Kinetics, chemical

Chemical reactions take time to occur. Some reactions, such as the rusting of iron or the changing of color in leaves, occur relatively slowly, requiring days, months, or years to complete. Others, such as the combustion reaction that generates the thrust for a rocket, as in the chapter-opening photograph, happen much more rapidly. As chemists, we need to be concerned about the speed with which chemical reactions occur as well as the products of the reactions. For example. 

So far, we have focused on the beginning and end of chemical reactions We start with certain reactants and see what products they yield. This view is useful but does not tell us what happens in the middle—that is, which chemical bonds are broken, which are formed, and in what order these events occur. The speed at which a chemical reaction occurs is called the reaction rate. Reaction rates can occur over very different time 

Cape Canaveral, Florida, in August 2011. The spacecraft is mounted to an Atlas V rocket, which at launch uses the very rapid combustion of kerosene and liquid oxygen fo generafe ifs fhrust. 

1 FACTORS THAT AFFECT REACTION RATES We see that four variables affect reaction rates concentration, physical states of reactants, temperature, and presence of catalysts. These factors can be understood in terms of the collisions among reactant molecules that lead to reaction. 

2 REACTION RATES We examine how to express reaction rates and how reactant disappearance rates and product appearance rates are related to the reaction stoichiometry. 

Physicochemical systems with coupled processes having different length scales can exhibit stationary spatially periodic structures. These arise from symmetry-breaking instabilities. 

Louis Pasteur, bom Dec. 27, 1822, in Dole, France, died Sep. 28, 1895, in Saint-Qoud, near Paris. 

A common example is the Belousov - Zhabotinsky reaction [24], Beautiful patterns of chemical wave propagation can be created in a chemical reaction - diffusion system with a spatiotemporal feedback. The wave behavior can be controlled by feedback-regulated excitability gradients that guide propagation in the specified directions [25, 26]. 

Lolka came to the United States in 1902 and wrote a number of theoretical articles on chemical oscillations. In his paper dated 1910 [27] he concludes 

Eine Reaktion, welche diesem Gesetze folgte, scheint zurzeit mcht bekannt zu sein. Der 

Chemical kinetics is the study of reaction mechanisms and rates. As of yet, there are no unifying principles of kinetics, which means kinetics is a complicated field with many opposing theories as to how reactions proceed. Additionally, the mathematics of kinetics is complicated and well beyond the scope of MCAT. MCAT will address kinetics only in its simplest form. Keep in mind that kinetics deals with the rate of a reaction typically as it moves toward equilibrium, while thermodynamics deals with the balance of reactants and products after they have achieved equilibrium. Kinetics tells us how fast equilibrium is achieved, while thermodynamics tells us what equilibrium looks like. The two disciplines are intricately related, but they should not be confused. 

Chemical kinetics is the branch of chemistry concerned with the rates of chemical reactions [3, 14, 19, 36-41]. Many chemical reactions involve the formation of unstable intermediate species (e.g., free radicals). Chemical kinetics is the study of the mechanisms involved in obtaining a rate expression for the chemical reaction (the reaction pathway). In most instances, the reaction rate expression is not available and should be determined experimentally. Chapter 3 covers the definitions and relations used in reactor analysis and design. 

Chemical kinetics may be considered to be the macroscopic version of chemical dynamics. Dynamics is concerned with determining the details of 

Fourier analysis are not as useful. This is where wavelet analysis can make an impact. 

The de-noising feature can also be applied to traditional linear kinetics. This was demonstrated by Fang and Chen for voltammetry [63]. Despite its use 

Chemical Kinetics Is the Study of the Rates at Which Chemical Reactions Occur 713 

The Rate Law Gives the Dependence of the Reaction Rate on the Reactant Concentrations 720 

Integrated Rate Laws Specify the Relationship Between Reactant Concentration and Time 723 

The Arrhenius Equation Gives the Temperature Dependence of Rate Constants 736 

The Reaction Mechanism Is the Sequence of Elementary Steps That Lead to Product Formation 744 

Consider the simple first-order chemical reaction, A — B. The corresponding kinetic equation, 

Let Pin, Z) be the probability that the number of A molecules in the system at time Z is n. We can derive a master equation for this probability by following a procedure similar to that used in Section 7.3.1 to derive Eq. (7.3) or (8.69)  

Unlike in the random walk problem, the transition rate out of a given state n depends on n The probability per unit time to go from n+1 to n is A (/j+1), and the probability per unit time to go from n to n — 1 is kn. The process described by Eq. (8.83) is an example of a birth-and-death process. In this particular example there is no source feeding molecules into the system, so only death steps take place. 

For detailed discussion and more examples see D. A. McQuarrie,. 4 Stochastic Approach to Chemical Kinetics, J. Appl. Probability 4, 413 (1967). 

Problem 8.8. How should Eq. (8.83) be modified if molecules J are inserted into the system with the characteristic constant insertion rate ka (i.e. the probability that a molecule A is inserted during a small time interval At is kal8l) l 

The effect of chemical kinetics on mass transport in incompressible flows is summarized by the reaction term r in Eq. (89). Applied to a chemical species a, it describes the rate of disappearance of this species per unit volume  

The reaction rate determines how fast the concentration of a chemical species a increases or decreases due to chemical reactions. It depends on temperature and on the concentrations of other chemical species involved in the reaction. Consider the case of a simple reaction  

Chemists seek to understand the details of chemical reactions  

These topics are the subject of that branch of chemistry called chemical kinetics. 

The reversible kinetic rate law for nth-order chemical reaction is 

If the reactor is well stirred, then the molar densities of reactant A and product B in the kinetic rate law are expressed in terms of conversion / via stoichiometry and the steady-state mass balance with convection and chemical reaction  

If one adopts a plug-flow thermal energy balance on the reactive fluid within a differential CSTR at high-heat-transfer Peclet numbers, then equation (3-37) yields  

If the kinetic rate law E is not a function of position throughout the well-mixed reactor, and (il conduction = - 2input is the differential rate of thermal energy 

The rate of heat generation G (T) due to exothermic chemical reaction in the CSTR, with units of calories per second, is 

Many environmental fate processes, such as the degradation of pollutant chemicals, are not usefully modeled as equilibrium chemistry problems because the rate of the reaction is more important to quantify than the final composition of the system. For example, even though it may be known that at equilibrium a certain chemical will be fully degraded, it is crucial to know whether degradation will take seconds, years, or perhaps centuries. 

The rate at which a chemical reaction occurs may be limited by the frequency of collisions between the reacting atoms or molecules, as well as the 

Such a chemical reaction, in which molecules are not colliding with other atoms or molecules, is called a first-order reaction because the rate at which chemical concentration changes at any instant in time is proportional to the concentration raised to the first power. Certain chemical processes, such as radioactive decay, are described by first-order kinetics. In the absence of any other sources of the chemical, first-order kinetics may lead to exponential decay or first-order decay of the chemical concentration (i.e., the concentration of the parent compound decreases exponentially with time)  

Note that the integration of Eq. [1-18] leads to Eq. [1-19], From Eq. [1-19] can be derived an expression, known as the half-life (t1/2), which represents the amount of time it takes for the parent compound to decay to half its initial concentration  

Note that half-lives convey exactly the same information as first-order decay rate constants, but may be more intuitive to some users. Half-lives for different decay processes may be easily compared to determine which decay mechanism is the most significant they may also be compared with a transport 

Because the general principles of chemical kinetics apply to enzyme-catalyzed reactions, a brief discussion of basic chemical kinetics is useful at this point. Chemical reactions may be classified on the basis of the number of molecules that react to form the products. Monomolecular, bimolecular, and termolecular reactions are reactions involving one, two, or three molecules, respectively. 

Reactions may also be classified on a kinetic basis by reaction order, which may be zeroth-order, first-order, second-order, third-order, or pseudo-first-order, contingent on how the reaction rate is affected by the concentration of the reactants. The rate equation for the reaction A — P may be written as 

In first-order reactions, tv2 is independent of the initial concentration of reactant Aq. It should be noted that tV2 is not one-half the time required for the reaction to 

For the reaction A + A — P, we can write —dAJdt = k(A)(A) = k A2. In this case, the rate involves a higher power of the concentration (n = 1 + 1 = 2) and the reaction is second-order. For the reaction A -I- B — P, the rate is proportional to the first power of each reactant and —d(A)(B)/dt = k(A)(B). The reaction is first-order with respect to A or B, but the overall reaction is second-order, because the right-hand side of the equation contains the product of two concentrations. The value of tV2 will depend on the initial reactant concentration(s), and the second-order rate constants have the dimensions of reciprocal concentration times time. Reactions of higher order, such as third order, are relatively rare, and their rates are proportional to the product of three concentration terms. 

Whenever a reactant is present in large excess, its concentration is virtually constant during the course of the reaction. Thus, in a second-order reaction A + B — P in which the concentration of B is very high and that of A is low, the reaction may appear to be first-order, because its rate will be nearly proportional to the concentration of A. This is an apparent or pseudo-first-order reaction. Pseudo-first-order reactions are common among biochemical reactions in which water is one of the reactants. Since the concentration of water is 55.5 M and far in excess of everything else, the reaction appears to behave like a first-order reaction. An example is the hydrolysis of an ester, 

In reaction kinetics, the object of interest is the progress of a chemical reaction measured in terms of the change in concentration of the corresponding reactants and products. The change of a component s amount in time is defined as the reaction rate of this component. Assuming constant volume this equals the concentration of this component. The reaction rate r, of component j can be expressed as the change in concentration Cj over time  

The change in concentration typically depends on the reaction time, the concentration of the other components, the temperature, and reaction specific properties. Temperature 

If one molecule of Ai is split into the products, it follows Ui = -1. For the change in concentration of the reactant follows  

In reality, most chemical reactions consist of numerous elementary reactions combined in reaction networks that are much more difficult to describe. An interesting elementary reaction is a 2-step sequential reaction which is relevant e.g. in modelling thermal separation processes  

This simple system of differential equations can be solved iteratively. The solution of (2.9) is given in (2.8) and, hence, can be substituted in (2.10)  

Nobody, I suppose, could devote many years to the study of chemical kinetics without being deeply conscious of the fascination of time and change this is something that goes outside science into poetry.. . .  

The children who live in my neighborhood (including my own kids) have a unique way of catching lizards. Armed with cups of ice water, they chase one of these cold-blooded reptiles into a comer, and then take aim and pour the cold water directly onto the lizard s body. The lizard s body temperature drops and it becomes virtually immobilized—easy prey for little hands. The kids scoop up the lizard and place it in a tub filled with sand and leaves. They then watch as the lizard warms back up and becomes active again. They usually release the lizard back into the yard within hours. I guess you could call them catch-and-release lizard hunters. 

Unlike mammals, which actively regulate their body temperature through metabolic activity, lizards are ectotherms—their body temperature depends on their surroundings. When splashed with cold water, a lizard s body simply gets colder. The drop in body temperature immobilizes the lizard because its movement depends on chemical reactions that occur within its muscles, and the rates of those reactions— how fast they occur—are highly sensitive to temperature. In other words, when the temperature drops, the reactions that produce movement in the lizard occur more slowly therefore, the movement itself 

Pouring ice water on a lizard slows it down, making it easier to catch. 

The rates of chemical reactions, and especially the ability to control those rates, are important not just in reptile movanent but in many other phenomena as well. For example, a successful rocket launch depends on the rate at which fuel bums—too quickly and the rocket can explode, too slowly and it will not leave the ground. Chemists must always consider reaction rates when synthesizing compounds. No matter how stable a compound might be, its synthesis is impossible if the rate at which it forms is too slow. As we have seen with reptiles, reaction rates are important to life. In fact, the human body s ability to switch a specific reaction on or off at a specific time is achieved largely by controlling the rate of that reaction through the use of enzymes (biological molecules that we explore more fully in Section 13.7). 

The rate at which a reaction proceeds is governed by the principles of chemical kinetics, which is one of the major topics of this book. Chenucal kinetics allows us to understand how reaction rates depend on variables such as concentration, temperature, and pressure. Kinetics provides a basis for manipulating these variables to increase the rate of a desired reaction, and minimize the rates of undesired reactions. We will study kinetics first from a rather empirical standpoint, and lat from a more fundamental point of view, one that creates a link with the details of the reaction chenustry. Catalysis is an extremely important tool within the domain of chemical kinetics. For example, catalysts are required to ensure that blood clots form fast enough to fight serious blood loss. Approximately 90% of the chemical processes that are carried out industrially involve the use of some kind of catalyst in order to increase the rate(s) of the desired reaction(s). Unfortunately, the behavior of heterogeneous catalysts can be significantly and negatively influenced by the rates of heat and mass transfer to and from the sites in the catalyst whrae the reaction occurs. We will approach the interactions between catalytic kinetics and heat and mass transport conceptually and qualitatively at first, and then take them head-on later in the book. 

LEARNING OBJECTIVES 20-2 20-3 Measuring Reaction Rates Effect of Concentration on Reaction 20-8 Theoretical Models for Chemical Kinetics 

1 Express the rate of reaction in terms Rates The Rate Law 20-9 The Effect of Temperature on 

2 Expiain how a graph of concentration 20-6 Second-Order Reactions 20-11 Catalysis  

3 Describe how the method of initiai rates can be used to determine the rate iaw for a reaction. 

4 Use the rate iaw for a zero-order reaction, together with experimentai data, to obtain the rate constant for a zero-order reaction. 

As described in the first part of this chapter, chemical thermodynamics can be used to predict whether a reaction will proceed spontaneously. However, thermodynamics does not provide any insight into how fast this reaction will proceed. This is an important consideration since time scales for spontaneous reactions can vary from nanoseconds to years. Chemical kinetics provides information on reaction rates that thermodynamics cannot. Used in concert, thermodynamics and kinetics can provide valuable insight into the chemical reactions involved in global biogeochemical cycles. 

In chemical kinetics an empirical relationship is used to relate the overall rate of a process to the concentrations of various reactants. A common form for this expression is 

This equation is known as the rate law for the reaction. The concentration of a reactant is described by A cL4/df is the rate of change of A. The units of the rate constant, represented by k, depend on the units of the concentrations and on the values of m, n, and p. The parameters m, n, and p represent the order of the reaction with respect to A, B, and C, respectively. The exponents do not have to be integers in an empirical rate law. The order of the overall reaction is the sum of the exponents (m, n, and p) in the rate law. For non-reversible first-order reactions the scale time, tau, which was introduced in Chapter 4, is simply 1 /k. The scale time for second-and third-order reactions is a bit more difficult to assess in general terms because, among other reasons, it depends on what reactant is considered. 

It is important to stress that the empirical rate law must be determined experimentally, with 

The temperature dependence of a rate is often described by the temperature dependence of the rate constant, k. This dependence is often represented by the Arrhenius equation, /c = Aexp(- a/i T). For some reactions, the temperature relationship is instead written fc = AT exp(- a/RT). The A term is the frequency factor for the reaction, which reflects the number of effective collisions producing a reaction. a is known as the activation energy for the reaction, and is a measure of the amount of energy input required to start a reaction (see also Benson, 1960 Moore and Pearson, 1981). 

A chemical reaction hardly ever takes place as the stoichiometric equation appears to indicate. Both the equation and the free energy considerations (p. 171) are concerned only with the initial and final states of the system, although the overall reaction to which they refer may proceed by a series of related steps that cannot be investigated separately and are inferred from reaction-rate studies. The term kinetics implies the experimental and theoretical study of the way reactions take place and the rate at which they proceed. Here the subject is reviewed only very briefly, though it is of considerable importance in inorganic chemistry. 

The study of reaction rates presents difficulties not encountered in investigations concerned only with the original and final states of a chemical system. The progress of a reaction can be followed by (a) physical methods such as the observation of changes in electrical conductance, colour, volume, ultra-violet absorption or optical activity, or the measurement of the gas evolved, (b) chemical methods leading to the determination of reactants and products, (c) radiochemical methods in which the transfer of radioactive material is observed. 

Until recently little work has been done on the rates of ionic reactions in solution, principally because these are usually so fast as to make measurement difficult. Working with metal ions in non-aqueous solutions at very low temperatures, Bjerrum and Poulsen (1952) found, for example, that NP+ reacted at a rate which was measurable with dimethylglyoxime in methanol at -75°. Awtrey and Connick (1951) applied Hartridge and Roughton s dynamic flow method (1923) to the reaction between SOg - and I3- in aqueous solution. Bell and Clunie (1952) have developed a thermal method for studying reactions occurring in a few seconds. 

When substances react in the gaseous phase or in solution their concentrations fall and the reaction rate decreases. Experiment shows that the reaction rate is jointly proportional to various powers of the concentrations of the individual reactants. In the reaction 

Zero order reactions also occur, the rate being independent of the measured concentrations of reactants. This often indicates an intermediate step that almost wholly determines the overall rate, in which an intermediary species is present in small and sensibly constant concentration. 

The type of kinetic model to be used depends on the type of reaction considered. For a homogeneous reaction occurring in the bulk of the fluid, a power-law kinetic model is often appropriate (see, e.g., [79]). In such models the rate of a certain reaction depends on a product of powers of the species concentration. On the other hand, heterogeneously catalyzed reactions are often conducted in microreactors. In a strict sense, power-law kinetics does not capture the dynamics of such processes over the full range of pressure, temperature and concentrations. Rather, a more complicated kinetic model of, e.g., Langmuir-Hinshelwood type [80] would have to be used. Nevertheless, power-law kinetics is frequently applied to heterogeneously catalyzed processes in a limited parameter range to simplify the description. 

Independent of the specific modeling strategy, the kinetic equations often exhibit a nonlinear dependence on the species concentrations and a reaction rate rapidly increasing with temperature. In combination with the transport equations for mass, momentum and heat, the resulting numerical problem is usually challenging due to the nonlinearities and the multitude of time scales involved. For this reason, methods are needed to eliminate some of these difficulties and to simplify the numerical structure. 

The recommendations given here are based on previous IUPAC recommendations [l.c,k and 27], which are not in complete agreement. Recommendations regarding photochemistry are given in [28] and for recommendations on reporting of chemical kinetics data see also [69]. 

It is thus recommended to specify in any given context exactly which activation energy is meant and to reserve (Arrhenius) activation energy only and exactly for the quantity defined in the table. 

Our previous discussion of chemical equilibria and chemical thermodynamics allows us to assess whether or not a chemical reaction will proceed in a certain direction, and what the concentrations of the reactants and products will be when a system is in chemical equilibrium. In this chapter we are concerned with how fast reactants are converted into products, some of the factors upon which the rate of conversion depends, and the sequence of steps by which the conversion occurs. These subjects are the province of chemical kinetics. 

General Rate is the change in a property divided by the time required for the change to occur. 

The average reaction rate may be determined for any reactant or product. 

Rates for different reactants and products may have different numerical values. 

For a general chemical reaction, a A + 6B — cC + dD, the unique average reaction rate is the average reaction rate of reactant or product divided by its stoichiometric coefficient used as a pure 

Laboratory studies are very important for providing basic knowledge to scale-up of batch reactions. Modeling a batch system is very important as well. When the batch reaction and system are well understood, a large scale-up factor may be applied while still maintaining safe operations. 

Phenomena Phase Phase Fixed Fixed Bed Moving Bed 

1 = critical/very important 2 = necessary/important 3 = desirable/some importance 4 = little value/unimportant  

For a gas-liquid reaction which is gas-phase controlling, the chemical kinetics must be well understood. The importance of laboratory studies must therefore be emphasized. However, for successful scale-up, pilot plant studies are very critical because of the difficulties in reliably modeling gas behavior on a small scale (due to hydrodynamics) and its influence on reaction rates. 

Scale-up equations for liquid, gas-liquid, and solid-liquid systems are detailed in [167,199,201,206]. 

In previous chapters, we considered questions like How much energy does a reaction liberate or consume and In which direction will a reaction proceed We then asked questions like To what extent will a reaction proceed in that direction, before it stops and even Why do reactions occur at all In this chapter, we look at a different question How fast does a reaction proceed Straightaway, we make assumptions. Firstly, we need to know whether the reaction under study can occur there is no point in looking at how fast it is not going if a reaction is not thermodynamically feasible So we first assume the reaction can and does occur. 

Secondly, we assume that reactions can be treated according to their type, so reaction order is introduced and discussed in terms of the way in which concentrations vary with time in a manner that characterizes that order. 

Finally, the associated energy changes of reaction are discussed in terms of the thermodynamic laws learnt from previous chapters. Catalysis is discussed briefly from within this latter context. 

An equivalency can be demonstrated between the concept of time constant ratios and the new dimensionless parameters as they appear in the model equations. The concept of time constants is discussed in Section 2.2. 

Thus the variables in this example can be interpreted as follows 

1 Transfer time constant Convection rate Ki t Residence time constant Transfer rate 

IcCbo Transfer time constant Reaction rate 

Further examples of the use of dimensionless terms in dynamic modelling applications are given in Sections 1.2.5.1, 4.3.6.1 and 4.3.7 and in the simulation examples KLADYN, DISRET, DISRE, TANKD and TUBED. 

In a molar-scale diatomic reaction mixture (i = kL) the number of reactants in the valence state, at a given temperature, is given by an equation such as (7.6) with tq = (2Vq)eff = Ea, called the activation energy. The mole 

In the case of a complex reaction the rate constant is the product of rate constants for several elementary steps. The progress of such reactions is usually presented along a reaction coordinate - a hypothetical parameter that depends on all internuclear distances of relevance to the mechanism whereby reactants are converted into products. 

All theories of chemical kinetics and reaction mechanisms are based on eqn. (7.8) by an estimation of the relevant partition function. 

This happens in the early spring in the Antarctic or in the months of September and October. The Cl atoms then catalytically destroy ozone, resulting in the rapid loss of ozone that we call the ozone hole. As time goes by, the Antarctic stratosphere warms up, the ice crystals melt, Cl starts to be inactivated, and the ozone hole heals. 

Standard Free Energy of Hydrolysis (AG ) of Some Organophosphates 

The study of chemical reaction rates is called chemical kinetics. Whereas thermodynamics deals with the relative energy states of reactants and products, kinetics deals with how fast a reaction occurs and with the chemical pathway (mechanism) it follow s. 

The instantaneous velocity, v, at some time t ( to) is expressed as follows  

Alternatively, one can consider the change in concentration of the reactants. Thus, 

The rate constant expresses the proportionality between the rate of formation of B and the molar concentration of A and is characteristic of a particular reaction. The units of k depend on the order of the reaction. For zero order, they are moles liter time (time is frequently given in seconds). For first order, they are time , and for second order, liters moles time , etc. The units of k are whatever is needed for i[B]/ it to have the unit of moles liters time . If = 0, v = fc this means that in a zero order reaction, [B] changes at a constant rate independent of the concentration of reactants, which is especially important in enzyme kinetics. A plot of [B] versus t for such a reaction is a straight line. A somewhat more complicated example is 

11 From the units of the rate constant, k, it follows that the reaction is first order, thus rate= [NjOj]. 

15 Because the rate increased in direct proportion to the concentrations of both reactants, the rate is first order in both reactants. 

17 (a) Use experiments 1 and 4 to show that [C] is independent of the rate. Use experiments 2 and 4 to solve for the order with respect to A  

The rate constant is then determined from the first-order integrated rate law. 

if the reaction is second order, a plot of 1/[HI] against time should give a straight line of slope k.As can be seen from the graph, the data fit the equation for a second-order reaction quite well. The slope is determined by a least squares fit of the data by the graphing program. 

PURPOSE OF EXPERIMENT Determine the orders of reagents and a specific rate constant by measuring the rate of oxygen evolution from the iodide-catalyzed decomposition of hydrogen peroxide, H2O2. 

The rate of a chemical reaction can be determined by following the rate at which one of the products is formed or the rate at which one of the reactants is consumed. The rate of a given reaction depends on the concentration of reagents and the temperature. The specific dependence of the rate of reaction on the concentration of the reagents is summarized by the rate law, which has the general form 

In this experiment you will study the rate of decomposition of hydrogen peroxide catalyzed by iodide ion, I , or really by 13 produced at the very beginning of the reaction by the oxidation of I by H2O2. The following equation summarizes the reaction 

You will follow the rate of reaction by monitoring the rate of oxygen evolution, and investigate how changes in the concentration of H2O2 and I affect the rate of oxygen evolution. You will summarize your kinetic data with an appropriate rate law. Your problem is to determine the numerical values of the specific rate constant, k, and exponents of the concentration of reagent terms, m and n. 

Initial rates will have been determined from three experiments using molar concentrations of H2O2 and I which can be calculated. Then two pairs of simultaneous equations that have identical terms for the molar concentration of either H2O2 or 1 can be written. The exponents, m and n, can be calculated by solving these two pairs of simultaneous equations. Finally, the specific rate constant, k, can be determined after substituting m and n and appropriate reactant concentrations into the original rate expression. For example, suppose that the initial rates shown in TABLE 23.1 are determined for the general reaction 

The quantity disappears in this subtraction, as do other composition-inde- 

LePree tested eq. [5.5.49] with the decarboxylative dechlorination ofN-chloroamino 

The success shown by this kinetic study of a unimolecular reaction unaccompanied by complications arising from solvent effects on pH or water concentration (as a reactant) means that one can be confident in applying the theory to more complicated systems. Of course, an analysis must be carried out for such systems, deriving the appropriate functions and making chemically reasonable approximations. One of the goals is to achieve a practical level of predictive ability, as for example we have reached in dealing with solvent effects on solubility. 

The quantity AG (g), disappears in this subtraction, as do other composition-independent quantities. We make use of eq. [5.5.13] and eq. [5.5.19] to obtain a function having six parameters, namely Kj , K2 , gA, and gA. This function is made manageable by 

A hot platinum wire glows when held over a concentrated ammonia solution. The oxidation of ammonia to produce nitric oxide and water, catalyzed by platinum, is highly exothermic. This reaction is the first step toward the synthesis of nitric acid. The models show ammonia, oxygen, nitric oxide, and water molecules. 

In previous chapters, we studied basic definitions in chemistry, and we examined the properties of gases, liquids, solids, and solutions. We have discussed molecular properties and looked at several types of reactions in some detail. In this chapter and in subsequent chapters, we will look more closely at the relationships and the laws that govern chemical reactions. 

How can we predict whether or not a reaction will take place Once started, how fast does the reaction proceed How far will the reaction go before it stops The laws of thermodynamics (to be discussed in Chapter 18) help us answer the first question. Chemical kinetics, the subject of this chapter, provides answers to the question about the speed of a reaction. The last question is one of many answered by the study of chemical equihbrium, which we will consider in Chapters 14, 15, and 16. 

Chemical kinetics is the area of chemistry concerned with the speeds, or rates, at which a chemical reaction occurs. The word kinetic suggests movement or change in Chapter 5 we defined kinetic energy as the energy available because of the motion of an object. Here kinetics refers to the rate of a reaction, or the reaction rate, which is the change in the concentration of a reactant or a product with time (M/s). 

16-1 The Rate ofa Reaction Factors That Affect Reaction Rates 16-2 Nature of the Reactants 16-3 Concentrations of Reactants  

Time The Integrated Rate Equation 16-5 Collision Theory of Reaction Rates 16-6 Transition State Theory 16-7 Reaction Mechanisms and the Rate-Law Expression 

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Although methanol itself is not particularly harmful, accidental or intentional ingestion of methanol can cause headache, nausea, blindness, seizures, and even death. In the liver, methanol is metabolized by the enzyme alcohol dehydrogenase (ADH) to yield formaldehyde  

The formaldehyde is subsequently converted to formic acid by another enzyme, aldehyde dehydrogenase (ALDH). 

Formic acid is the species responsible for the toxic effects of methanol poisoning— including metabolic acidosis, in which the blood becomes dangerously acidic. 

In 2000, the FDA approved the drug fomepizole, marketed under the name Antizol, for the treatment of methanol poisoning. Fomepizole (C4H6N2) has an affinity for ADH approximately 8000 times that of methanol and is used to treat methanol toxicity without the intoxication and CNS depression caused by ethanol. Dialysis is still used to remove methanol from the blood. 

Understanding chemical kinetics can make it possible to minimize the damage done by unde.sirable reactions. It also enables us to enhance the speed of desirable reactions. 

The same (has the same munerical value) for any reactant or product in a given reaction For the reaction above, the imique average reaction rate is defined as  

The minus signs for the terms involving reactants are required because the concentrations of reactants decrease as the reaction time increases. 

Note Two critical underlying assumptions for generating a unique reaction rate are that the overall reaction time is slow with respect to the buildup and decay of any intermediate and that the stoichiometry of the reaction is maintained throughout. 

Most reactions slow down as they proceed and reactants are depleted. 

Methanol is an ingredient in gas-line antifreeze. When ingested, it is converted to formic acid, the substance responsible for methanol s toxic and potentially deadly effects. 

The three reactions arc first order in their respective reactants and products. Concentration units are mole fractions. 

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Rate effects may not be chemical kinetic ones. Benson and co-worker [84], in a study of the rate of adsorption of water on lyophilized proteins, comment that the empirical rates of adsorption were very markedly complicated by the fact that the samples were appreciably heated by the heat evolved on adsorption. In fact, it appeared that the actual adsorption rates were very fast and that the time dependence of the adsorbate pressure above the adsorbent was simply due to the time variation of the temperature of the sample as it cooled after the initial heating when adsorbate was first introduced. 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

The foundations of the modem tireory of elementary gas-phase reactions lie in the time-dependent molecular quantum dynamics and molecular scattering theory, which provides the link between time-dependent quantum dynamics and chemical kinetics (see also chapter A3.11). A brief outline of the steps hr the development is as follows [27],... 

Lam S H and Goussis D A 1988 Understanding complex chemical kinetics with computational singular perturbation 22nd Int. Symp. on Combustion ed M C Salamony (Pittsburgh, PA The Combustion Institute) pp 931-41... 

Maas U and Pope S B 1992 Simplifying chemical kinetics intrinsic low-dimensional manifolds in composition space Comb. Flame 88 239... 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

As these examples have demonstrated, in particular for fast reactions, chemical kinetics can only be appropriately described if one takes into account dynamic effects, though in practice it may prove extremely difficult to separate and identify different phenomena. It seems that more experiments under systematically controlled variation of solvent enviromnent parameters are needed, in conjunction with numerical simulations that as closely as possible mimic the experimental conditions to improve our understanding of condensed-phase reaction kmetics. The theoretical tools that are available to do so are covered in more depth in other chapters of this encyclopedia and also in comprehensive reviews [6, 118. 119],... 

Larson R S and Kostin M D 1982 Kramers theory of chemical kinetics curvilinear reaction coordinates J. Chem. Phys. 77 5017-25... 

As a final point, it should again be emphasized that many of the quantities that are measured experimentally, such as relaxation rates, coherences and time-dependent spectral features, are complementary to the thennal rate constant. Their infomiation content in temis of the underlying microscopic interactions may only be indirectly related to the value of the rate constant. A better theoretical link is clearly needed between experimentally measured properties and the connnon set of microscopic interactions, if any, that also affect the more traditional solution phase chemical kinetics. 

A3.12 Statistical mechanical description of chemical kinetics RRKM... 

Steinfeld J I, Francisco J S and Hase W L 1999 Chemical Kinetics and Dynamics 2nd edn (Upper Saddle River, NJ Prentice-Hall)... 

Montroll E W and Shuler K E 1958 The application of the theory of stochastic processes to chemical kinetics Adv. Chem. Phys. 1 361-99... 

Scott S K 1994 Oscillations, Waves and Chaos in Chemical Kinetics (Oxford Oxford University Press) A short, final-year undergraduate level introduction to the subject. 

Unwin P R and Compton R G 1989 Comprehensive Chemical Kinetics vol 29, ed R G Compton (Lausanne Elsevier)... 

Montenegro M I 1994 Research in Chemical Kinetics vol 2, ed R G Compton and G Flancock (Amsterdam Elsevier)... 

Carrington T and Polanyi J C 1972 Chemiluminescent reactions Chemical Kinetics, Int. Rev. Sc/. Physical Chemistry senes 1, vol 9, ed J C Polanyi (London ButtenA/orths) pp 135-71... 

The key to experimental gas-phase kinetics arises from the measurement of time, concentration, and temperature. Chemical kinetics is closely linked to time-dependent observation of concentration or amount of substance. Temperature is the most important single statistical parameter influencing the rates of chemical reactions (see chapter A3.4 for definitions and fiindamentals). 

The approach is ideally suited to the study of IVR on fast timescales, which is the most important primary process in imimolecular reactions. The application of high-resolution rovibrational overtone spectroscopy to this problem has been extensively demonstrated. Effective Hamiltonian analyses alone are insufficient, as has been demonstrated by explicit quantum dynamical models based on ab initio theory [95]. The fast IVR characteristic of the CH cliromophore in various molecular environments is probably the most comprehensively studied example of the kind [96] (see chapter A3.13). The importance of this question to chemical kinetics can perhaps best be illustrated with the following examples. The atom recombination reaction... 

Figure C3.1.8. Schematic diagram of a transient kinetic apparatus using iaser-induced fluorescence (LIF) as a probe and a CO2 iaser as a pump source. (From Steinfeid J I, Francisco J S and Fiase W L i989 Chemical Kinetics and. Dynamics (Engiewood Ciiffs, NJ Prentice-Fiaii).)...

Oref I 1995 Superoollisions Advances in Chemical Kinetics and Dynamics vol 2B, ed J Barker (Greenwioh, CT JAI Press) pp 285-98... 

Complex chemical mechanisms are written as sequences of elementary steps satisfying detailed balance where tire forward and reverse reaction rates are equal at equilibrium. The laws of mass action kinetics are applied to each reaction step to write tire overall rate law for tire reaction. The fonn of chemical kinetic rate laws constmcted in tliis manner ensures tliat tire system will relax to a unique equilibrium state which can be characterized using tire laws of tliennodynamics. 

In chemical kinetics, it is often important to know the proportion of particles with a velocity that exceeds a selected velocity v. According to collision theories of chemical kinetics, particles with a speed in excess of v are energetic enough to react and those with a speed less than v are not. The probability of finding a particle with a speed from 0 to v is the integral of the distribution function over that interval... 

G. D. Billing, K. V. Mikkelsen, Advanced Molecular Dynamics and Chemical Kinetics John Wiley Sons, New York (1997). 

Although the Arrhenius equation does not predict rate constants without parameters obtained from another source, it does predict the temperature dependence of reaction rates. The Arrhenius parameters are often obtained from experimental kinetics results since these are an easy way to compare reaction kinetics. The Arrhenius equation is also often used to describe chemical kinetics in computational fluid dynamics programs for the purposes of designing chemical manufacturing equipment, such as flow reactors. Many computational predictions are based on computing the Arrhenius parameters. 

There are many potential advantages to kinetic methods of analysis, perhaps the most important of which is the ability to use chemical reactions that are slow to reach equilibrium. In this chapter we examine three techniques that rely on measurements made while the analytical system is under kinetic rather than thermodynamic control chemical kinetic techniques, in which the rate of a chemical reaction is measured radiochemical techniques, in which a radioactive element s rate of nuclear decay is measured and flow injection analysis, in which the analyte is injected into a continuously flowing carrier stream, where its mixing and reaction with reagents in the stream are controlled by the kinetic processes of convection and diffusion. 

The earliest examples of analytical methods based on chemical kinetics, which date from the late nineteenth century, took advantage of the catalytic activity of enzymes. Typically, the enzyme was added to a solution containing a suitable substrate, and the reaction between the two was monitored for a fixed time. The enzyme s activity was determined by measuring the amount of substrate that had reacted. Enzymes also were used in procedures for the quantitative analysis of hydrogen peroxide and carbohydrates. The application of catalytic reactions continued in the first half of the twentieth century, and developments included the use of nonenzymatic catalysts, noncatalytic reactions, and differences in reaction rates when analyzing samples with several analytes. 

Despite the variety of methods that had been developed, by 1960 kinetic methods were no longer in common use. The principal limitation to a broader acceptance of chemical kinetic methods was their greater susceptibility to errors from uncontrolled or poorly controlled variables, such as temperature and pH, and the presence of interferents that activate or inhibit catalytic reactions. Many of these limitations, however, were overcome during the 1960s, 1970s, and 1980s with the development of improved instrumentation and data analysis methods compensating for these errors. ... 

Every chemical reaction occurs at a finite rate and, therefore, can potentially serve as the basis for a chemical kinetic method of analysis. To be effective, however, the chemical reaction must meet three conditions. First, the rate of the chemical reaction must be fast enough that the analysis can be conducted in a reasonable time, but slow enough that the reaction does not approach its equilibrium position while the reagents are mixing. As a practical limit, reactions reaching equilibrium within 1 s are not easily studied without the aid of specialized equipment allowing for the rapid mixing of reactants. 

A final requirement for a chemical kinetic method of analysis is that it must be possible to monitor the reaction s progress by following the change in concentration for one of the reactants or products as a function of time. Which species is used is not important thus, in a quantitative analysis the rate can be measured by monitoring the analyte, a reagent reacting with the analyte, or a product. For example, the concentration of phosphate can be determined by monitoring its reaction with Mo(VI) to form 12-molybdophosphoric acid (12-MPA). 

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