Some Basic Ideas and Examples

1.1 INTRODUCTION

Physical chemistry is widely perceived as a collection of largely independent topics, few of which appear straightforward. This book aims to remove this misconception by basing it securely on the atoms and molecules that constitute matter, and their properties. We shall concentrate on just two aspects and we focus mainly on thermodynamics, which although extremely powerful is one of the least popular subjects with students. A briefer account describes how reactions occur. We shall nevertheless encounter the major building blocks of physical chemistry, the foundations that, if understood, together with their inter-dependence, remove any mystique. These include statistical thermodynamics, thermodynamics and quantum theory.

The way that physical chemistry is taught today reflects the historical process by which understanding was initially obtained. One subject led to another, not necessarily with any underlying philosophical connection but largely as a result of what was possible at the time. All experiments involved very large numbers of molecules (although when thermodynamics was first formulated the existence of atoms and molecules was not generally accepted) and people attempted to decipher what happened at a molecular level from their results. This was very indirect. Nowadays the existence and properties of atoms and molecules are established and experiments can even be performed on individual atoms and molecules. This provides the opportunity for a different way of looking at the subject, building from these properties to deduce the characteristic behaviour of large collections of them, which is more in keeping with how chemistry is taught at school level. Similarly, our understanding of how reactions occur has come from observations of samples containing huge numbers of molecules and we have tried to deduce what happens at molecular level from them. Yet it is now possible to observe reactions between individual pairs of molecules, and we can reverse the procedure and start from these observations to understand reactions in bulk. It is the object of this book to demonstrate the possibility of a molecular approach to thermodynamics and reaction dynamics. It is not intended as an introduction to these subjects but rather is offered as an aid to understanding them, with some prior knowledge assumed.

We start with the properties of atoms and molecules as deduced from thermodynamic measurements and from spectroscopy. This is, paradoxically, the historical approach but it establishes straightaway that the properties are directly connected to the thermodynamics and it is artificial to separate the two. But once the connection is established we show how it can be exploited to give real insight into various problems. In this chapter we introduce the fact that the energy levels of atoms and molecules are quantised and use some simple ideas to establish the effectiveness of our general approach before proceeding to their origins in the second chapter.

1.2 ENERGIES AND HEAT CAPACITIES OF ATOMS

In the gas phase, atoms move freely in space and frequently collide, at a rate that depends upon the pressure of the gas. At atmospheric pressure (~105 N m-2) and room temperature they move approximately 100 molecular diameters between collisions, at average velocities about equal to that of a rifle bullet (300 m s-1). In elastic collisions some atoms effectively stop whilst others gain increased velocity (cf. collisions of billiard balls) so that instead of all the atoms having a single velocity they have a wide distribution of velocities. This is the familiar Maxwell distribution (Figure 1.1) that results from classical Newtonian mechanics. In it all velocities are possible but some are more probable than others. The most probable velocity depends upon the temperature, as does the width of the distribution.

A moving atom of mass m possesses a kinetic energy of 1/2mu2, where u is its velocity. Since in the whole collection of atoms in a gas there is no restriction to the velocity of an atom, there is no restriction to its energy either. Using the Maxwell distribution (see below), the average energy of an atom can be shown to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the mean square velocity of the atoms in the sample, k is Boltzmann's constant (k = R/NA is the universal gas constant and NA the Avogadro number) and T is the absolute temperature. To obtain the total energy, E, of a mole of gas we simply multiply by the total number of atoms, NA, and obtain (3/2)RT. This is the energy due to the motion (translation) of the atoms in the gas, the 'translational energy'.

Remarkably, although the kinetic energy of an individual atom depends upon its mass, the prediction is that the total energy of the gas in the sample does not. It seemed so outrageous when first made that it had to be tested, but how? We have calculated the absolute quantity, E, but have no way of measuring it directly. But there is a closely related property that we can measure. This is the heat capacity of the system, defined as the amount of heat required to raise the temperature of a given quantity of gas (here 1 mole) by 1 K. Different values are obtained if this measurement is made keeping the volume of the gas constant (with a heat capacity defined as Cv) or keeping its pressure constant (Cp) since in the latter case energy is expended in expanding the gas against external pressure. Here we consider just what is happening to the energy of the gas itself, and must use the former. Writing the definition in mathematical form, as a partial differential (a differential with respect to just one variable, here T),

CV = ([partial derivative]E/[partial derivative]T)v = ([partial derivative](3/2RT)/[partial derivative]T)v = 3/2R = 12.47 J K-1 mol-1

The subscript on the bracket reminds us that we are dealing with a constant volume system.

This is again remarkable. It says that for all monatomic gases, regardless of their precise chemical nature or mass, the molar heat capacity is the same, and independent of temperature. Experiment shows this to be correct. For example, He, Ne, Ar and Kr were early shown to have precisely this value over the temperature range 173–873 K, the range then investigated.

It is now worth examining in more detail where the result that the mean energy of a monatomic gas is independent of its nature comes from. To obtain this average over the whole range of velocities we must multiply the kinetic energy of an atom at a given velocity by the probability that it has this velocity, and integrate over the whole velocity distribution normalised to the total number of atoms present. This probability function (dN/N) is the Maxwell distribution of velocities.

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