Principles of Reaction Kinetics

Rate equations are exactly solved for the reversible consecutive reaction of the first-order and the time- dependence of concentrations is analytically determined for species in the reaction. With the assumption of pseudo first-order reaction, the calculation applies and determines the concentration of product accurately and explicitly as a function of time in the unimolecular decomposition of Lindemann and in the enzyme catalysis of Michaelis-Menten whose rate laws have been approximated in terms of reactant concentrations by the steady-state approximation. Mechanisms and rate laws of most chemical reactions can be described based on reversible, consecutive or parallel processes whose reaction rates are determined in detail as functions of time provided that they consist of the first-order or pseudo first-order steps. Concentrations in terms of time for species involved in these simple processes are easily obtained mathematically and the exact results are found in usual chemical kinetics texts. However many realistic reac- tions in fact contain the variety of combinations of the three rudimentary processes aforementioned and this complicates solving rate equations analytically and makes it difficult determining time-dependence of rates accurately. Approxi- mate methods thus have been developed and introduced to handle these reactions. A typical example is the steady-state approximation (SSA) which is often used to describe reactions in which reversible processes are combined with consecutive steps. It has been developed from the first-order reversible consecutive reaction which is represented by . If the rate of consumption of B is much greater than its rate of formation in the reaction scheme, the concentration of B may be presumed to change little with time and can be estimated in terms of reactant concentration from the assumption. The overall rate of reac- tion is then evaluated using the approximate concentration without detailed integration of rate equations. This simple but efficient technique is widely applied in reactions consist- ing with many steps to determine rates approximately. Ex- amples are the unimolecular decomposition known for Lin- demann mechanism and the enzyme catalysis of Michaelis- Menten. 1 The rate of reaction , if it is obtained fully as a function of time, can be extended under the pseudo first-order reaction condition to precisely determine the time-dependent behavior of reactions obeying mechanism of Lindemann or of Michaelis-Menten. Analytical solution to the rate equations of these reactions, however, has not been reported and the detailed time-dependence of concentrations is not known yet within the author's knowledge. In this work, rate equations of the first-order reversible consecutive reaction are exactly solved and two sample calculations are performed with different set of rate con- stants to show how concentrations depend on time in detail. The derivations are applied to reactions of the named mech- anisms to calculate concentrations of products analytically and are used to check the approximate rate laws by SSA for validity and limitations. Then a brief summary concludes the article.

Chapter 3 - Rate laws

Elements, Atoms, and Electrons Matter and Elements Atoms Atomic Structure Isotopes The Periodic Table Electron Structure of Atoms Covalent Bonding and Molecules Interactions between Atoms Covalent Bonds Are Formed by Sharing Outer Electrons Formulae of Compounds Covalent Bonds Formed by Combining Atomic Orbitals Single Overlap, the Sigma-Bond Double Overlap, the Pi-Bond Molecules with s- and p-Bonds Hybrid Molecular Orbitals Forces Within and Between Molecules Ionic Bonding Polar Covalent Bonds Dipole-Dipole Forces The Hydrogen Bond van der Waals Forces The Hydrophobic Effect Coordinate Bonds Chemical Reactions Rates of Reaction Factors Affecting Rate of Reaction Rate Equations Integrated Forms of Rate Equations Zero-Order Reactions Integrated Form of the Zero-Order Rate Equation First-Order Reactions The Integrated Form of the First-Order Rate Equation Second-Order Reactions Integrated Forms of Second-Order Rate Equations Pseudo-First-Order Reactions Reversible Reactions Equilibrium Water The Water Molecule Ice Water Solutions The Mole Concept Calculating Molar Masses Molarity Colloidal Solutions Diffusion and Osmosis Acids, Bases, and Buffers Ionisation of Water The Hydrogen Ion Acids and Bases Strong Acids and Strong Bases Weak Acids and Weak Bases Ka and Kb Relationship between Ka and Kb pH, pOH, pKw, pKa, pKb Solutions of Weak Acids and Bases Salts and Salt Hydrolysis Buffer Systems Calculating the pH Values of Buffers Indicators Titrations Gases Pressure Measurement of Pressure Ideal Gas Laws Partial Pressures Solubility of Gases Diffusion in Gases Aliphatic Carbon Compounds Simple Molecules Containing Carbon Organic Compounds Alkanes and Alkyl Groups Alkenes Alcohols Thiols Aldehydes and Ketones Carboxylic Acids Amines Lipids, Sugars, and Linkages between Reactive Groups Fatty Acids Esters Glycerol Esters Hemiacetals and Hemiketals Simple Sugars Chirality in Simple Sugars Straight-Chain Sugars Spontaneously Form Rings Sugar Hydroxyls Can Be Chemically Modified Sugars Are Joined Together by Glycosidic Bonds Aromatic Carbon Compounds and Isomerism Benzene Bioactive Aromatic Compounds Isomerism Structural Isomerism Chain, Positional, and Functional Group Isomerism Tautomerism Stereoisomerism Geometrical Isomerism Optical Isomerism Organic and Biological Reaction Mechanisms Reactive Sites and Functional Groups Describing Reaction Mechanisms Bimolecular Nucleophilic Substitution Electrophilic Addition to a Nonpolar Double Bond Elimination to Form an Alkene Nucleophilic Addition to a Polar Double Bond Free Radical Reactions Carbon-Carbon Bond Formation in Biosynthesis Sulphur and Phosphorus The Electron-Shell Structure and Valency of Phosphorus and Sulphur Sulphur The Thiol Group and Thiol Esters Phosphate, Pyrophosphate, and Polyphosphate Phosphate Esters The Role of Phosphate Esters and ATP in Cellular Energy Metabolism Oxidation and Reduction Reactions Oxidation Is Linked to Reduction The Chemical Changes in the REDOX Process Splitting REDOX Reactions Standardising REDOX Half-Reactions Predicting Electron Flow Free Energy and Standard Reduction Potentials Redox Reactions and Nonstandard Conditions Metals in Biology General Properties of Metals in Biology Some Properties of Alkali Metals The Alkaline Earth Metals Transition Metals Role of Metal as Oxygen Carrier Metals Facilitate Biocatalysis The Role of Metal Ions as Charge Carriers The Toxicity of Metals Energy The First Law of Thermodynamics Units of Energy Measurement of Energy Internal Energy, U, and Enthalpy, H Calorimetry Hess's Law Enthalpies of Formation The Second Law of Thermodynamics Free Energy Interaction of DeltaH with TDeltaS Reactions and Equilibrium DeltaG and Equilibrium Activation Energy The Effect of the Temperature on Reaction Rate The Arrhenius Equation Catalysis Enzyme Catalysis Kinetics of Enzyme Reactions Finding Vmax and KM Light Light Is Part of the Electromagnetic Spectrum Wavelength and Frequency The Quantum Theory of Light The Absorption of Light The Relationship between Light Absorption and Concentration The Spectrophotometer The Fate of Absorbed Light Appendix: Derivations of Equations Index

The reaction kinetics of reductive pyrite (FeS 2 ) dissolution by H 2 is important to predict the long-term aqueous and gaseous sulfide release in underground hydrogen storage and other engineered subsurface sites. We investigated the rates of pyrite reduction based on sulfide formation as a function of temperature (60-150 °C), H 2 partial pressure (0-150 bar), and pH in situ (∼ 4-10) in deoxygenated 0.03 M NaCl solutions. The experiments used natural pyrite powder (ø = 50-100 µm) and were run in hydrothermal batch reactors made of either titanium or Dursan®-coated 316L stainless steel. After experimental durations of ∼ 700-900 h, dissolved S -II (aq) concentrations measured by methylene blue spectrophotometry ranged from 10 -5 to 10 -3 M. Long-term pyrite dissolution and concomitant elevated S -II (aq) concentrations were controlled by the formation of secondary pyrrhotite (Fe 1-x S) and magnetite (Fe 3 O 4 ), which prevented the ion activity product of FeS 2 from achieving rapid saturation. Conversely, in the absence of pyrrhotite and magnetite precipitation, the solutions rapidly equilibrated with respect to pyrite, resulting in low S -II (aq) concentrations. Rates of reductive pyrite dissolution were determined at 12 h from total S -II (aq+gas) concentration vs. time data and were found to increase with temperature and H 2 -partial pressure. The rate dependency on pH at 90 °C, 7 bar P H2 was regressed either with an asymmetric ‘V’-shaped (two rate-pH domains) or a ‘U’-shaped (three rate-pH domains) relation. For this reason, two separate kinetic rate laws were derived, based on the ‘U’-regression ( E a = 35.2 kJ mol -1 ): r = 10 - 5.42 e - 35208 / R T P H 2 0.37 1 - Q K eq or the ‘V’-regression ( E a = 29.4 kJ mol -1 ): r = 10 - 5.13 e - 29370 / R T a H + 0.13 P H 2 0.37 1 - Q K eq The performance of both rate laws was validated by reproducing the experimental S -II (aq) concentrations in kinetic models. This comparison determined their applicability to be valid from 60 to 120 °C, 0-150 bar P H2 , and pH ∼ 6.7-8.6 (‘U’ rate law) or pH ∼ 4.1-8.6 (‘V’ rate law). The rate laws are compatible with geochemical reactive transport codes and will enhance the understanding of geochemical fluid-rock-gas interactions in H 2 -bearing subsurface environments.

Overview: Introduction. Natural Waters as Nonequalibrium Systems. Scope of This Book. Rate Expressions for Chemical Reactions: Introduction. Rate Equations for Elementary Reactions. Rate Equations for More Complicated Reactions. Determination of Rate Equations and Rate Parameters from Experimental Data. Relationships between Rate Equations and Reaction Mechanisms. Experimental Aspects of Kinetic Studies. Theoretical Aspects of Kinetics and Effects of Physical Conditions on Reaction Rates: Introduction. The Arrhenquation and Related Temperature Relationships. Properties of Water and Reactants in Aqueous Solution. Encounter Theory for Reactions in Solution. Transition State/Activated Complex theory. Effects of Pressure on Rates of Reactions in Solution. Effects of Ionic Strength on Reaction Rates. Ranges of Values for Kinetic Parameters: Summary and perspective . Kinetics of Chemical Reactions in Aquatic Systems: From Homogeneous Catalysis to Reactions at Interfaces: Homogeneous Catalysis. Introduction. Types and Mechanisms of Homogeneous Catalysis. Rate Equations for Catalyzed Reactions. Kinetics of Chemical Reactions in Aqueous Solution: Kinetics of Dissociation and Hydrolysis Reactions in Natural Waters. Kinetics of Chlorination Reactions. Inorganic Redox Reactions. Processes at Interfaces. Gas Transfer Kinetics. Mineral Dissolution and Formation. Reactors, Mass Transport, and Process Models: Reactor Theory. Fundamentals of Chemical Reactors. Principles of Mass Balance Analysis. Comparative Behavior of CFSTRs and PFRs. Applications of Reactor Models to Natural Aquatic Systems. Equilibrium versus Kinetic Models for Open Systems. Nutrient Models. Models of Organic Contaminant Transport and Fate. CFSTR Models for Sulfate and Alkalinity on Lakes. Kinetics of Multi-Compartment Systems. Nature of Multi-Compartment Systems. Analysis of Compartment Models by Matrix Methods. The Inverse Problem. Kinetics of Biochemical Reactions and Microbial Processes in Natural Waters. Biochemical Kinetics: Enzyme Catalyzed Reactions. Reactions Involving a Single Substrate. Multisubstrate Reactions. Enzyme Inhibition. Theoretical Aspects of Enzyme Catalysis. Kinetics of Microbial Processes. Kinetics of Nutrient-Limited Microbial Growth. Continuous Culture of Microorganisms. Effects of Temperature on Microbial Processes. Kinetic Models of Heterotrophic Growth in Aquatic Systems. Kinetic Models for Autotrophic Growth and Nutrient Uptake. Diffusion-Limited Transport and Growth. Microbial Dieoff and Disinfection Kinetics. Prediction Methods for Reaction Rates and Compound Reactivity. Linear Free energy Relationships. Introduction. The Bronsted Relationship. Sigma Relationships for Organic Substituent Effects. LFERs for Redox Reactions. Applications of LFERs to Reactions in Natural Waters. Property-Activity and Structure-Activity Relationships in Environmental Biology and Chemistry. Nature and Importance of PARs and SARs. Physical-Chemical Measures of Compound Availability and Transportability. Linear Solvation Energy Relationships. A Simple Property-Activity Model for Biological Systems. Property-Activity Relationships for Aquatic Organisms. Quantitative Structure-Activity Relationships. Use of Statistical Techniques to Develop and Analyze PARs and SARs. Summary. Photochemical Reactions in Natural Waters: Introduction. Kinetics of Photochemical Reactions. Inorganic Photochemistry of Natural Waters. Nature and Photochemistry of Natural DOM. Photochemistry of Organic Contaminants in Natural Waters. Photochemistry on Suspended Oxide Particles. References. Index.

The identification of undergraduate chemistry students’ conceptual difficulties and common mistakes with basic concepts and problems in chemical kinetics provided the aim for this study, which involved 2nd-year/4th semester students who had passed the chemical kinetics component of a physical chemistry course. The study involved the analysis, evaluation and interpretation of students’ answers to the final examination in chemical kinetics. Three achievement groups, for the various topics, were identified: Group A, high achievement (mean ≈ 85%): (a) the steps in a chain-reaction mechanism, (b) integrated 1st- and 2nd-order rate laws; and (c) the Lindemann–Hinshelwood mechanism. Group B, intermediate achievement (mean ≈ 74%): (a) half-life, (b) instantaneous rate and the extent of reaction variable (ξ), (c) the Michaelis–Menten mechanism, and (d) theoretical rate law not asking for a final formula. Group C, low achievement (mean ≈ 54%): (a) experimental rate law and the reaction rate constant on the basis of an experimental-data table, (b) extracting the theoretical rate law, and (c) the Arrhenius equation. Students’ errors and misconceptions have also been identified. Successful students tended to respond well to straightforward questions on the theory of the subject, but had difficulties when solving problems. It is essential that teachers understand the potential of their students, especially possible misconceptions they may hold, and the teaching approaches that may contribute to overcoming the student difficulties. Problems in chemical kinetics can be very demanding both in terms of algebraic manipulations and conceptually. Teaching should focus on problem solving, with the emphasis on students themselves trying to solve the problems.

Stochastic simulation of chemical chaos

The kinetics of the alcoholysis-polycondensation of cottonseed oil (COSO), glycerol (GL), phthalic anhydride (PA), Maleic anhydride (MA) and Succinic acid (SA) leading to the formation of alkyd resins were investigated. Three grades of alkyd samples formulated to have oil content of 50% were prepared with COSO, GL, PA (I), MA (II) and SA (III) respectively. The extent of polycondensation was monitored by measuring the volume of water of esterification evolved, determining the pH and acid value of aliquots of the reaction mixture at various intervals of time. The extent of reaction (Pn) and number-average degree of polymerization (�� �� ) were calculated from end-group analysis of aliquots of reaction mixture withdrawn at various intervals of time. The rate of water evolution for COSO MA and COSO SA alkyds respectively were in the order of 10 -1 cm 3 /min, while that of COSO PA alkyd was in the order of 10 -2 cm 3 /min. The pH of the in-process samples increases with decreased in acid values. The initial reaction rates followed second order kinetics and thereafter deviations were observed. The �� �� calculated in the region of deviation from second order kinetics suggests the occurrence of chain branching at relatively short intervals along the polymer chain. The second order rate constants for all the alkyd samples were of the order 10 -4 g(mgKOH) -1 min -1 .

General rate equation theory for gas–solid reaction kinetics and its application to CaO carbonation

Autocatalytic reactions are in a certain contrast with the linear algebra of reaction stoichiometry, on whose basis rate equations respecting the permanence of atoms are constructed. These mathematical models of chemical reactions are termed conservative.Using a non-equilibrium thermodynamics-based theory of chemical kinetics, this paper demonstrates how to properly introduce an autocatalytic step into a (conservative) rate equation. Further, rate equations based on chemical potentials or affinities are derived, and conditions for the consistency of rate equations with entropic inequality (the second law of thermodynamics) are illustrated.

Hydrogenation of p-isobutyl acetophenone using a Ru/Al 2O 3 catalyst: reaction kinetics and modelling of a semi-batch slurry reactor

Catalyst performance of MoS 2 and WS 2 for the CO 2-reforming of CH 4 Suppression of carbon deposition

The Maximum Entropy Principle has been used to model complex chemical reaction processes. The maximum entropy principle has been employed by the Rate-Controlled Constrained-Equilibrium (RCCE) method to determine concentration of different species during non-equilibrium combustion process. In this model, it is assumed that the system evolves through constrained equilibrium states where entropy of the mixture is maximized subject to constraints. Mixture composition is determined by integrating set of differential equations of constraints rather than integration of differential equations for species as is done with detailed kinetics techniques. Since the number of constraints is much smaller than the number of species present, the number of rate equations required to describe the time evolution of the system is considerably reduced. This method has been used to model the stoichiometric mixture of the formaldehyde-oxygen combustion process. In this study 29 species and 139 reactions has been used, while keeping the energy and volume of the system constant. Calculations have been done at different sets of pressures and temperatures, ranging from 1 atm to 100 atm, and from 900 K to 1500 K respectively. Three fixed elemental constraints: conservation of elemental carbon, elemental oxygen and elemental hydrogen and from one to six variable constraints were used. The four to nine rate equations for the constraint potentials (Lagrange multipliers conjugate to the constraints) were integrated and as expected, RCCE calculations gave correct equilibrium values in all cases. Only 8 constraints were required to give very good agreement with detailed calculations. Ignition delay times and major species concentrations were within 0.5% to 5% of the values predicted by detailed chemistry calculations. Adding more constraints improved the accuracy of the mole fractions of minor species at early times, but had only a little effect on the ignition delay times. Rate-Controlled Constrained-Equilibrium calculations reduced the computation time by 50% when using eight constraints.

A comprehensive laboratory experimental study was conducted on the application of electroflotation in removal of paint from automotive paint system. In 40 min testing time and under 44 A/m2 current density, the total suspended solids (TSS) removal rates ranged from 90.39 to 97.43%. The first and second-order rate constants were developed. The kinetics of the TSS removal from the automotive paint wastewater was best described with the second-order rate constants. Two statistical methods were applied to identify the key factors for electroflotation performance among the Initial TSS Concentration, Current Density, water pH, Zeta Potential and Electrical Conductivity. It was confirmed that in both methods, the Initial TSS Concentration and Current Density were the most significant variables. Further, through Response Surface Methodology and Stepwise Regression analysis, equations of the removal rate of TSS vs reaction time and applied current density were produced for different wastewater samples.

The rate of iodination of acetone has been measured as a function of temperature in the binary solvent isobutyric acid (IBA) + water near the upper consolute point. The reaction mixture was prepared by the addition of acetone, iodine, and potassium iodide to IBA + water at its critical composition of 38.8 mass % IBA. The value of the critical temperature determined immediately after mixing was 25.43 degrees C. Aliquots were extracted from the mixture at regular intervals in order to follow the time course of the reaction. After dilution of the aliquot with water to quench the reaction, the concentration of triiodide ion was determined by the measurement of the optical density at a wavelength of 565 nm. These measurements showed that the kinetics were zeroth order. When at the end of 24 h the reaction had come to equilibrium, the critical temperature was determined again and found to be 24.83 degrees C. An Arrhenius plot of the temperature dependence of the observed rate constant, k(obs), was linear over the temperature range 27.00-38.00 degrees C, but between 25.43 and 27.00 degrees C, the values of k(obs) fell below the extrapolation of the Arrhenius line. This behavior is evidence in support of critical slowing down. Our experimental method and results are significant in three ways: (1) In contrast to in situ measurements of optical density, the determination of the optical density of diluted aliquots avoided any interference from critical opalescence. (2) The measured reaction rate exhibited critical slowing down. (3) The rate law was pseudo zeroth order both inside and outside the critical region, indicating that the reaction mechanism was unaffected by the presence of the critical point.