The entire text has been revised and includes many new sections and an additional chapter on applications of kinetics. Chapters also include new problems, with answers to selected questions, to test the reader's understanding of each area. A solutions manual with answers to all questions is available for instructors. A useful text for both students and interested readers alike, Dr. House has once again written a comprehensive text simply explaining an otherwise complicated subject.
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It may take up to minutes before you receive it. Through extensive worked examples, Dr Wright, presents the theories as to why and how reactions occur, before examining the physical and chemical requirements for a reaction and the factors which can influence these.
Chemical Kinetics and Mechanism considers the role of rate of reaction. It begins by introducing chemical kinetics and the analysis of reaction mechanism, from basic well-established concepts to leading edge research. Organic reaction mechanisms are then discussed, encompassing curly arrows, nucleophilic substitution and E1 and E2 elimination reactions.
The book concludes with a Case Study on Zeolites, which examines their structure and internal dimensions in relation to their behaviour as molecular sieves and catalysts. The accompanying CD-ROM contains the "Kinetics Toolkit", a graph-plotting application designed for manipulation and analysis of kinetic data, which is built into many of the examples, questions and exercises in the text.
There are also interactive activities illustrating reaction mechanisms. The Molecular World series provides an integrated introduction to all branches of chemistry for both students wishing to specialise and those wishing to gain a broad understanding of chemistry and its relevance to the everyday world and to other areas of science. The books, with their Case Studies and accompanying multi-media interactive CD-ROMs, will also provide valuable resource material for teachers and lecturers. This book began as a program of self-education.
While teaching under graduate physical chemistry, I became progressively more dissatisfied with my approach to chemical kinetics. The solution to my problem was to write a detailed set of lecture notes which covered more material, in greater depth, than could be presented in undergraduate physical chemistry. These notes are the foundation upon which this book is built.
My background led me to view chemical kinetics as closely related to transport phenomena. While the relationship of these topics is well known, it is often ignored, except for brief discussions of irreversible thermody namics.
In fact, the physics underlying such apparently dissimilar processes as reaction and energy transfer is not so very different. The intermolecular potential is to transport what the potential-energy surface is to reactivity. Instead of beginning the sections devoted to chemical kinetics with a discussion of various theories, I have chosen to treat phenomenology and mechanism first. In this way the essential unity of kinetic arguments, whether applied to gas-phase or solution-phase reaction, can be emphasized.
Theories of rate constants and of chemical dynamics are treated last, so that their strengths and weaknesses may be more clearly highlighted. The book is designed for students in their senior year or first year of graduate school. A year of undergraduate physical chemistry is essential preparation. While further exposure to chemical thermodynamics, statistical thermodynamics, or molecular spectroscopy is an asset, it is not necessary.
A practical approach to chemical reaction kinetics—from basic concepts to laboratory methods—featuring numerous real-world examples and case studies This book focuses on fundamental aspects of reaction kinetics with an emphasis on mathematical methods for analyzing experimental data and interpreting results.
It describes basic concepts of reaction kinetics, parameters for measuring the progress of chemical reactions, variables that affect reaction rates, and ideal reactor performance. Mathematical methods for determining reaction kinetic parameters are described in detail with the help of real-world examples and fully-worked step-by-step solutions.
Both analytical and numerical solutions are exemplified. The book begins with an introduction to the basic concepts of stoichiometry, thermodynamics, and chemical kinetics. This is followed by chapters featuring in-depth discussions of reaction kinetics; methods for studying irreversible reactions with one, two and three components; reversible reactions; and complex reactions.
In the concluding chapters the author addresses reaction mechanisms, enzymatic reactions, data reconciliation, parameters, and examples of industrial reaction kinetics. Throughout the book industrial case studies are presented with step-by-step solutions, and further problems are provided at the end of each chapter. It is also an ideal textbook for undergraduate and graduate-level courses in chemical kinetics, homogeneous catalysis, chemical reaction engineering, and petrochemical engineering, biotechnology.
James House's revised Principles of Chemical Kinetics provides a clear and logical description of chemical kinetics in a manner unlike any other book of its kind. Clearly written with detailed derivations, the text allows students to move rapidly from theoretical concepts of rates of reaction to concrete applications.
Unlike other texts, House presents a balanced treatment of kinetic reactions in gas, solution, and solid states. The entire text has been revised and includes many new sections and an additional chapter on applications of kinetics. Chapters also include new problems, with answers to selected questions, to test the reader's understanding of each area. A solutions manual with answers to all questions is available for instructors. A useful text for both students and interested readers alike, Dr.
House has once again written a comprehensive text simply explaining an otherwise complicated subject. Provides an introduction to all the major areas of kinetics and demonstrates the use of these concepts in real life applications Detailed derivations of formula are shown to help students with a limited background in mathematics Presents a balanced treatment of kinetics of reactions in gas phase, solutions and solids Solutions manual available for instructors.
Skip to content. Chemical Kinetics. An Introduction to Chemical Kinetics. Chemical Kinetics and Reaction Dynamics. Using the method of calculating rate constants may make it impossible to distinguish between experimental errors in the data while graphical presentation of the data may reveal a trend or curvature of the plot, which suggests that another rate law is applicable. Finally, when the calculated rate constants are displayed as shown in Table 3. In all cases where it is possible to do so, a reaction should be studied over several half-lives in order to obtain data that are amenable to kinetic analysis.
The data shown in Table 3. Taking the logarithm of both sides of Eq. If the reaction is carried out using two diVerent [A]o values, a ratio of two equations having the form of Eq. While this quick, approximate method is valid, it is not generally as accurate as more detailed methods of data analysis because it is based on only two data points.
As developed here, it applies only to reactions that obey an nth-order rate law in one reactant. From the initial rates determined from the slopes, the data shown in Table 3.
Initial rate, [A]o , M ln [A]o M 1 min 1 ln rate 1. A plot of the values obtained for ln R versus ln [A]o such as that shown in Figure 3. Linear regression of the preceding data, which were determined graphically from Figure 3.
Since the intercept is equal to ln k, this value corres- ponds to a k value of 0. This exercise shows that when suitable data are available, the method of initial rates can be successfully employed to determine a reaction order and rate constants. Initial rates can also be used in another way. Subsequently, the process can be repeated using diVerent initial concentrations of A and B. For convenience in the second experiment, the initial concentrations of A and B will be taken as twice what they were in the Wrst experiment so that 2[A]o and 2[B]o.
This rate law can be treated by the integral methods that were described in Chapter 2 to determine n. The procedure can be repeated by making the initial con- centration of A large compared to [B]o so that m can be determined. Flooding is essentially making the conditions of the reaction such that it becomes a pseudo nth-order process in one reactant by using a larger concentration of the other reactant.
Under these conditions, the concentration of OH is suYciently large that the reaction appears to be Wrst order in t CH3 3 CBr, but is actually a pseudo Wrst-order process. Many hydrolysis reactions appear to be independent of [H2 O] only because water is usually present in such a large excess. Of course, not all reactions can be studied by the method of Xooding because a very large excess of a reactant may cause the reaction to take place in a diVerent way.
Techniques and Methods 87 3. A somewhat better way to apply the logarithmic method is to carry out the reaction using several diVerent starting concentrations of A while keeping the concentration of B constant. The procedure can be repeated to Wnd m, the order of the reaction with respect to B. It can be studied kinetically by monitoring the production of I3 , which gives the familiar blue color with starch as an indicator. When the S2 O23 is exhausted, the I3 that is produced by I reacting with S2 O28 interacts with the starch to produce a blue color.
In this way, the amount of I3 produced can be monitored, which makes it possible to determine in an indirect way the amount of S2 O28 that has reacted. This reaction can be used to illustrate the application of the logarithmic method. In the study described here, the Wrst of three runs had an initial concentration of S2 O28 and I of 0. In the second run, the concentration of S2 O28 was 0. In the Wnal run, the concentration of S2 O28 was 0.
Using Eq. In this example, exactly doubling the concentrations makes it possible to deduce the reac- tion order by inspection, but the method described is a general one that can be applied under other conditions.
Consequently, the application of pressure to a chemical system is equivalent to performing work on the system in a manner that is somewhat analogous to changing the temperature of the system. The principle of Le Chatelier enables us to predict the eVects of changing conditions on a system at equilibrium. For example, increasing the tem- perature causes the system to shift in the endothermic direction. Likewise, increasing the pressure on a system at equilibrium causes the system to shift in the direction corresponding to smaller volume.
For chemical reactions, we have repeatedly assumed that a small but essentially constant concentration of the transition state is in equilibrium with the reactants. It is the concentration of the transition state that determines the magnitude of the rate constant. In Section 2. If the transi- tion state occupies a smaller volume than the reactants, increasing the pressure will shift the equilibrium toward the formation of a higher con- centration of the transition state, which will increase the rate of the reaction.
If the transition state occupies a larger volume than the reactants, increasing the pressure will decrease the concentration of the transition state and decrease the rate of the reaction. As will be discussed in Chapter 5, the eVect of internal pressure caused by the solvent aVects the rate of a reaction in much the same way as does the external pressure. Because volumes of liquids change slightly due to their compressibility, molality or mole fraction should be chosen to meas- ure concentrations rather than molarity.
Under most conditions, the diVer- ence is negligible. Therefore, the volume of activation can be determined if the reaction is carried out to determine the rate constant at several usually quite high pressures.
While such plots are sometimes approximately linear, they often exhibit some degree of curvature, which indicates that the value of DVz is somewhat pressure dependent.
To deal with this situation, we need either a theoretical approach to determine DVz or perhaps a graphical procedure to obtain an empirical relationship. This amount of energy would be equivalent to that involving only a very small change in temperature. Therefore, in order to accomplish a change equivalent to that brought about by a modest change in temperature, an enormous change in pressure is required.
The interpretations of volumes of activation are not always unambigu- ous, but generally if DVz is negative, the rate of the reaction increases as pressure is increased. This signiWes that the transition state occupies a smaller volume than the reactants, and its formation is assisted by increasing the applied pressure. The reason for this rather large negative change is that ions are strongly solvated, which leads to a compacting and ordering of the solvent surrounding the ions.
Therefore, DVz is made up of two parts: 1 an intrinsic part, DVzint , which depends on changes in molecular dimensions as the transition state forms and 2 a solvation part, DVzsolv , which depends on the nature and extent of solvation of the transition state.
If forming the transition state involves forming ions, DVzsolv will be negative because of the ordering of the solvent that occurs in the vicinity of the charged ions. This phenomenon is known as electrostriction. Studying the eVect of pressure on the rate of a reaction can yield infor- mation about the mechanism that is diYcult to obtain by any other means.
One reaction in which rate studies at high pressure have yielded a consider- able amount of information is a linkage isomerization reaction, which has been known for over a century. A simplistic view of this process suggests that the Co—ONO bond could break and then the nitrite ion could reattach by bonding through the nitrogen atom to give CoNO2 linkages.
It was reported that the linkage isomer- ization takes place more rapidly at high pressures. These values indicate that the transition state occupies a smaller volume than the reactants.
It is generally accepted that this reaction does not take by a bond-breaking bond-making mechanism. This observation indicates that the coordinated NO2 never leaves the coordination sphere of the metal ion. Support for a structure that has NO2 bonded to the metal ion by both O and N atoms has been obtained by studying the reaction photochemically in the solid state and quenching the solid to very low temperature. Studies of this type require specialized equipment, but they frequently yield a great deal of information about reaction mechanisms.
Undoubtedly, there are many other reactions that have not been studied in this way that should be. This time is usually very short compared to the time that the reaction is followed during a kinetic study. In this period, often called the transient or pre—steady state period, the kinetic rate laws developed earlier do not represent the reaction very well, and diVerent experimental techniques must be employed to study such processes.
One technique, developed in by Hartridge and Roughton for studying the reaction between hemoglobin and oxygen, makes use of a continuous-Xow system. The two reacting solutions were forced under constant pressure into a mixing chamber as illustrated by the diagram shown in Figure 3. After the liquids mix and the reaction starts, the mixture Xows out of the mixing chamber to a point where a measuring device is located. A suitable measuring device for many reactions may be a spectrophotometer to determine the concentration of a reactant or product from absorption measurements.
The length of time that the reaction has been taking place is determined by the distance from the mixing chamber to the observation point. Presently used continuous-Xow systems can study reactions fast enough to have a half-life of 1 ms. Many types of Xow equipment have been developed, and diVerent methods of introducing the sample have been devised.
In the stopped-Xow technique, the solutions are forced from syringes into a mixing chamber. After a very short period of Xow, perhaps a few ms, the Xow is stopped suddenly when the observation cell is Wlled by an opposing piston that is linked to a sensing switch that triggers the measuring device see Figure 3.
Small volumes of solutions are used, and the kinetic equations for modeling the reactions are equivalent to those used in conventional methods in which concentration and time are measured.
Commercial stopped-Xow apparatus is available with several modiWcations in the designs. Both stopped-Xow and continuous-Xow techniques are useful for studying fast reactions that have half-lives as short as a few milliseconds. In the next section, relaxation methods that can be used to study very fast reactions that have half-lives as short as 10 10 to 10 12 sec will be described.
Classical techniques generally rely on mixing of reactants and can be used for studying reactions that take place on a timescale of approximately a few seconds or longer. Flow techniques described in Section 3. In contrast, extremely fast reactions in solution may take place on a timescale as short as 10 10 to 10 12 sec. A time of this magnitude corresponds to the time necessary for diVusion to occur over a distance that represents the distance separating them at closest approach.
That distance is typically on the order of 10 4 to 10 5 cm so the time necessary for diVusion to occur is approximately 10 10 to 10 12 sec. If long-range diVusion is involved, the reaction rate will be dependent of the viscosity of the solvent. Relaxation techniques are designed so that mixing rates and times do not control the reaction. Instead, they utilize systems that are at equilibrium under the conditions of temperature and pressure that describe the system before some virtually instantaneous stress is placed on the system.
The stress should not be a signiWcant fraction of the half-life of the reaction. After the stress disturbs the system, chemical changes occur to return the system to equilibrium. This relieving of the stress is the reason why the term relaxation is applied to such experiments. Several relaxation techniques diVer primarily in the type of stress applied to the system. For example, the shock tube method makes use of two chambers that are separated by a diaphragm. The reactants are on one side of the diaphragm where the pressure is much lower than on the other, which contains a gas under much higher pressure.
The high-pressure gas is known as the driver gas, and when the diaphragm is ruptured, it expands rapidly into the chamber containing the reactants. This expansion generates a shock wave that results in rapid heating of the reactant gases.
The reaction between the gases occurs as the system reestablishes equilibrium. Changes in concentrations of the reactants or products are followed by using a spectrophotometer. Shock tube techniques can be used to study gas phase reactions that occur on the timescale of 10 3 to about 10 6 sec.
Another means of producing an instantaneous stress on a system at equilibrium is by irradiating it with a burst of electromagnetic radiation. Known as Xash photolysis, this technique is based on the fact that absorption of the radiation changes the conditions so that the system must relax to reestablish equilibrium.
As it does so, the changes can be followed spectro- photometrically. The technique known as temperature jump commonly referred to as T-jump involves rapidly heating the system to disturb the equilibrium.
Heating is sometimes accomplished by means of electric current or micro- wave radiation. When a rapid change in pressure is used to disturb the system, the technique known as pressure jump shortened to P-jump results. Techniques and Methods 97 Keep in mind that a change in pressure accomplishes P-V work that is very small compared to the eVects of changing the temperature for liquids see Section 3. Consequently, the P-jump technique is normally used to study reactions in gaseous systems.
Kinetic analysis of a relaxing system is somewhat diVerent than for classical reactions as will now be described. From the stoichiometry of the reaction, we know that the change in the concentration of A is equal to that of B and negative that of C.
For example, a kinetic study of the hydrolysis of ethyl acetate was described in Chapter 2. The reaction was found to obey a second-order rate law that is Wrst-order in two reactants, but there is still a question to be answered: Which C—O bond breaks? This question can be answered only if the oxygen atom is made distinguishable from those in the bulk solvent, water. Then, hydrolysis of the ester will produce diVerent products depending on which bond breaks.
One of the important types of reactions exhibited by coordination compounds is that known as an insertion reaction, in which an entering ligand is placed between the metal ion and a ligand already bound to it.
A well-known reaction of this type for which tracer studies have yielded important mechanistic information is the CO insertion that occurs in [Mn CO 5 CH3 ]. The reaction actually proceeds by a Wrst step that involves a group transfer of one of the CO ligands already bound to the metal. It is this ligand that is inserted between the Mn and the CH3 group.
Without the use of an isotopic tracer, there would be no way to distinguish one CO molecule from another. Therefore, the nitrate ion is not totally decomposed, and one of the oxygen atoms is found attached to the same nitrogen atom that it was initially bonded to.
Although this result is not totally surprising, it is still interesting to see how tracer techniques can be used to answer questions regarding how reactions take place. Based on the observations just described, it is possible to postulate a mechanism for the decomposition of NH4 NO3 that is consistent with these observations.
The proposed mechanism can be shown as follows. The crystal Weld stabilization energy in such complexes is 24Dq, and substitution reactions occur by a dissociative pathway that is normally slow. It seems unlikely that the Co OH bond would be so easily broken in this case because other Co ligand bonds are rather inert from a kinetic standpoint.
The reaction appears to take place by a mechanism that can be shown as follows. A classic example of the use of isotopically labeled compounds in organic chemistry involves the identiWcation of the benzyne intermediate by J. Roberts and coworkers Employing 14 C in the reactant C—chlorobenzene produces results that provide a way to explain which carbon atom the NH2 group attaches to. As illustrated earlier, the use of a tracer gives information about the type of intermediate formed that is not easily obtained in any other way.
While only a few examples of the use of labeling techniques have been cited here, the reactions chosen represent drastically diVerent types. These examples show how the use of tracers in elucidating reaction mechanisms has been of great value. In many cases, the results obtained are simply not obtainable by any other means because there is no other way to distinguish between atoms that are otherwise identical.
For example, it is the diVerence in rates of electrolysis that allows D2 O to be obtained by the electrolysis of water, even though the relative abundance of D compared to H is This phenomenon is known as the kinetic isotope eVect.
A mathematical treatment of isotope eVects is rather laborious and unnecessary, but we can show how they arise in a straightforward way. It is known that the greater the relative diVerence in the mass of two isotopes, the greater the kinetic isotope eVect.
Suitable preparation and detection proced- ures must be available, and a radioactive isotope must have a suitable half- life for the isotopically labeled materials to be employed. This limits somewhat the range of atoms that are useful in studying kinetic isotope eVects on reaction rates.
Only in the case of the hydrogen isotopes is the relative mass eVect this large. The eVect of the reduced mass on the zero-point vibrational energy is easily seen. As a result, the zero-point vibrational energy is greater for the H—H bond than it is for the H—D bond.
Because the H2 molecule already resides in a higher vibrational energy state than does the HD molecule, it requires corresponding less energy to dissociate the H2 molecule. Accordingly, a reaction that requires the dissociation of these molecules will take place more rapidly for H2 than for HD. This suggests that the diVerences in the nature of the X—H bonds should give rise to a kinetic isotope eVect when reactions occur at these bonds. Consider two reactant molecules that are identical except that one of them contains a diVerent isotope at the reactive site.
If the bonds in the reactant molecules that link the two isotopic atoms in their positions are not broken in forming the transition state, the extent to which isotopic labeling aVects the rate will be less than when those bonds are completely broken. If the formation of the transition state does not alter the bond holding the isotopic atoms, there will be no isotope eVect. However, if during the formation of the transition state, the bond to the diVerent isotopes in the reacting molecules becomes stronger, there will be an inverse isotope eVect.
This results from the fact that as the bond becomes stronger in the transition state, the heavier isotope will give a transition state having a lower zero-point vibrational energy. Because this gives an overall lowering of the energy of the transition state relative to the reactants, there will be a rate increase in the case of the heavier isotope. Earlier in this chapter, we described the reaction of chlorobenzene with amide ion to produce aniline.
The mechanism for this reaction involves the removal of H by NH2 to form the benzyne intermediate. As expected, the rate of breaking the X—H bond is higher than that for breaking the X—D bond. To this point, we have presumed that the bond breaking actually occurs as the transition state forms. Because quantum mechanically it is possible for barrier penetration to occur, tunneling must be considered as a possible reaction pathway. The transmission of a particle through a potential energy barrier is one of the basic models of quantum mechanics.
The transparency decreases as the height of the barrier, U, increases. The transparency increases as the energy of the particles, E, increases. The transparency is greater for particles of smaller mass, M. The transparency decreases as the thickness of the barrier, x, in- creases.
That is, the particle would not pass over or through the barrier because it has an energy that is lower than the height of the barrier.
From the discussion just presented, it can be seen that particles having lower mass have a greater probability of penetrating a barrier if all other factors are equal.
Likewise, the higher the energy of the particle, the higher the transmission coeYcient. Both of these factors favor barrier penetration by H over that by D, so reactions that involve tunneling also show the expected kinetic isotope eVect, which predicts that the lighter isotope reacts faster. Although we have considered the separation of only diatomic molecules, the conclusions reached are still generally valid for more complex mol- ecules.
Bending vibrations are altered during a bond-breaking reaction, but because bending vibrations normally involve considerably lower energies than do stretching vibrations, they can usually be ignored in a qualitative approach to isotope eVects.
There may also be other eVects produced by isotopic substitution at positions other than the reactive site in the molecule. These eVects are usually much smaller than primary isotope eVects, and they are referred to as secondary isotope eVects. A very large number of reactions have been studied using kinetic isotope eVects to obtain information about the transition states, and the information obtained has signiWcantly increased knowledge of how reactions take place.
For further details, the following references should be consulted. VI, in A. Weissberger, Ed. Numerous chapters dealing with all aspects of kinetics in over pages. Bernasconi, G. This book deals with many aspects of reactions in solution and with solvent eVects on reaction rates.
Caldin, E. Fast Reactions in Solution, Blackwell, Oxford. Espenson, J. Chemical Kinetics and Reaction Mechanisms, 2nd ed. The second edition of a well-known book on mechanistic chemistry. House, J. Fundamentals of Quantum Chemistry, 2nd ed. A basic quantum mechanics text that illustrates the appli- cations of quantum mechanical models such as barrier penetration.
Loupy, A. A book that discusses how microwaves can be used to enhance reactions. Mares, M. Acta 27, Melander, L. A standard reference in the Weld of isotope eVects. Nicholas, J. An introduction to the theory and practice in the study of gas phase reactions. Roberts, J. Chemical Kinetics and Dynamics, 2nd Ed.
Wentrup, C. Products, the initial rate varies with initial concentra- tions as follows. Write the question to be answered. Next, decide which atom could be replaced by a diVerent isotope and show how the mechanism could be elucidated by the use of a labeled compound.
Products, the following data were obtained for the initial rates, Ri. Techniques and Methods 5. Acta, , 27, : P, bar k, sec 1 P, bar k, sec 1 1 The reaction 2 2 C5 H10!
Use these data and the logarithmic method to determine the order with respect to the catalyst. CHAPTER 4 Reactions in the Gas Phase In the previous chapters, we have considered reactions on an empirical basis in terms of several concentration-time relationships that apply to many types of chemical systems.
These units molecules, ions, atoms, radicals, and electrons must be involved in some simple step at the instant of reaction. These steps through which individual units pass are called elementary reactions. The sequence of these elementary reactions constitutes the mechanism of the reaction. In many cases, there must be energy transfer between the reacting molecules. For reactions that take place in the gas phase, molecular colli- sions constitute the vehicle for energy transfer, and our description of gas phase reactions begins with a kinetic theory approach to collisions of gaseous molecules.
In simplest terms, the two requirements that must be met for a reaction to occur are 1 a collision must occur and 2 the molecules must possess suYcient energy to cause a reaction to occur. It will be shown that this treatment is not suYcient to explain reactions in the gas phase, but it is the starting point for the theory.
As a result, it is the rate constant that contains information related to the collision frequency, which determines the rate of a reaction in the gas phase. For molecules that undergo collision, the exponential is related to the number of molecular collisions that have the required energy to induce reaction.
The pre-exponential factor, A, is related to the fre- quency of collisions. The collision frequency between two diVerent types of molecules can be calculated by means of the kinetic theory of gases.
In this discussion, in which collisions are occurring between molecules of A and B, we will consider the molecules of B as being stationary and A molecules moving through a collection of them. A diagram showing this situation is shown in Figure 4.
In 1 second, a molecule of A travels a distance of nAB where nAB is the average molecular velocity of A relative to B and it will collide with all molecules of B that have centers that lie within the cylinder. The preceding result is for a single molecule of A.
To obtain the total number of collisions between molecules of A and B, ZAB , the result must be multiplied by CA , the number of molecules of A per cm3.
We must now consider these other factors as will now be described. One factor that has been ignored to this point is that although a collision frequency can be calculated, the collision between the molecules must occur with suYcient energy for the reaction to occur. As we have previ- ously seen, that minimum energy is the activation energy. Figure 4. Over a narrow range of temperature, this dependence on temperature is not usually observed. However, the area under the curve corresponding to molecules having energies greater than Ea is in- creased slightly see Figure 4.
As was illustrated in Chapter 2, an increase in tem- perature of can double or triple the rate of a reaction. When reaction rates calculated using collision theory are compared to the experimental rates, the agreement is usually poor. In some cases, the agreement is within a factor of 2 or 3, but in other cases the calculated and experimental rates diVer by to The discrepancy is usually explained in terms of the number of eVective collisions, which is only a fraction of the total collisions owing to steric requirements.
To compensate for the diVerence between calculated and observed rates, a steric factor, P, is introduced. If we speculate about the structure of this three-body species, we realize that repulsions will be minimized if the structure is linear.
That the transition state is linear in this case follows from the fact that to form a bent transition state would bring the terminal atoms closer together, which would increase repulsion. To relate the energy of this system to the bond distances is now the problem. While we might ap- proach this problem in a number of ways, one simple approach is to extend a relationship used for a diatomic molecule to include a second bond.
The bond energy of a diatomic molecule varies with the bond length as shown in Figure 4. The energy is most favorable at the bottom of the potential well which corresponds to the equilibrium bond length. One equation that models the kind of relationship shown in Figure 4.
Attraction between the atoms increases as they get closer together the energy be- comes more negative , but at distances smaller than ro , repulsion increases and becomes dominant at very short internuclear distances. For a linear triatomic transition state, it is assumed that a second potential energy curve results so that the total energy is a function of two bond distances.
Therefore, a diagram can be constructed that shows energy on one axis usually chosen to be the vertical axis , one of the bond distances on another, and the second bond distance on the third axis, which gener- ates a three-dimensional energy surface.
The path representing the changes in conWguration as the reaction takes place is called the reaction coordinate. While more exact calculations based on the variation method and semi-empirical pro- cedures provide results that are in qualitative agreement with experimental results, especially for simple molecules, the details of these methods will not be presented here.
It is suYcient to point out that ab initio calculations have largely replaced the older type of calculations. Another facet of the potential energy barrier to reaction is that of quantum mechanical tunneling. Classically, an object must have an energy at least equivalent to the height of a barrier in order to pass over it.
Quantum mechanically, it is possible for a particle to pass through a barrier even though the particle has an energy that is less than the height of the barrier. The tunneling coeYcient also referred to as the transmission probability or transparency of a barrier is determined by the height and thickness of the barrier and the mass and energy of the particle.
For a given barrier, the transparency decreases as the mass of the particle increases so that tunneling is greater for light atoms, i.
However, the transparency increases as the energy approaches the barrier height see Section 3. The potential energy surface may be almost symmetrical if the diatomic molecule AB is very similar to BC. In a more general case, the reactant and product molecules will have considerably diVerent bond energies so the potential energy surface will not be as nearly symmetrical. In such a case the product molecule lies at a lower energy than the reactant showing that the reaction is exothermic.
Reactions in the Gas Phase An alternative method of showing a potential energy surface is based on the same principle as that used to prepare a topographical map. In a topographical map, lines connect points of equal altitude creating contours that have speciWc altitudes. Where the contour lines are closely spaced, the altitude is changing abruptly, and where the contour lines are widely separated the surface is essentially Xat. Slices through the surface at speciWc constant energies of the transition state provide the contour lines.
This case corresponds to the reaction in which the molecules BC and AB have similar bond energies. Transition state theory, developed largely by Henry Eyring, takes a somewhat diVerent approach. We have already considered the potential energy surfaces that provide a graphical energy model for chemical reactions.
Transition state theory or activated complex theory refers to the details of how reactions become products. The term phase space is applied to the coordinate and momentum space for the system. In order for a reaction to occur, the transition state must pass through some critical conWguration in this space.
Because of the nature of the potential function used to express the energy of the system as a function of atomic positions, the system energy possesses a saddle point.
The Boltzmann Distribution Law governs the concentration of that transition state, and the rate of reaction is proportional to its concentration. The concentration of the transition state is not the only factor involved, since the frequency of its dissociation into products comes into play because the rate at which it decomposes must also be considered.
When it does separate, one of the 3N 6 vibrational degrees of freedom for a linear molecule it is 3N 5 is lost and is transformed into translational degrees of freedom of the products. Central to the idea of transition state theory is the assumption that the transition state species is in equilibrium with the reactants. According to this procedure, it is assumed that there is a certain distance, d, at the top of the barrier, which must be the distance where the transition state exists.
It is within this distance that a vibrational mode of the complex is transformed into translational motion of the products. The rate of passage of the transition state through distance d is related to the molecular velocity in one direction.
This expression for k is similar to that obtained from collision theory. An approximate rate constant, ka , can be calculated from probability that the reactants in the distribution of quantum state will collide and react in accord with the collision frequency.
The approximate constant is greater than the measured rate constant, k. One approach to improving transition state theory with respect to calculating the rate constant is to alter the conWguration of the transition state used in the energy calculations in order to eVect a change in ka.
In fact, the calculations are performed in such a way that the calculated rate constant is a minimum and thereby approaches the observed k. In practice, a series of transition states is considered and the calculations are performed to obtain the desired minimization. It is of some consequence to choose the reaction path with respect to the energy surface. Generally, the path chosen is the path of steepest descent on either side of the saddle point.
This path represents the path of minimum energy. Accordingly, the rate constant is minimized with respect to a parameter related to conWguration of the transition state in the same way that energy is minim- ized with respect of variables in a trial wave function. Although this topic will not be described further here, details have been published in several places for example, see Truhlar, Many decomposi- tions, e. However, some such reactions do appear to be second-order at low gas pressure.
In , Lindemann proposed an explanation of these obser- vations. Molecules transfer energy as a result of molecular collisions. The activation of molecules by collision can thus be accomplished. However, the activated molecule need not react immediately, and, in fact, it may become deactivated by undergoing subsequent collisions before it reacts. For reaction to occur, the activated molecule that has increased vibrational energy must have some bond activated to the point where bond rupture occurs.
Solving Eq. Therefore, at relatively high pressure where [A] is high, the reaction appears to be unimolecular Wrst-order in [A]. Under these conditions, the increase in vibrational energy can cause bond rupture and decomposition.
Thus, the observed bimolecular dependence at low pressure and the unimolecular dependence at high pressure are predicted by a mechanism involving activation of molecules by collision. The activation of reactant molecules by collision was described earlier. However, this is not the only vehicle for molecular activation.
It is possible for a non-reactant gas a so-called third body to cause activation of molecules of the reactant. These results are in accord with experience for the unimolecular decomposition of a large number of gaseous compounds. Ozone decomposes by a mechanism that appears to be somewhat diVerent from that described earlier, but it provides a rather simple appli- cation of the steady state approximation. The overall reaction is 2O3 g! The approach of Lindemann is based on collisional activation of mol- ecules as a result of energy transfer.
Hinshelwood Nobel Prize in extended this approach to include changes in vibrational energies that can be distributed internally to supply suYcient energy to the bond being broken. This approach provided a better Wt to observed kinetics in the region of low pressure. In the late s, O. Rice and H. Ramsperger as well as L. Kassel developed an approach now known as the RRK theory to unim- olecular decomposition reactions which is based on statistically treating the molecules as coupled oscillators.
In this way, energy is presumed to be distributed about the energized molecule until it vibrates in a way that results in bond rupture. Marcus J. Az k3 Az! This is presumed to occur when the energy at the reactive site becomes as large as E a , the activation energy. Many of the details of the Marcus theory can be found in the book by Nicholas Reactions following free radical mechanisms have reactive intermediates containing unpaired electrons which are produced by homolytic cleavage of covalent bonds.
A method of detecting free radicals was published in , and it is based on the fact that metals such as lead react with free radicals.
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