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.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

where

dN/N = 4π(m/2πkT)3/2M exp(-mu2/2kT)u2du (1.4)

Thus the expression for E, clearly and understandably, contains the mass, m, and the velocity, u. Yet due to its mathematical form and since the integral is real (and is evaluated between these upper and lower limits) its value (3/2kT for motion in three dimensions, Equation 1.1) does not. The expression may look formidable but the integral has a standard form (it is a Gaussian function) and its evaluation is straightforward; see Appendix 1.1. It follows that the same result is obtained for any form of energy (not necessarily translational) that can be expressed in the same mathematical form, 1/2ab2, where 'a' is a constant and 'b' is a variable that can take any value within a Maxwellian distribution. Such a term is known as a 'squared term'.

From experience, gases are homogeneous and possess the same properties in all three directions in space; for example, the pressure is the same in all directions. The motion of the gas atoms in the three perpendicular Cartesian directions is independent and we say that they have three 'translational degrees of freedom'. Resolving the velocity into these directions and using Pythagoras gives, with obvious notation,

u2 = u2x + u2y + u2z (1.5)

with an analogous result for their means. In the gas the mean square velocities in the three directions are equal. The form of the Maxwell distribution we have used is that for motion in three dimensions and the average energy associated with each translational degree of freedom is consequently one-third of the value obtained. It then follows that for each degree of freedom whose energy can be expressed as a squared term we should expect an average energy of 1/2kT per atom. This is an important result of classical physics and is the quantitative statement of 'the Principle of the Equipartition of Energy'.

We stress that it has resulted from the Maxwell distribution in which there is no restriction of the translational energy that an atom (or molecule) can possess. From everyday experience this seems eminently reasonable. We can indeed make a car travel at a continuous range of velocities without restriction (and luckily personal choice of how hard we press the accelerator rather than collisions make a whole range possible if we consider a large number of cars!). But is this true of molecules that might possess other sources of energy besides translation? We should not assume so, but again put it to experimental test. We shall find later that we have to re-examine the case of translational energy too.

1.3 HEAT CAPACITIES OF DIATOMIC MOLECULES

The heat capacity, Cv, of a sample is directly related to its energy, and can be measured. We expect gaseous diatomic molecules, like atoms, to move freely in independent directions in space so that translational energy should confer upon the sample a heat capacity of 3/2R = 12.47 J mol-1 K-1. If this was the only source of energy that molecules possess then the heat capacity should have this value, and be independent of temperature. This turns out to be wrong on both counts. For example, the measured heat capacities (in J mol-1 K-1) of dihydrogen and dichlorine at various temperatures are given in Table 1.1.

All these values are substantially greater than expected from translational motion. Through the direct relationship between Cv and E this implies that there must be additional contributions to the energy of the sample. We note also that for each gas the value increases with temperature, with a tendency for it to become constant at high temperatures for dichlorine, and that over the whole range of temperatures in the table the heat capacity of dichlorine exceeds that of dihydrogen.

So what forms of energy can a diatomic molecule have that an atom cannot? The obvious physical difference is that in the molecule the centre of mass is no longer centred on the atoms. This implies that if there are internal motions in the molecule as it translates through space these have associated energies. The first, and most obvious, possibility is that the molecule might rotate. A sample containing rotating molecules might therefore possess both translational and rotational energy, and we need to assess the latter. The simplest, and quite good, model for molecular rotation is to treat the diatomic molecule as a rigid rotor (Figure 1.2) with the atoms as point masses (m1 and m2) separated from the centre of mass of the molecule by distances r1 and r2. Classical physics shows the rotational energy to be 1/2Iω2, where I is its moment of inertia and ω the angular velocity (measured in rad s-1). We immediately recognise this as a 'squared term'.

Rotation might occur about any of three independent axes which in general might have different moments of inertia, although for a diatomic molecule two are equal. Taking the bond as one axis (z), these are those about axes perpendicular to it through the centre of mass and their moments of inertia are defined by

Ix = Iy = m1r21 m2r22 (1.6)

We note that the distances are measured in the direction perpendicular to the axes of rotation, here along the z-axis. This implies that the moment of inertia for rotation about the z-axis is zero because the point masses and the centre of mass all lie on a straight line, and no perpendicular distance in the x or y directions separates them. We conclude that only two of the three rotational degrees of freedom contribute to the energy of the molecule, both through squared terms in the angular velocity. Using the Equipartition Principle we predict that their contribution to the energy will be 2 x 1/2RT J mol-1. This implies that, together with the translational contribution, the total energy of the molecule should be 5/2 RT J mol-1 and Cv should be 5/2R J mol-1 K-1 It should not vary as the temperature is changed.

This has the value 20.78 J mol-1 K-1, which, interestingly and significantly (see later), is very close to the value observed for dihydrogen at 350 K, but Table 1.1 shows Cv to increase with temperature. However, for dichlorine it is still much too low at this temperature compared with experiment. Once again we conclude that the actual energy is greater than we thought, and that the molecule must have another form of internal motion associated with it. This is vibration.

In a vibration the atoms continuously move in and out about their average positions (Figure 1.3). As they move outwards the bond is stretched, as would be a spring, and this generates a restoring force, which if Hooke's Law is obeyed is proportional to the displacement from the equilibrium positions, and the atoms return through these positions. In this model (also quite good) the molecule behaves as a simple harmonic oscillator with continually interchanging kinetic (KE, from the motion of the atoms) and potential (PE, from stretching the bond) energies. The total energy is the sum of the two, and is conserved in an isolated gas molecule.

Evib = (KE + PE)vib (1.7)

At any instant the kinetic energy is given classically by 1/2µv2 where µ is the 'reduced mass' of the molecule (defined as m1m2/(m1 + m2)) and v is the instantaneous velocity of the atoms, whilst the potential energy is 1/2kx2, where k is the bond force constant (Hooke's Law constant) and x is the instantaneous displacement from the average position of each atom. A diatomic molecule can only vibrate in one way, in the direction of the bond, but because of having to sum the contributions from both forms of energy this one degree of vibrational freedom contributes two squared terms to the total energy, through the Equipartition Principle, 2 x 1/2 RT J mol-1. Once again we have assumed that, in using this Principle, there are no limitations on (now) the vibrational energy that a molecule can possess.

The total energy of the molecule is, therefore, predicted to be the sum of the translational (3/2RT), rotational (RT) and vibrational (RT) contributions, giving 7/2RT J mol- and Cv = 7/2R = 29.1 J mol-1 K-1, greater than before but still independent of temperature. This is precisely the value obtained experimentally for dichlorine at 1000 K but it is much higher than that of dihydrogen at the same temperature. The heat capacities of both are still predicted, wrongly, to be independent of temperature.

It is now instructive to plot Cv against T for a diatomic molecule (shown diagrammatically in Figure 1.4). The value jumps discontinuously between the three calculated values, corresponding to translation alone, translation plus rotation and finally translation plus rotation plus vibration, over small temperature ranges (near the characteristic temperatures for rotation and vibration, θrot and θvib, Section 2.5.1). These temperatures depend on the precise gas studied, and the changes occur at higher temperatures for molecules consisting of light atoms than for those that contain heavy ones. Only at the highest temperatures are the values those predicted by Equipartition. But the contribution from translation alone is evident at temperatures close to absolute zero, but not extremely close to it when this contribution falls to zero. In this plot the translational contribution is easily recognised through its unique value but which of rotation or vibration contributes at the lower temperature is obtainable only through further experiment or theory; the rotational contribution appears at the lower temperature.