Part 1

Chemical Kinetics

Clive McKee and Michael Mortimer

INTRODUCTION

Movement is a fundamental feature of the world we live in; it is also inextricably linked with time. The measurement of time relies on change — monitoring the swing of a pendulum, perhaps — but conversely, any discussion of the motion of the pendulum must involve the concept of time. Taken together, time and change lead to the idea of rate, the quantity which tells us how much change occurs in a given time. Thus, for example, for our pendulum we might describe rate in terms of the number of swings per minute. Or, to take a familiar example from everyday life, we refer to a rate of change in position as speed and measure it as the distance travelled in a given time (Figure 1.1).

The study of movement in general is the subject of kinetics and chemical kinetics, in particular, is concerned with the measurement and interpretation of the rates of chemical reactions. It is an area quite distinct from that of chemical thermodynamics which is concerned only with the initial states of the reactants (before a reaction begins) and the final state of the system when an equilibrium is reached (so that there is no longer any net change). What happens between these initial and final states of reaction and exactly how, and how quickly, the transition from one to the other occurs is the province of chemical kinetics. At the molecular level chemical kinetics seeks to describe the behaviour of molecules as they collide, are transformed into new species, and move apart again. But there is also a very practical side to the subject which is quickly appreciated when we realize that our very existence depends on a balance between the rates of a multitude of chemical processes: those controlling our bodies, those determining the growth of the animals and plants that we eat, and those influencing the nature of our environment. We must also not forget those changes that form the basis of much of modern technology, for which the car provides a wealth of examples (see Box 1.1). Whatever the process, however, information on how quickly it occurs and how it is affected by external factors is of key importance. Without such knowledge, for example, we would be less well-equipped to generate products in the chemical industry at an economically acceptable rate, or design appropriate drugs, or understand the processes that occur within our atmosphere.

Historically, the first quantitative study of a chemical reaction is considered to have been carried out by Ludwig Wilhelmy in 1850. He followed the breakdown of sucrose (cane sugar) in acid solution to give glucose and fructose and noted that the rate of reaction at any time following the start of reaction was directly proportional to the amount of sucrose remaining unreacted at that time. For this observation Wilhelmy richly deserves to be called 'the founder of chemical kinetics'. Just over a decade later Marcellin Berthelot and Péan de St Gilles made a similar but more significant observation, In a study of the reaction between ethanoic acid (CH3COOH) and ethanol (C2H5OH) to give ethyl ethanoate (CH3COOC2H5) they found the measured rate of reaction at any instant to be approximately proportional to the concentrations of the two reactants at that instant multiplied together. At the time, the importance of this result was not appreciated but, as we shall see, relationships of this kind are now known to describe the rates of a wide range of different chemical processes. Indeed, such relationships lie at the heart of empirical chemical kinetics, that is an approach to chemical kinetics in which the aim is to describe the progress of a chemical reaction with time in the simplest possible mathematical way.

By the 1880s, the study of reaction rates had developed sufficiently to be recognized as a discipline in its own right. The 21 December 1882, issue of the journal Nature noted,

'What may perhaps be called the kinetic theory of chemical actions, the theory namely, that the direction and amount of any chemical change is conditioned not only by the affinities, but also by the masses of reacting substances, by the temperature, pressure, and other physical circumstances — is being gradually accepted, and illustrated by experimental results.'

Over a century later, chemical kinetics remains a field of very considerable activity and development; indeed nine Nobel prizes in Chemistry have been awarded in this subject area. The most recent (1999) was to A. H. Zewail whose work revealed for the first time what actually happens at the moment in which chemical bonds in a reactant molecule break and new ones form to create products. This gives rise to a new area: femtochemistry). The prefix femto (abbreviation 'f') represents the factor 10-15 and indicates the timescale, which is measured in femtoseconds, of the new experiments. As some measure of how short a femtosecond is, while you read these words light is taking about 2 million femtoseconds (2 x 106 fs) to travel from the page to your eye and a further 1000 fs to pass through the lens to the retina.

QUESTION 1.

In an empirical approach to chemical kinetics, what would be the simplest mathematical way of representing the information obtained by Marcellin Berthelot and Pean de St Gilles for the reaction between ethanoic acid and ethanol?

1.1 A general definition of rate

So far, we have tended to use the term rate in a purely qualitative way. However, it is important for later discussions to introduce a more quantitative definition. In one sense, rate is the amount of one thing which corresponds to a certain amount (usually one) of some other thing. For example, governments, financial markets and holidaymakers in foreign countries may be concerned about exchange rates: the number of dollars, euros or other currency that can be bought for one pound sterling. More frequently, however, and as we have mentioned earlier, the concept of rate involves the passage of time. This is particularly so in the area of chemical kinetics. We shall restrict our definitions of rate, therefore, to cases in which time is involved.

For a physical quantity that changes linearly with time, we can take as a definition:

rate of change = change in physical quantity in typical units/time interval in typical units (1.1)

For time, typical units are seconds, minutes, hours, and so on. If, for example, the physical quantity was distance then typical units could be metres and the rate of change would correspond to speed measured in, say, metres per second (ms-1). Since the physical quantity changes linearly with time this means that the change in any one time interval is exactly the same as that in any other equal interval. In other words a plot of physical quantity versus time will be a straight line and there is a uniform, or constant, rate of change.

Equation 1. I can be written in a more compact notation. If the physical quantity is represented by y, then it will change by an amount Δy during a time interval Δy, and we can write

rate of change of y = Δy/Δt (1.2)

This rate of change, Δy/Δt, corresponds mathematically to the slope (or gradient) of the straight line and, as already stated, has a constant value.

A very important situation arises when a rate of change itself varies with time. A familiar example is a car accelerating; as time progresses, the car goes faster and faster. In this case a plot of physical quantity versus time is no longer a straight line. It is a curve. At any particular time, the rate of change is often referred to as the 'instantaneous rate of change'. It is measured as 'the slope of the tangent to the curve at that particular time' and is represented by the expression dy/dt. (The notation d/dt can be interpreted as 'instantaneous rate of change with respect to time'.) It is not easy to draw the tangent to a curve at a particular point. If the real experimental data consist of measurements at discrete points then it is first necessary to assume that these points are linked by a smooth curve and then to draw this curve. Again, this is not easy to accomplish although reasonable efforts can sometimes be achieved 'by eye'. A better approach is to use appropriate computer software. Even so, the best curve that can be computed will always be an approximation to the true curve and will also depend on the quality of the experimental data; for example in a chemical kinetic investigation on how well concentrations can be measured at specific times. The uncertainty in the value of the tangent that is computed at any point will reflect these factors.

A CLOSER LOOK AT CHEMICAL REACTIONS

Around the beginning of the nineteenth century, the early chemists concentrated much of their effort on working out the proportions in which substances combine with one another and in developing a system of shorthand notation for representing chemical reactions. As a result, when we now think of the interaction of hydrogen and oxygen, for example, we tend to think automatically in terms of a balanced chemical equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

This equation serves to identify the species taking part and shows that for every two H2 molecules and one O2 molecule that react, two molecules of water are formed. This information concerning the relative amounts of reactants and products is known as the stoichiometry of the reaction. This term was introduced by the German chemist Jeremias Benjamin Richter as early as 1792 in order to denote the relative amounts in which acids and bases neutralize each other; it is now used in a more general way.

Important as it may be, knowing the stoichiometry of a reaction still leaves open a number of fundamental questions:

• Does the reaction occur in a single step, as might be implied by a balanced chemical equation such as Equation 2.1, or does it involve a number of sequential steps?

• In any step, are bonds broken, or made, or both? Furthermore, which bonds are involved?

• In what way do changes in the relative positions of the various atoms, as reflected in the stereochemistry of the final products, come about?

• What energy changes are involved in the reaction?

Answering these questions, particularly in the case of substitution and elimination reactions in organic chemistry, will be the main aim of a large part of this book. As you will see the key information that is required is embodied in the reaction mechanism for a given reaction. Broadly speaking, this refers to a molecular description of how the reactants are converted into products during the reaction. It is important at the outset to emphasize that a reaction mechanism is only as good as the information on which it is based. Essentially, it is a proposal of how a reaction is thought to proceed and its plausibility is always subject to testing by new experiments. For many mechanisms, we can be reasonably confident that they are correct, but we can never be completely certain.

A powerful means of gaining information about the mechanism of a chemical reaction is via experimental investigations of the way in which the reaction rate varies, for example, with the concentrations of species in the reaction mixture, or with temperature. There is thus a strong link between, on the one hand, experimental study and, on the other, the development of models at the molecular level. In the sections that follow we shall look in some depth at the principles that underlie experimental chemical kinetics before moving on to discuss reaction mechanism.

However, it is useful to establish a few general features relating to reaction mechanisms at this stage. In particular we look for features that relate to the steps involved and the energy changes that accompany them.

2.1 Individual steps

If we consider the reaction between bromoethane (CH3CH2Br) and sodium hydroxide in a mixture of ethanol and water at 25 °C then the stoichiometry is represented by the following equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

where we have represented the states of all reactants as aqueous (aq). It is well established (and more to the point, no evidence has been found to the contrary) that this reaction occurs in a single step. We refer to it as an elementary reaction. For Reaction 2.2, therefore, the balanced chemical equation does actually convey the essential one-step nature of the process. The reaction mechanism, although consisting of only one step, is written in a particular way

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

The arrow sign (->) is used to indicate that the reaction is known (or postulated) to be elementary and, by convention, the states of the species involved are not included. (Arrow signs are also used in this course in a more general way, particularly for organic reactions, to indicate that one species is converted to another under a particular set of conditions. The context in which arrow signs are used, however, should always make their significance clear.)

The reaction between phenylchloromethane (C6H5CH2Cl) and sodium hydroxide in water at 25 °C

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

is of a similar type to that in Reaction 2.2. However, all of the available experimental evidence suggests that Reaction 2.4 does not occur in a single elementary step. The most likely mechanism involves two steps

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

A reaction such as this, because it proceeds via more than one elementary step, is known as a composite reaction. The corresponding mechanism, Reactions 2.5 and 2.6, is referred to as a composite reaction mechanism,, or just a composite mechanism. In general, for any composite reaction, the number and nature of the steps in the mechanism cannot be deduced from the stoichiometry. This point is emphasized when we consider that the apparently simple reaction between hydrogen gas and oxygen gas to give water vapour (Reaction 2.1) is thought to involve a sequence of up to 40 elementary steps.

The species [C6H5CH2]+ in the mechanism represented by Reactions 2.5 and 2.6 is known as a reaction intermediate. (This particular species, referred to as a carbocation, has a trivalent carbon atom which normally takes the positive charge. Carbocations are discussed in more detail in Part 2 of this book.) All mechanisms with more than a single step will involve intermediate species and these will be formed in one step and consumed, in some way, in another step. It is worth noting, although without going into detail, that many intermediate species are extremely reactive and short-lived which often makes it very difficult to detect them in a reaction mixture.

* What is the result of adding Equations 2.5 and 2.6 together?

* The addition gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Cancelling the reaction intermediate species from both sides of the equation gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In other words, adding the two steps together gives the form of the balanced chemical equation.

In general, for most composite mechanisms the sum of the various steps should add up to give the overall balanced chemical equation. (An important exception is a radical chain mechanism; see Further reading for a reference to these types of mechanism.)

It is a matter of general experience, that chemical reactions are not instantaneous. Even explosions, although extremely rapid, require a finite time for completion. This resistance to change implies that at the molecular level individual steps in a mechanism require energy in order to take place. For a given step, the energy requirement will depend on the species involved.

A convenient way to depict the energy changes that occur during an elementary reaction is to draw, in a schematic manner, a so-called energy profile; an example is given in Figure 2.1. The vertical axis represents potential energy which has contributions from the energy stored within chemical bonds as well as that associated with the interactions between each species and its surroundings. The horizontal axis is the reaction coordinate and this represents the path the system takes in passing from reactants to products during the reaction event.

An energy profile such as that in Figure 2.1 can be interpreted in two distinct ways; either as representing the energy changes that occur when individual molecular species interact with one another in a single event, or as representing what happens on a macroscopic scale, in which case some form of average has to be taken over many billions of reactions. It is useful to consider the molecular level description first.

If we take the elementary reaction in Equation 2.3 as an example then from a molecular viewpoint, the energy profile shows the energy changes that occur when a single bromoethane molecule encounters, and reacts with, a single hydroxide ion in solution. As these species come closer and closer together they interact and, as a consequence, chemical bonds become distorted and the overall potential energy increases. At distances typical of chemical bond lengths, the reactant species become partially bonded together and new chemical bonds begin to form. At this point the potential energy reaches a maximum and any further distortion then favours the formation of product species and a corresponding fall in potential energy. It is, of course, possible to imagine that a bromoethane molecule and a hydroxide ion, particularly in the chaotic environment of the solution at the molecular level, will approach one another in a wide variety of ways. Each of these approaches will have its own energy profile.