This chapter will introduce you to the analysis of a key class of kinetics experiments, namely initial rate experiments. We will consider only the simplest class of rate laws, those where the rate is proportional to some powers of the concentrations. This is a limited class, but one that comes up surprisingly often in practice. Initial rate experiments are also useful when a reaction has a complex rate law. However, complex rate laws are best understood in the context of a study of reactionmechanisms, so we put off this discussion for now.

Initial rate studies

Rate laws of chemical reactions are often complicated. They can depend on the concentrations of reactants, products, or other chemicals present in the system in almost arbitrary ways. It would be impractical to try to deal at once with all possible complications, so we often try to study reactions under conditions in which we can expect simpler behavior.

Perhaps the most widely used experimental simplification is the method of initial rates, which was briefly mentioned in Section 11.3. In this method, we study the reaction only for a very short time after initiation. During this time, the concentrations change very little so that a number of complications arising from changes in the reaction mixture over time are avoided. For instance, if we start with a mixture containing only reactants, then very little product will accumulate during the reaction and we can neglect the reverse reaction whose rate, according to the law of mass action, depends on the product concentrations.

To the student

I wrote this book for you.

When I came to the University of Lethbridge in 1995, I started teaching physical chemistry to a mixed class of chemistry and biochemistry students. I have been teaching versions of this course ever since. My first year here, I picked a book that I liked. Boy, was that a mistake! First of all, the book contained almost no examples that appealed to the biochemists, who were the majority of the students in the class. Second, it was filled with mathematical derivations, which I found very satisfying, but which sometimes obscured the concepts of physical chemistry for the students.

Having made this mistake, I started looking around for other textbooks. The ones that I liked the best, Barrow's Physical Chemistry for the Life Sciences and Morris's A Biologist's Physical Chemistry, were out of print at the time. We used a few different textbooks over the years, and some of them were very good books, but my students were never completely happy, and so I wasn't happy. In some cases, the books contained too much math and not enough insight. In others, too many equations were presented without derivation or explanation, which undermined the students' understanding of the material. I therefore set about writing a book for my students, and therefore for the broader community of life science students who need a term of physical chemistry.

The objective of this chapter is to go over a few of the basic concepts of quantum mechanics in preparation for a discussion of spectroscopy, which is in many ways the business end of quantum mechanics, at least for chemical scientists. We will also need a few quantum mechanical ideas from time to time in our study of thermodynamics and of kinetics.

Why should we learn quantum mechanics at all? Atoms and molecules are small, and their constituent parts, electrons, protons and neutrons, are even smaller. Early in the twentieth century, we learned that small things don't obey the laws of classical mechanics. A different kind of mechanics, quantum mechanics, is required to understand chemistry on a fundamental level. In fact, we need different mechanical theories to treat extremes of both size and speed. Figure 2.1 summarizes the situation. There isn't a sharp cut-off between the various sectors of this diagram. Also note that some of the theories are more general than others. We could in principle use quantum mechanics or general relativity to predict the trajectories of tennis balls, but it just isn't worth the effort, given that classical mechanics works perfectly well in this range of masses and speeds. On the other hand, classical mechanics doesn't give very good results for things that are either extremely large, or small, or fast.

Like it or not, to discuss phenomena on an atomic scale, we need quantum mechanics. Ordinary (non-relativistic) quantum mechanics is generally adequate, although electrons in heavy atoms sometimes reach relativistic speeds (approaching the speed of light, c), requiring relativistic quantum mechanics.

Thermodynamics tells you what reactions can happen, but not how fast they will happen. In fact, some reactions are so slow that they are never actually observed, even though they are thermodynamically allowed. Other reactions are stunningly fast. In this part of the book, we will discuss the factors that affect rates of reaction, and consider some basic theories that can be used to rationalize them. This chapter introduces a few definitions and ideas that we will need in our study of kinetics.

The business of kinetics

Chemical kinetics is the study of the rates of chemical reactions or, to put it in simpler language, of the speeds of chemical reactions. There are many different questions one can ask about how fast reactions go, and many different approaches one can take to answering these questions. Roughly though, we can break down the field into two major branches:

1. (1) Phenomenological kinetics: this is the part of kinetics that is concerned with measuring rates of reactions and with the relationship between rates and chemical mechanisms.

2. (2) Kinetic theory and dynamics: this field focuses on the relationship between rates of reactions and events on a molecular scale. In the best cases, we are able to predict a reaction mechanism and the rates of the reactions that make up that mechanism. More commonly, kinetic theory or dynamics let us relate molecular properties to observed reaction rates. Kinetic theory and dynamics are attempts to answer questions about why the rate of a reaction has a particular value, rather than just taking the rates as quantities determined by experiment, as we do in phenomenological kinetics.

The First Law of Thermodynamics tells us that energy is conserved. The Second Law, which was originally formulated to help us understand steam engines, tells us that heat can't be converted into work with 100% efficiency. Our study of the Second Law will lead us to define a new state function known as the entropy. In turn, the entropy will turn out to be pivotal to our understanding of what drives chemical reactions.

The Second Law of Thermodynamics

There are many different statements of the Second Law. My favorite, because it seems to convey most clearly the meaning of this law, is due to Kelvin:

In more modern language, we might say that it is impossible to continuously convert heat completely into work. The Second Law limits the efficiency of heat engines, machines that convert heat into work. Just as the First Law makes a certain kind of perpetual motion machine impossible, the Second Law makes it impossible to build a so-called perpetual motion machine of the second kind, namely one that produces work from heat with perfect efficiency.

I have mentioned several times in the past few chapters that complex reactions can have either simple or complex rate laws. We will now see how complex reactions can give rise to complex rate laws, but also how these rate laws often simplify to simple forms.

You may notice as you read this chapter that it seems to focus a lot on gas-phase chemistry, and that there are few biological examples and problems. This is a deliberate choice. First of all, gas-phase chemistry is of interest to biologists since the atmosphere ties us all together. My second reason for focusing on gas-phase chemistry here has to do with pedagogy: examples of complex mechanisms deriving from gas-phase chemistry are often simpler than those arising in biochemistry.We will get a good grounding in the basic methods for treating complex mechanisms in this chapter before moving on to enzyme kinetics in the next chapter.

Two-step mechanisms

In order to get our feet wet with complex reactions, we will try to resolve a puzzle: on the surface, unimolecular gas-phase reactions, i.e. reactions with the stoichiometry A → product(s), should be the simplest reactions around. However, these reactions always have complex rate laws: at low pressures, they display second-order kinetics, while at high pressures, they follow first-order kinetics with respect to the reactant. (At intermediate pressures, the rate can't be described by either a first- or a second-order rate law.)

Life is a far-from-equilibrium phenomenon, so one might reasonably question the utility of equilibrium theory to biochemists. There are a few reasons why you should care about equilibrium. One is that we still do a lot of experiments in test tubes, and under those conditions, systems eventually go to equilibrium. Another reason is that some reactions, like acid–base reactions, equilibrate much faster than others, so they can be treated as being in equilibrium. There is another, more subtle reason: we saw earlier that reversible processes, those in which the system is constantly in equilibrium, are the most efficient ones possible. Equilibrium therefore sets a limit to the efficiency of chemical processes, an idea we will use when we discuss how one reaction can drive another forward.

What does ΔrGm mean?

The molar free energy of reaction, ΔrGm (as well as other Δ quantities in thermodynamics), is a funny quantity. We typically think of it as the difference in free energy between reactants and products, since that's how we calculate it. However, its meaning is a bit more subtle than that. To make things specific, imagine that we have a reaction

reactants → P + some other products

in a system held at constant temperature and pressure. ΔrGm represents the change in free energy when we convert one molar equivalent of reactants to one mole of P under specified conditions (concentrations, pressures etc.). In other words, the composition of the system should not change as our mole of P is made. However, if we carry out this conversion, we will change the quantities of reactants and products, and thus the composition.

Thermodynamics is one of the most useful pieces of science you'll ever learn. Given a little knowledge of thermodynamics, one can understand and explain a broad range of natural phenomena and technological applications. In this book, for instance, we will touch on energy use in living organisms, energy production by fuel-burning engines and spontaneity of chemical processes. This list of topics hardly scratches the surface of the phenomena that can be understood using thermodynamics.

When chemists say “thermodynamics,” they usually mean “equilibrium thermodynamics,” also known as “classical thermodynamics.” This is a theory that emphasizes transformations between equilibrium states. This is important because it is difficult to even define some important quantities like temperature for systems that are out of equilibrium. However, it does mean that we have to be careful how we apply the theory, which requires a certain precision of thought, language and notation. The issues dealt with in the rest of this chapter may seem pedantic or trivial to you, but they are critical prerequisites to understanding anything else in thermodynamics.

The domain of classical thermodynamics

Classical thermodynamics is a theory of equilibrium states and of transformations between these states. What do we mean by equilibrium? A system is in equilibrium if its state, under the given conditions, does not change with time. This may seem like an obvious definition, but in thermodynamics, even “state” has a technical meaning: the state of a system is the collection of all its observable properties (pressure, volume, chemical composition etc.).