5: Moving Molecules and Chemical Kinetics
- Page ID
- 540243
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 5.1: Classical Model of Ideal Gas Pressure
- Classical physics can be used to derive the form of the ideal gas law, which can be combined with the root-mean-square average speed from the Maxwell-Boltzmann distribution of speeds (derived classically or via statistical mechanics) to recover the ideal gas law. The discussion of collisions with the walls to produce a pressure provides a place to begin understanding how collisions are involved in the properties of gases.
- 5.2: Collisions
- This page discusses particle interactions in gases, focusing on collision energy, cross-section, collision frequency, and mean free path. It describes particle motion as a random walk affected by collisions, represented probabilistically. The text introduces Gaussian distribution for displacement and diffusion, emphasizing the diffusion constant and equation.
- 5.3: The Frequency of Collisions with a Wall
- This page explains the derivation of gas pressure through the analysis of gas molecule collisions with container walls. It introduces key concepts like collision volume and number density, while exploring motion in three dimensions to determine collision frequency per unit area (\(z_w\)).
- 5.4: Transport of Matter and Energy in Perfect Gases
- In this section we will consider the transport of matter and energy in perfect gases to develop a microscopic particulate picture of the processes. Additionally, we will use this model to develop some quantitative predictions on how rates of transport depend on conditions.
- 5.5: Stokes-Einstein-Sutherland Equation
- A useful approximate description of the diffusion constant in a liquid is the Stokes-Einstein-Sutherland equation. This section shows one way of deriving this approximate expression.
- 5.6: The Rate of Bimolecular Gas-Phase Reaction Can Be Estimated Using Hard-Sphere Collision Theory and an Energy-Dependent Reaction Cross Section
- This page covers collision frequency in bimolecular gas-phase reactions using the Hard Sphere Model, linking collision frequency to reaction rates and introducing a new reaction cross-section that considers energy requirements. It highlights the model's temperature and energy thresholds while noting deviations from the Arrhenius equation.
- 5.7: A Reaction Cross Section Depends Upon the Impact Parameter
- This page critiques the flawed assumption that all collisions between Q and B particles result in reactions, noting that proper alignment is essential. It discusses the need to modify the reaction cross-section using the line-of-centers model, which takes collision angles into account. Effective collisions depend on the impact parameter, leading to a lower energy available for the reaction than the total kinetic energy.
- 5.8: The Rate Constant for a Gas-Phase Chemical Reaction May Depend on the Orientations of the Colliding Molecules
- This page discusses a revision of the hard-sphere model for particle collisions, highlighting that not all collisions lead to reactions due to insufficient energy and the importance of proper orientation, particularly for non-spherical particles. The traditional model's assumption of spherical particles overestimates effective collisions, leading to inaccuracies in the reaction rate constant \(A\).
- 5.9: The Internal Energy of the Reactants Can Affect the Cross Section of a Reaction
- This page discusses modifications to the collision model to account for the internal energy of polyatomic gas particles, including electronic, vibrational, and rotational components. It notes that molecules in high vibrational states could react without additional translational energy, influencing the reaction cross-section \(\sigma_r\).
- 5.10: A Reactive Collision Can Be Described in a Center-of-Mass Coordinate System
- This page discusses modeling bimolecular reactions of ideal gases with a center-of-mass coordinate system for enhanced accuracy in reaction kinetics. It explains how reactant molecules' kinetic energy is calculated before collisions, distinguishing between the kinetic energy of the center of mass and the relative motion of reactants. The model emphasizes that only the latter impacts the reaction outcome and addresses the conservation of mass and energy throughout the reaction process.
- 5.11: Reactions Can Produce Vibrationally Excited Molecules
- This page discusses the center of mass reaction model utilized to examine energy distribution in the reaction between fluorine and deuterium. It highlights that for the reaction to be successful, the vibrational energy of \(\ce{DF(g)}\) must be under 165 kJ/mol. The analysis shows that \(\ce{DF(g)}\) can occupy vibrational states ranging from \(v = 0\) to \(4\) while adhering to this energy limitation.
- 5.12: Reactive Collisions Can be Studied Using Crossed Molecular Beam Machines
- This page discusses crossed molecular beam experiments, which involve colliding gas-phase atom or molecule beams to study chemical reaction dynamics. Key features include detecting collisions, analyzing product velocities and angles, and using mass spectrometry for energy insights. Originating from Herschbach and Lee, the method has been enhanced with tools like quadrupole filters and supersonic nozzles, broadening its application.
- 5.13: The Velocity and Angular Distribution of the Products of a Reactive Collision
- This page discusses the speed distribution of DF molecules formed from the collision of F and D2, highlighting the connection between products' vibrational energy and their velocity. It notes that vibrational states influence translational energy and speed, analyzing data from crossed molecular beam experiments to explore angular distribution and velocity.
- 5.14: Not All Gas-Phase Chemical Reactions are Rebound Reactions
- This page describes two gas-phase reaction types: rebound reactions, where products bounce back (e.g., \(\ce{D_2}\) with \(\ce{F}\)), and stripping reactions, where products like \(\ce{KI}\) move in the same direction as reactants (e.g., \(\ce{K}\) and \(\ce{I2}\)). Stripping reactions show larger experimental collision cross-sections than predicted and involve electron transfer prior to collision.
- 5.15: The Potential-Energy Surface Can Be Calculated Using Quantum Mechanics
- This page covers Potential Energy Surfaces (PES), which illustrate the potential energy of atomic systems based on their configurations, favoring stable structures at energy minima. It discusses PES in the context of exchange reactions involving hydrogen and deuterium, detailing specific reaction pathways and highlighting transition states.
- 5.16: Equilibrium Constants in Terms of Partition Functions
- This page discusses a gas phase chemical reaction's equilibrium properties, focusing on the equilibrium constant (K_c) and chemical potential. It details the calculation of K_c for the formation of HCl from H₂ and Cl₂ using molecular partition functions, including contributions from various states. At 650 K, the reaction is spontaneous with a K_c of about 2.26 x 10^11, attributed to strong H—Cl bond formation despite breaking H—H and Cl—Cl bonds.
- 5.17: Statistical Mechnanical Transition State Theory
- Transition state theory can be formulated in terms of statistical mechanics. This section outlines how that can be done.
- 5.18: Chain Reactions
- The page describes the concept of chain reactions, which consist of initiation, propagation, and termination steps, particularly when radicals are involved. It uses the reaction between H_2 and Br_2 to produce HBr as an example. A proposed mechanism is analyzed, and steady-state approximations are applied to derive expressions for radicals involved. The resulting expression aligns with the experimentally determined rate law.
- 5.22: Atoms and Molecules can Physisorb or Chemisorb to a Surface
- This page explores the energetics and kinetics of adsorption, focusing on the potential energy curve of adsorbates near surfaces. It contrasts physisorption, which involves weak van der Waals forces, with chemisorption, characterized by stronger chemical bonds. The text outlines the adsorption process of hydrogen molecules (H2), discussing two pathways: direct chemisorption and precursor-mediated chemisorption, which includes initial physisorption.
- 5.23: Isotherms are Plots of Surface Coverage as a Function of Gas Pressure at Constant Temperature
- This page examines the equilibrium between gas molecules and adsorbed species on solid surfaces, introducing the Langmuir isotherm, which connects surface coverage to gas pressure at constant temperature. The isotherm predicts increased surface coverage with rising pressure until a monolayer is achieved. The derivation involves temperature-dependent constants and assumptions about surface site energies, emphasizing the importance of constant enthalpy of adsorption for the model's validity.
- 5.24: Using Langmuir Isotherms to Derive Rate Laws for Surface-Catalyzed Gas-Phase Reactions
- This page discusses the kinetics of heterogeneously-catalyzed reactions, emphasizing unimolecular decomposition and bimolecular processes using Langmuir isotherm expressions. It covers how pressure affects reaction rates, with distinct behaviors observed in first-order kinetics at low pressures and zero-order at high pressures.
- 5.25: The Structure of a Surface is Different from that of a Bulk Solid
- This page discusses the influence of solid surface structure, including defects, on chemisorption and physisorption kinetics and thermodynamics. It highlights the importance of techniques such as Secondary Electron Microscopy (SEM) for topography and Scanning Auger Microscopy (SAM) for compositional analysis.
- 5.26: The Haber-Bosch Reaction Can Be Surface Catalyzed
- This page discusses the Haber-Bosch process, an industrial method for synthesizing ammonia from nitrogen and hydrogen gases. It highlights the contributions of Karl Bosch and Fritz Haber, along with operational parameters like high pressure and intermediate temperatures. The process utilizes a catalyst and focuses on maximizing ammonia yield through recycling unreacted gases. Ongoing research aims to enhance catalysts and further comprehend the reaction's energetics.