separate particles colliding simultaneously with proper orientation and sufficient energy is exceptionally low.

Catalysts and Reaction Energetics

• Function of a Catalyst:
  • Primary Purpose: Increases the overall rate of a chemical reaction.
  • Mechanism of Action: Lowers the activation energy barrier ($E_a$).
  • Holds reactants in close spatial proximity, raising collision frequency.
  • Positions reactants in the exact spatial orientation needed for effective collision.
  • Uses a combination of proximity and orientation control.
• Thermodynamic Invariance:
  • Catalysts do not alter the overall energetics ($\Delta H$) of a reaction.
  • Catalyzed endothermic reactions remain endothermic; catalyzed exothermic reactions remain exothermic.
  • A catalyst only alters the height of the energy barrier ($E_a$) between reactants and products.
• Types of Catalysts:
  • Biological Catalysts (Enzymes): Highly specialized proteins capable of increasing reaction rates by factors up to hundreds of millions of times ($10^8 \times$) compared to uncatalyzed rates.
  • Non-biological (Chemical) Catalysts: Synthetic or inorganic catalysts that significantly enhance reaction rates, though generally with lower specificity and magnitude than biological enzymes.

Fundamentals of Thermodynamics and Energy Forms

• Scope of Thermodynamics:
  • Thermodynamics examines the quantitative relationships existing between heat, work, temperature, and energy.
  • Focuses on changes in Enthalpy ($H$), Entropy ($S$), and Gibbs Free Energy ($G$).
• Energy Classifications:
  • Energy: The physical capacity to supply heat or perform work.
  • Potential Energy: Energy stored within an object due to its position, structure, or composition.
  • A roller coaster car stopped at the absolute highest point of a track possesses maximum potential energy.
  • A tank filled with propane gas possesses high chemical potential energy due to its flammable chemical bonds.
  • Kinetic Energy: Energy associated with motion and velocity.
  • A roller coaster car accelerating down a slope possesses high kinetic energy.
  • Ignited propane gas powering a burner converts chemical potential energy into high kinetic/thermal energy.
• Chemical Definition of Work ($W$):
  • Physics Definition: Force multiplied by displacement ($W = F \times d$).
  • Chemistry Application: Expansion work defined as pressure times volume change ($W = -P \Delta V$). Units of $F \times d$ and $P \times V$ are identical.
  • Work can involve changes in mechanical, electromagnetic (e.g., emitting light), or gravitational variables.

Heat, Enthalpy, and Spontaneity

• Thermal Energy and Heat ($q$):
  • Thermal Energy: A form of kinetic energy associated with the microscopic motion of particles.
  • High thermal energy corresponds to rapidly moving molecules, perceived as high temperature.
  • Low thermal energy corresponds to slowly moving molecules, perceived as low temperature.
  • Temperature terms ("hot" and "cold") are relative. For humans, $50\,^\circ\text{C}$ is hot and $0\,^\circ\text{C}$ is cold; relative to liquid nitrogen, $0\,^\circ\text{C}$ is extremely hot.
• Heat Calculations:
  • Heat transfer within a single phase:     $q = m \times c \times \Delta T$     (where $m$ is mass, $c$ is specific heat capacity, and $\Delta T$ is temperature change).
  • Heat transfer during a phase change:     $q = n \times \Delta H_{\text{phase change}}$     (where $n$ is amount of substance and $\Delta H_{\text{phase change}}$ is molar enthalpy of transformation).
• Internal Energy ($U$) and the First Law of Thermodynamics:
  • Energy and matter cannot be created or destroyed; they can only transform.
  • Internal energy $U$ represents total system energy capacity to supply heat or perform work.
  • Enthalpy Relationship:     $H = U + PV$
  • Negative Enthalpy Change ($\Delta H < 0$): Exothermic process; the system releases heat energy to its surroundings.
  • Positive Enthalpy Change ($\Delta H > 0$): Endothermic process; the surroundings provide heat energy to the system.
• Chemical Spontaneity:
  • Definition: A spontaneous process or reaction occurs naturally without continuous external energy input.
  • Spontaneous processes may require an initial spark or activation energy ($E_a$) to start, but proceed continuously on their own once initiated.
  • Non-spontaneous processes require continuous, ongoing external energy input to proceed.
  • Example of Spontaneity: Dissolving sodium chloride (table salt) in room-temperature water without heating or mechanical stirring.
• Clarifications Regarding Spontaneity:
  • Spontaneity is not dictated by heat flow: both endothermic and exothermic processes can be spontaneous.
  • Spontaneity is not equivalent to reaction speed: a process can be highly spontaneous yet proceed at an extremely slow rate.
• Reversibility of Chemical Enthalpy:
  • Decomposition of hydrogen peroxide:     $2H_2O_2(aq) \rightarrow 2H_2O(l) + O_2(g)$     This reaction is spontaneous and exothermic, with an enthalpy of reaction $\Delta H \approx -94\,\text{kJ/mol}$ (or $\approx -95\,\text{kJ/mol}$). Stored hydrogen peroxide spontaneously decomposes over several years.
  • Reverse combination reaction:     $2H_2O(l) + O_2(g) \rightarrow 2H_2O_2(aq)$     This reverse process is non-spontaneous and endothermic, requiring a continuous energy input. The magnitude of enthalpy change remains identical, but the sign is inverted ($\Delta H \approx +94\,\text{kJ/mol}$ or $\approx +95\,\text{kJ/mol}$).
  • Phase changes exhibit identical sign-reversal rules (e.g., $\Delta H_{\text{sublimation}} = -\Delta H_{\text{deposition}}$).

Questions & Discussion

• Question on Activation Energy in Endothermic Reactions:
  • Prompt: In an endothermic reaction where products are higher in energy than reactants, does activation energy still go up, or can it be below product levels?
  • Response: Activation energy must always go up. An initial input of energy beyond the final energy level of the products is always required to reach the transition state.
• Question on Arrhenius Plotting Curves:
  • Prompt: How should a small energy of activation be represented relative to a large activation energy on a reaction coordinate diagram?
  • Response: As long as the hill height is drawn smaller than the comparison curve, it accurately represents a lower activation energy, corresponding to a shorter energy barrier for reactants to cross over.
• Question on Long-Term Preservation:
  • Prompt: Where do we store items when we want to preserve them over long periods of time?
  • Response: In a freezer, because lowering the temperature reduces thermal energy and drastically decreases the kinetic exponential factor, slowing decomposition reactions.
• Question on Catalyst Function:
  • Prompt: What is the precise purpose of a catalyst in a reaction?
  • Response: A catalyst speeds up the rate of a chemical reaction by lowering the activation energy barrier.