Chemical Kinetics

Chemical Kinetics - Introduction

Chemical kinetics, a topic in several chemistry courses, illustrates the connection between mathematics and chemistry. Chemical kinetics deals with chemistry experiments and interprets them in terms of a mathematical model. The experiments are perfomed on chemical reactions as they proceed with time. The models are differential equations for the rates at which reactants are consumed and products are produced. By combining models with experiments, chemists are able to understand how chemical reactions take place at the molecular level.

Composite Reactions or Reaction Mechanisms

Oscillating Chemical Reactions:

History of oscillating reactions.
Example chemical oscillator: Lotke-Volterra.
Example chemical oscillator: Brusselator.
Example chemical oscillator: Oregonator.

Thermogravimetric Analysis

TGA refers to the process of monitoring the mass of a sample while it is heated. The rising temperature causes chemical reactions to occur and may result in loss of mass. Background for TGA and illustrative examples are found by clicking here .

Glossary of Terms

• Stoichiometry determines the molar ratios of reactants and products in an overall chemical reaction. We express the stoichiometry as a balanced chemical equation. For kinetics it is convient to write this as products minus reactants: npP + nqQ - naA - nbB (instead of the conventional equation naA + nbB ---> npP + nqQ). This indicates that na and nb moles of reactants A and B, resp., produce np and nq moles of products P and Q.
• The rate of a chemical reaction is defined in such a way that it is independent of which reactant or product is monitored. We define the rate, v, of a reaction to be v = (1/ng) d[G]/dt where ng is the signed (positive for products, negative for reactants) stoichiometric coefficient of species G in the reaction. Namely, v = (-1/na) d[A]/dt = (1/np) d[P]/dt, etc.
• It is convenient to refer to the extent of reaction. As the reactants are sonsumed and the products are produced, their concentrations change. If the initial concentrations of A, B, P and Q are [A], [B], [P] and [Q], resp., then the extent of reaction is defined: x = -([A]-[A]0)/na = -([B] - [B]0)/nb = ([P]-[P]0)/np = ([Q]-[Q]0)/nq. Alternately, each species concentration is a function of the extent of reaction: [A] = [A]0 - nax, etc.
• Many reactions follow elementary differential rate laws such as v = k f([A], [B], ...) where f([A], [B], ...) is a function of the concentrations of reactants and products. That is, the rate varies as the concentrations change. A proportionality constant, k, is called the rate constant of the reaction.
• When the rate law has the special form of a product (or quotient) of powers, f([A], [B], ...) = [A]a [B]b [P]p [Q]q then a is the order of the reaction with respect to A, b is the order w.r.t. B, etc. Note that order may be positive, negative, integer, or non-integer. Further, the sum a + b + p + q is the overall order of the reaction rate law.
• NOTE: there is no necessary relation between orders and stoichiometric coefficients. That is, a might differ from na.
• Reaction rate constants are usually temperature dependent; the rate of a reaction usually increases as the temperature rises. The temperature dependence often follows Arrhenius' equation: k(T) = A exp(-Ea/RT) where T is the absolute temperature, R the universal gas constant, Ea is the activation energy (specific to each reaction), and A is the "pre-exponential" or "frequency" or "entropy" factor.
• One objective of chemical kinetics is to solve the differential rate law d[G]/dt = k f([A], [B], ...), and thereby express each species concentration as a function of time: [G](t). Since solution requires integration, we call it the integrated rate law.
• A reaction mechanism is a set of steps at the molecular level. Each step involves combinations or re-arrangements of individual molecular species. The steps in combination describe the path or route that reactant molecules follow to reach the product molecules. The result of all steps is to produce the overall balanced stoichiometric chemical equation for reactants producing products.

Extensive collection of physical chemistry problem solutions using MathCad can be found at the scicomp site.

References

• Physical Chemistry by K.J. Laidler & J.H. Meiser [Houghton Mifflin Co., 1995] Chapters 9 and 10.
• Physical Chemistry by P. Atkins [W.H. Freeman & Co., 1994] Chapters 25 and 26.
• More references will be found under subtopics in chemical kinetics.

CODEE, the Consortium for Differential Equations Experiments, has been revitalized. CODEE was quite active in the 1990s in spreading differential equations activities, information, and software tools. In particular, CODEE formed the organization for the ODE Architect software. Recently, an NSF project headed by Darryl Young of Harvey Mudd College has reinvigorated CODEE.
New activities! There are two new activities concerning the Idaho plan for wolf management, and a model for an insurgency. See the project page for details.

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This project is supported, in part, by the National Science Foundation. Opinions expressed are those of the authors, and not necessarily those of the Foundation.