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Molecular Physical Chemistry
A Concise Introduction
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Molecular Physical hemistry
A
oncise
Introduction
K A
McLauchlan
University
o
Oxford
U K
dv ncing
the chemic l
sciences
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ISBN 0-85404-619-4
catalogue record for this book is available from the British Library
he Royal Society of Chemistry 2004
ll rights reserved
Apart fr om air dealing-for the purposes
o
research fo r non-commercial purposes or fo r
private stu dy criticism or review as permitted under the Copyrig ht Designs and Pa tents
Act
1988
and the Copy right and Related Righ ts Regulations 2003 this puhlicution m ay
not be reproduced stored or transm itted in an y or m or by any means without the prior
permission in writing of Th e Ro yal Society of Chemistry or in the case of reproduction in
accordance with th e terms of licences issued by the Cop yrigh t Licensing Age ncy in the U K
or in accordance with the terms o f t h e licences issued by the appropriate Rep rodu ction
Rights Organization outside the U K . Enquiries concerning reproduction outside th e terms
stated here should be sent t o T he Royal Socie ty of Chem istry at the address printed on this
Page.
Published by The Royal Society
of
Chemistry
Thomas Graham House Science Park Milton Road
Cambridge CB4
OWF
UK
Registered Charity Number 207890
or
further information see our web site at www.rsc.org
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Printed by TJ International Ltd Padstow Cornwall UK
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reface
To
write any b ook is an indulgence and my excuse for having do ne
so
is
tha t have tried to provide underg raduates with a text that differs in
approa ch from any other I kn ow o n its subject ma tter. It is an attempt to
provide the reader with a n understanding of thermodynamics and to a
lesser extent) reactions based upo n ato m s and molecules an d their pr op-
erties rather th an o n one based up on the historical development of these
subjects. Th is is possible as a result
of
innovative experiments performed
on very small numb ers of ato m s an d molecules that have been performed
in the last decade o r so
This book makes no pretence to be a primary source of its subject
matter. Rather it attempts to give molecular insight into the familiar
equ ation s of therm odyna mics, for example, and should be read in con-
junctio n with the excellent Physical Chem istry texts th at a lready exist. It
is an aid to understanding, no more and n o less. But hope that those
deterred by the elegant but possibly dry approaches found in other
books, which develop the subject without the properties of molecules
considered, will find this m ore to their liking. Th e subjects are imp ortan t
to the whole understand ing
o
Physical Chem istry an d provide the unde r-
lying philosophical structure th at binds its apparently separate subjects
together.
I a m indebted to all my students
or
what they have taught me an d for
the sheer pleasure of kno w ing them . But my g reatest deb t is to Jo an , for
everything imp ort an t in my life.
V
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Contents
Chapter Some Basic Ideas and Exam ples
1 1 Introduction
1.2 Energies and Heat Capacities of Atom s
1.3 Heat Capacities of Diatomic M olecules
1.4 Spectroscopy an d Qu antisation
1.5 Summary
1.6 Fu rthe r Implications from Spectroscopy
1.7 Nature of Quantised Systems
1.7.1 Boltzmann Distribu tion
1.7.2 Two level Systems
1.7.3 Two level Systems with Degeneracies Gr eate r tha n
Unity; Halogen Atoms
1.7.4 Molecular Exam ple: NO gas
Appendix 1 1 The Eq uipartition Integral
Appendix 1.2 Term Symbols
Problems
Chapter
2
Partition Functions
2.1 Mo lecular Partition Fun ction
2.2 Boltzman n Distribution
2.3 Canonical Partit ion Fun ction
2.4 Sum mary of Partition Fun ctions
2.5 Evaluation of Molecular Partition Functions
2.5.1 Electronic Pa rtition Fu nc tion
2.5.2 Vibrational Partition Fu nction
2.5.3 Th e Rotational Partit ion Function
2.5.2.1
2.5.3.1
2.5.3.2
Vibrational Heat Capacity
of
a Diatomic Ga s
Ro tational Heat Capacity of a Diatomic G as
Ro tational Partition Functions in the Liquid
State
1
2
5
10
14
14
16
18
19
22
24
26
26
28
3
30
33
36
39
39
40
41
44
46
51
53
vii
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viii
Contents
2.5.4
Translational Partition Function
54
2.5.4.1 The Translational Heat Capacity
55
Appendix
2.1
Units
57
Problems 58
2.6
Overall Molecular Partition Function for a Diatomic Molecule 56
Chapter
3
Thermodynamics
3.1 Introduction
3.2 Entropy
3.3
3.4 State Functions
3.5
Thermodynamics
3.5.1 First Law
Entropy at 0 K and the Third Law
of
Thermodynamics
3.5.1.1
3.5.2 Second Law
Thermodynamic Functions and Partition Functions
Absolute Entropies and the Entropies of
Molecular Crystals at 0 K
3.6 Free Energy
3.7
3.8 Conclusion
Problems
Chapter Applications
4.1 General Strategy
4.2 Entropy
of
Gases
4.2.1
Entropies of Monatomic Gases
4.2.2 Entropy of Diatomic and Polyatomic Molecules
Two level Systems; Zeeman Effects and Magnetic Resonance
4.3.1 Internal Energy of Two level Systems
4.3.2
Curie Law
Pauli Principle and Ortho and Para hydrogen
4.6.1 Isotope Equilibria
4.3
4.4 Intensities of Spectral Lines
4.5
4.6 Chemical Equilibria
4.7 Chemical Reaction
4.8 Thermal Equilibrium and Temperature
Problems
Chapter
5
Reactions
5.1 Introduction
5.2 Molecular Collisions
5.2.1 Collision Diameters
5.2.2 Reactive Collisions
9
59
59
61
63
63
64
61
69
70
72
74
74
77
77
77
78
79
80
80
81
82
84
88
92
93
97
99
1 1
101
101
102
105
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Contents
ix
5.3
Collision Theory
5.4 Energy Considerations
5.4.1
Activation Energy
5.4.2
Disposal
of
Energy and Energy Distribution in
Molecules
5 5 Potential Energy Surfaces
5 6
Summary
Problems
Answers to Problems
Some Useful Constants and Relations
Further Reading
108
110
110
111
112
115
116
8
2
22
Subject Index
23
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CHAPTER
Some Basic Ideas and Examples
1.1
INTRODUCTION
Phy sical chem istry is widely perceived a s a collection of largely indepen-
dent topics, few of which appear straightforward. This book aims to
remove this misconception by basing it securely on the atom s a n d mol-
ecules that c onstitute matter, an d their properties. W e shall concen trate
o n just two aspects and we focus mainly o n thermodynam ics, 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 chem istry,
the foun dation s that, if unde rstood , togethe r with their inter-dependence ,
remove any mystique. These include statistical thermodynamics, ther-
mody namics and qua ntum theory.
T he way t ha t physical chem istry is tau gh t tod ay reflects the historical
process by which und erstan ding w as initially obtain ed. On e 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 ther-
modynamics was first formulated the existence of atoms and molecules
was not generally accepted) and people attempted to decipher what
hap pen ed a t a mo lecular level from their results. Th is was very indirect.
Nowadays the existence and properties of atoms and molecules are
established and experiments can even be performed o n individual atom s
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 -hu ge num bers of m olecules a nd we have tried to
deduce w hat h app ens a t molecular level from them. Yet it is now possible
to o bserve reactions between individual p airs of molecules, an d we can
reverse the procedure and start from these observations to understand
1
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Chapter
1
reactions in bulk. It is the object of this boo k to d em ons trate th e possibil-
ity of a molecular app roac h t o thermody nam ics an d reaction dynam ics.
It is not intended as a n introduction to these subjects but rathe r is offered
as an aid to understand ing them, with some prior know ledge assumed.
W e start with the properties of ato m s and molecules as deduced from
thermod ynam ic measurem ents an d from spectroscopy. This is, paradoxi-
cally, the historical approach but it establishes straightaway that the
properties are directly connected t o the thermod ynam ics an d it is artifi-
cial to separate the two. B ut once the conne ction is established we show
how it ca n be exploited to give real insight into various problem s. In this
chapter we introduce the fact that the energy levels of atoms and mol-
ecules are quantised an d use som e simple ideas to establish the effective-
ness of our general approach before proceeding to their origins in the
second chapter.
1.2
ENERGIES A N D HEAT CAPACITIES O F ATOMS
In the gas phase, ato m s move freely in space a nd frequently collide, at a
rate tha t de pends up on the pressure of the gas. At atmo spheric pressure
l o 5
N
m-2) and room temperature they move approximately
100
mo lecular diam eters between collisions, a t average velocities abo ut eq ual
to that
of
a rifle bullet
(300
m
s-l).
In elastic collisions some atoms
effectively stop whilst others gain increased velocity cJ: collisions of
billiard balls) so that instead of all the a to m s having a single velocity they
have a wide distribution
of
velocities. This is the familiar Maxwell dis-
tribution (F igure
1.1)
ha t results from classical Ne wto nian mechanics. In
it all velocities are possible but som e are more probable than others. The
mo st probable velocity depends upon the temperature, as does the width
of the distribu tion.
A moving atom
of
mass m possesses a kinetic energy of +mu2,where u is
its velocity. Since in the whole collection of atoms in a gas there is no
restriction t o th e velocity of a n a to m , there is no restriction to its energy
either. Using the Maxwell distribu tion (see below), the average energy of
an ato m can be shown to be
where
2
s the mean square velocity of the atoms in the sample, k is
Boltzmann’s co nstan t
k
=
R/N,
where
R
is the universal gas constant
and NA the Avogadro number) and T is the absolute temperature. To
obta in the to tal energy, E , of a mole of gas we simply multiply by th e to tal
number
of
atoms,
NA,
an d obta in
( )RT.
This is the energy due to the
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Some Basic Ideas and Exam ples
3
Number of molecules
1000
2000
Velocity/rns-
Figure 1.1 The Maxwell distribution of velocities of molecules in a gas at 273 and 1273 K .
A s
the temperature is increased th e most probable velocity moves t o a higher
value and the distribution widens, reflecting a greater range o molecular velo-
cities.
m otion (translation ) of the a tom s in the gas, the 'translation al energy'.
Remarkably, although the kinetic energy of an individual atom de-
pen ds up on its mass, the prediction is tha t the to tal energy of the gas in
the sample does not.
It
seemed so outrageous w hen first made tha t it had
to be tested, but how? We have calculated the abso lute 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 am ou nt of heat required to raise the tem peratu re of a given
qu an tity of gas (here
1
mole) by 1 K. Different values are ob tain ed if this
measurement is made keeping the volume of the gas constant (with a heat
cap acity defined as C,) o r keeping its pressure co nstan t (C,) since in th e
latter case energy is expended in expanding the gas against external
pressure. H ere we consider jus t w hat is hap pen ing 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
variab le, here T),
(1.2)
, =
g),
lT),
( R
T,
= R
= 12.47
J
K - mol-I
The subscript on the bracket reminds us that we are dealing with a
constant volume system.
This is again rem arkable. It says that for all mo natom ic gases, regard-
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4
Chapter 1
less of their precise chemical natur e o r mass, the m ola r heat capacity is
the same, a n d independen t of temp erature. Experiment show s 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 wo rth exam ining in more detail where the result tha t the m ean
energy of a m ona tom ic gas is indepen den t of its natur e com es from. T o
ob tain this average ove r the whole range of velocities we m ust mu ltiply
the kinetic energy of a n a to m a t a given velocity by the prob ability tha t it
has this velocity,
normalised t o the
tion (dN /N) is the
where
and integrate over the whole velocity distribution
total number of atoms present. This probability func-
Maxwell dis tribu tion of velocities.
Th us the expression for E , clearly and understanda bly, contains the mass,
m,
and the velocity, u. Yet due to its mathematical form and since the
integral is real (an d is evaluated between these up per an d low er limits) its
value ($kT for m otion in three dimensions, Eq uatio n 1.1) does no t. The
expression m ay look formidable bu t th e integral has a stand ard form (it is
a Gaussian function) and its eva luation is straightforward; see Appendix
1.1. It follows tha t the sam e result is obta ined for any form of energy (no t
necessarily translational) that can be expressed in the sa me m athem atical
form,
+ab2,
where ‘a’ is a co ns tan t an d
‘b’
is a variable tha t can take any
value within a Maxwellian distribution. Such a term is known as a
‘squared term’.
F ro m experience, gases are hom ogene ous and possess the sam e prop-
erties in all three directions in space; for example, the pressure is the sam e
in all directions. The mo tion of the ga s ato m s in the three pe rpendicular
Cartesian directions is independent and we say that they have three
‘tran slatio na l degrees of freedom ’. Resolving the ve locity in to these direc-
tions and using P ytha gora s gives, with obvious no tation,
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
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Some Basic Ideas and Examples
5
distribution we have used is tha t for m otion in three dimensions an d 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
shou ld expect an average energy of
i k T
per atom . This is an im portant
result of classical physics and is the q uantita tive statem ent of 'the Prin -
ciple of the Eq uip artitio n of Energy'.
W e stress that it h as resulted from the Maxwell distribution in which
there is no restriction of the translational energy that an ato m (o r mol-
ecule) can possess. F ro m everyday experience this seems em inently rea-
sonable. We can indeed make a car travel at a continuous range of
velocities w itho ut restriction (an d luckily perso nal cho ice of how ha rd we
press the accelerator ra the r th an collisions m ake a w hole range possible if
we consider a large number of cars ). But is this true of molecules that
migh t possess othe r sources of energy besides translation ? We sh ould no t
assume
so,
but again put it to experimental test. We shall find later that
we have to re-examine the case of translation al energy too.
1.3
HEAT CAPACITIES
OF
DIATOMIC MOLECULES
Th e heat capacity, C,, of a samp le is directly related to its energy, an d can
be measured. We expect gaseou s diatom ic molecules, like atom s,
to
move
freely in independent directions in space
so
that translational energy
should confer upo n th e sample a heat capacity
of R
= 12.47
J
mol-' K- .
If this was the on ly source of energy tha t m olecules possess then the hea t
capacity shou ld have this value, and be independe nt of tempe rature. This
turns ou t to be w rong on both counts. Fo r example, the m easured heat
capacities (in
J
mol-1
K-') of
dihydrogen and dichlorine at various
tempe ratures are given in Ta ble 1.1.
All these values are substantially greater than expected from transla-
tional motion. Through the direct relationship between C, and
E
this
implies tha t there must be additional con tributions to the energy of the
sample. W e no te also tha t for each gas th e value increases with tem pera-
ture, with a tendency for it to becom e consta nt a t high tem peratures for
Table
1.1
He at capacities ( J mol-'
K - ' )
of dihydr ogen and dichlorines at
diflerent tem peratures
T ( K )
Molecule
298 400 600 800 1000
20.52
20.8
7 21.01 2 1.30
21.89
c 2 25.53 26.49
28.29
28.89
29.10
H2
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6 Chapter
1
dichlorine, an d t ha t over the whole range of temp erature s in the table the
heat capac ity of dichlorine exceeds tha t of dihyd rogen.
So what forms of energy can a diatom ic molecule have that an ato m
can not? Th e obv ious physical difference is th at in the m olecule the ce ntre
of mass is n o longer centred on the atom s. This implies that if there a re
internal m otion s in the m olecule a s 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 a n d rotatio nal energy, and we need
to assess the latter. The simplest, and quite goo d, model for molecular
rotation is to trea t the diatom ic molecule as a rigid roto r (Figure 1.2)with
the atom s as point masses
( m ,
and
m,)
sepa rated from the centre of mass
of the molecule by distances r l and r 2 . Classical physics shows the
rotation al energy to be
02,
where
I
is its mom ent of inertia an d ci the
ang ular velocity (measu red in rad
s-').
We imm ediately recognise this as a
'squared term'.
Rotation might occur about any of three independent axes which in
general might have different m om ents of inertia, althou gh for a d iatom ic
molecule two are equal. Taking the bon d as on e axis
( z ) ,
hese are those
about axes perpendicular to it through the centre of mass and their
m om ents of inertia a re defined by
(9
(ii) (iii)
Figure
1.2 Rota tions o j a diatomic molecule. T he atoms are treated as point masses, m , and
m2,
with their centres lying along the axis
of
the molecule with th e distances
r l
and r2 measured hetween these centres and the centre of mass ( C M ) of the
molecule. The molecule can r otate about three axe s, in the plane of the paper (i),
about the bond axis
(ii)
and out o the plane
(iii).
The moments o inertia for
rotations
(i)
and
(iii)
are non-zero and equal but the moment of inertia o r rotation
(ii)
is zero since there is no perpendicular distance between the point masses and
the C M along the bond axis. This rotation therefore does not contribute to the
total rotational energy
o the molecule.
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Some Basic Ideas and
Examples
7
We note th at the distances are measured in the direction p erpendicular to
the axes of rota tion , here along th e z-axis. This implies that the mo m en t
of
inertia for rotation ab ou t the z-axis is zero because the point masses and
the centre of mass all lie on a straigh t line, an d n o perpendicular distance
in the
x
or
y
directions separates them. We conclude th at only two of the
three rotational degrees of freedom contribute to the energy of the
molecule, both throu gh squared terms in the an gular velocity. Using the
Equipartition Principle we predict that their contribution to the energy
will be
2 x i R T J
mol--'. Th is implies th at , toge ther with the translation al
con tribution , the tota l energy of the molecule sho uld be
:RT J
mol-land
C,
should be
R
J mol-' K-l. It should no t vary as the tem perature is
changed.
This has the value
20.78 J
mol-I
K-',
which, interestingly and signifi-
cantly (see later), is very close to the v alue ob served for dihyd rogen a t 350
K, but Table 1.1 shows
C,
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
interna l mo tion associated w ith it. Th is is vibratio n.
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, an d this gen erates a restoring force, which
i Equilibrium
position
+ f - -
Figure 1.3 Vibr ation of a diatomic molecule. T he diagram shows at th e top the (point mass)
atom s at their distance o closest approach when the y start to move apart again,
in the centre at their average positions, and at the bo ttom when the bond
is
fullest
stretched and the elasti city
o
the bond brings the a tom s back towards each other
once more. A t the two extrem e positions the a toms are momentarily stationary
and the molecule possesses the potential energy obtained from stretching
or
compressing the bond o nly, but they then start to move, transforming potential
energy into kinetic energy, a process complete just
as
the atoms pass through
their equilibrium positions.
I f
the bond stretching obeys Hooke's Law (the
restoring for ce generated by moving aw ay fr om the equilibrium position is
proportional to the distance moved) then Simple Harmonic M otio n results.
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8
Chapter 1
if Hooke's Law is obeyed is prop ortion al t o the displacement from the
equilibrium positions, an d the ato m s return th roug h these positions. In
this model (also quite good ) the m olecule behaves as a simple harm onic
oscillator with co ntinually interch ang ing kinetic
(KE,
from the m otio n of
the atoms) and potential (PE, from stretching the bond) energies. The
total energy is the sum of the two, a nd is conserved in a n isolated gas
molecule.
Evib= ( K E + PQ
At a ny instan t the k inetic energy is given classically by m2 where ,u is the
'reduced mass' of the molecule (defined as
m, m2/ (m ,
+
m2))
nd
v
is the
instan taneo us velocity of the a tom s, whilst the potential energy is i k x 2 ,
where
k
is the bo nd force con stant (Hooke's Law co nstan t) an d 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
bon d, but because
of
having to sum the contributions from bo th forms of
energy this one degree
of
vibrational freedom contributes two squared
terms to the total energy, throu gh the Equ ipartition Principle, 2 x
RT J
m ol? On ce again we have assumed that, in using this Principle, there
are no limitations on (now) the vibrational energy that a molecule can
possess.
Th e total energy of the molecule is, therefore, pred icted t o be the su m of
the translational (;RT), rotational (RT)and vibrational (RT)contribu-
tions, giving ZRT
J
mol- ' and
C,
=
ZR
=
29.1
J
mol-'
K- ,
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
tha n tha t of dihydroge n at the same temperature. The heat capacities of
bo th are still predicted, wrongly, to be indep end ent of temp erature .
It is now instructive to plot C, against T for a diatomic molecule
(shown diagrammatically in F igure 1.4).The value jum ps discontinuously
between the three calculated values, corresponding to translation alone,
translation plus rotation an d finally translation plus rotation plus vibra-
tion, over small temp erature ranges (near the characteristic tem peratures
for rotation an d vibration, Or and
Ov ib
Section
2.5.1).
These temp eratures
depend on the precise gas studied, and the changes occur at higher
tem peratu res for molecules consisting of light ato m s tha n for those t ha t
contain heavy ones. Only a t the highest tempe ratures are the values those
predicted by E quipartition. But the contribution from translation alone
is evident at tem peratu res close to abso lute zero, bu t n ot 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
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Som e Basic Ideas and Examp les
9
Figure 1.4
rotation
3 5
2.5
1.5
Cvf R
Vibration
otation
Translation
ii
293
0
ot
TIK
Schematic diagram of how
C,
for a diatomic molecule varies with temperature. I t
rises sharply fr om near 0 K and soon reaches 3R/2, expected for translational
mo tion, where it remains until a higher temperature (near O r , , is reached when
the rotational degrees offre edo m contribute another
R
to th e overall value. T his
happens well below room temperature (293 K ). fo r all diatomic molecules. A t still
higher temperatures the vibrational motion eventually contributes another
R,
again near a rather well-defined temperature (
O v i b .
Th e actual values
of
these
temperatures vary with t he precise molecule concerned, and are lower fo r heavy
molecules (e.g. Cl ,) than light ones
(e.g. H 2 ) .
Th e beginnings of the vibrational
contribution occur below room temperature for C1, but the u ll contribution is not
apparent until well above it.
or vibration con tributes at the lower temp erature is obtainable
only through further experiment or theory; the rotational contribution
appea rs at the lower temperature.
W e conclude tha t molecules exhibit very different behaviour from th at
we predicted using classical theory an d we m ust exam ine where we might
have gon e wrong. All the basic equations for rotation al an d v ibrational
energy are well established in classical physics and ar e no t as sum ptions .
O n e possibility might be th at the rotationa l an d vibrational energies are
correctly given classically but that they do not have Maxwellian dis-
tributions. But also we have made what in the classical world seems a
wholly unexceptional assu mp tion,
i.e.
that there are no restrictions a s to
the energies a molecule m ight possess in its different degrees of freedom.
Since the predictions d o not conform t o the experimental observations
this might be wrong.
W e speculate that rather th an being able to possess
any
values of their rotatio na l an d vib rationa l energies, the molecule may
be able to possess only specijic values of them . Th is is confirmed directly
by spectroscopy, see below. W e describe the energies as ‘quantised’. This
is the basic realisation from which mu ch
of
physical chemistry flows.
Translational energy seems no t to be quantised bu t ac tually is; experi-
me nts have to be performed a t very close to 0
K
to observe this. It is why
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10 Chapter
1
C, in Figure
1.4
goes to zero a t this tem peratu re. The fact th at different
diatom ic molecules possess the tw o further forms of energy ab ove cha rac-
teristic temp erature s th at differ from o ne molecule to the next is an aspect
of energy-quantised systems that we shall have to understand. But all
systems behave classically at high enough temperatures, which may,
however, be below room temp erature. F o r example, all diatomics (save
H2)exhibit their full rotational contribution below room temperature.
But only the heaviest molecules exhibit the full vibratio nal co ntributio n
below very high temp erature s.
T ha t the lim iting classical beh aviou r is often observed in real systems,
especially for po lyatom ic molecules, is ultimately w hy we are norm ally
unaw are of the quantised natu re of the world th at su rroun ds us. But the
world is quantised in energy and we need t o und erstand an d exploit the
properties
of
matter that this implies. Much of the new technology in
everyday use depends o n it.
It
is fascinating and significant that this conclusion of paramount
importance was indicated as a possibility through the interpretation of
classical thermod ynam ic measurements, emphasising tha t a conne ction
exists between the therm ody nam ics of systems an d the prope rties of the
individual atom s and molecules tha t com prise them.
1.4
SPECTROSCOPY
AND QUANTISATION
Q ua nt isa tio n of energy show s itself very directly in the optica l sp ec tra of
atoms and molecules and initially we consider the electronic spectra of
atom s. Wh en a sam ple of ato m s is excited in a flame, for example, it emits
radiation to yield a ‘line spectrum’ (Figu re 1.5). This is in fact a series of
images of the exit slit in a spectrometer corresponding to a series of
different discrete frequencies of light emitted by the atoms. If the atom
behaved classically this would n ot be
so
since the electrostatic a ttrac tion
between the electron an d nucleus w ould accelerate one tow ards the other,
an d acc ording to classical physical laws the ato m would em it light over a
continuous frequency range until the electron was annihilated on en-
cou ntering th e nucleus. Th e explan ation is tha t the energies possible for
the electrons in a n a tom are themselves quantised, and the frequencies in
the emission spe ctrum co rrespond to the electron jump ing between levels
of different energy, acco rding to the B ohr co ndition ,
where h is Planck’s con stant,
v
the frequency of the light and
E ,
and
E ,
are the energies of tw o of the levels (Fig ure
1.6).
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Some Basic Ideas and Examples
11
Figure 1.5
Figure
1.6
120
100
k/nm
Ultraviolet region of the emission spectrum
of
hydrogen atoms (the Lyman
series). Atoms in the sample have been excited by an electric discharge to a
number of higher atomic energy levels and they emit light at digerent discrete
,frequencies as the electrons return t o the lowest level (principle qua ntum number,
n
=
. Th is displays the quantised nature o the atomic energy levels directly.
Other series
of
lines are observed in the visible and infrared regions
of
the
electromagnetic spectrum, corresponding to electron jumps to diferent lower
quantised states. The positions
of
the lines allow the frequ encie s
of
the emitted
light t o be measured.
The origin of a line in a spectrum o frequency
v.
In a H atom at thermal
equilibrium with its surroundings the electron is in the lower energy level and it
can absorb the specific amount of energy hv to ju m p t o a higher energy level. I f
however, the atom has been excited
so
as to put the electron in the upper energy
level then it can emit the sume energy and return t o the lower level. I n a real atom
there are many pairs of energy levels between which spectroscopic transitions can
occur.
An early trium ph of the Schrod inger eq ua tion was tha t it rationalised
why the levels are quantised and allowed the energies to be calculated,
with the difference frequencies agreeing with the observed ones.
Asso-
ciated with each level is a mathematical function, the wave function,
known as an orbital .
The C, measurements discussed above gave no indication tha t m on-
atom ic gases could possess electronic energy besides translationa l energy.
This must m ean that
ouer
the
temperature range studied
the atoms do not
hav e any. But we have seen in Fig ure 1.4 th at d ifferent sources of energy
m ake their contributions a t different tempe ratures and at a high eno ugh
temp erature there would indeed be an electronic con tribution to the
C ,
of atom s. We shall see later th at the crucial factor is the energy sepa ration
between the qu antised energy levels com pared with th e ‘thermal energy’
(given by
k T )
in the system, and the lower atom ic orbitals are separated
from each other by large energy gaps. For some atoms, such as the
halogens (see below) an electronic contribu tion is evident even a t q uite
low temperatures, bu t this is rathe r unusual.
In molecules, too, discrete energy levels and molecular orbitals exist
with differing electronic energies. But associated with each and every
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12 Chapter
1
electronic level there is a set of vibrational and rotational ones (Figure
1.7).
Th roug h the Bo hr condition spectroscopy allows us to measure the
energy gaps directly, whilst q ua ntu m mechanics also allows us to calcu-
late the energy levels, an d the ga ps between them. T his confirms tha t the
energies molecules possess are quan tised. Molecular sp ectra are norm ally
observed in a bso rption , with the m olecules absorbing certain frequencies
from white light flooding through a sample of them. Whereas emission
spec tra arise w hen electrons fall from higher energy levels into w hich the
substance has been excited by heat or electricity, absorption spectra
dep end on species in their lower energy levels jum ping to h igher ones.
They therefore give inform ation o n th e lowest energy levels of the mo l-
ecules. Experim entally a huge range of trans ition frequencies is invo lved,
varying from th e m icrowave (far infrared) to the u ltraviolet regions of the
electromagnetic spectrum. The former occur between energy levels that
are closest in energy, those d ue t o rotatio n. At higher frequencies, in the
infrared, the light has sufficient energy to cause jumps between vibra-
tional levels, bu t these all have m ore closely spaced associated rota tion al
ones, an d under certain selection rules changes occur to both the vibra-
tional an d rotatio nal energies simultaneously. Th e spectra are know n as
‘vibration-rotation’ ones (an example is given in F igure
2.6,
Chapter 2).
Finally, at mu ch high er frequencies, in the ultraviolet, electrons can ju m p
between the vibrational and rotational levels of different electronic en-
ergy states and the sp ectra reflect sim ultaneous changes in all three types
ot
energy.
Through Equation (1.8) there is a direct relationship between fre-
quency an d energy difference an d we conclude tha t in terms of the gaps
between the energy levels
AE(e1ectronic)>> AE(vibration) > AE(rotation)(and
>>
AE(trans1ation)) (1.9)
This confirms o u r conclusions from
C,
measurements (Figure
1.4).
At
0 K
Cv is zero, but as the temperature is increased translational motion
rapidly makes the contribution expected from classical physics. At a
higher temperature the effects of rotation become apparent, and at a
higher one still, vibration (and at very high temperatures electronic
energy contributions might appear if dissociation does not take place
first).
M oti on s corresponding to th e lower energy gaps make their contribu-
tions at lower temperatures than those with higher energy gaps.
It is important to distinguish between the absolute values of the
various types of energy and the separations between the energy levels.
Th us the g aps between tran slationa l levels are miniscule but even at very
low temperature there is a contribution of ( )RT mol-‘ to the energy.
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Some
Basic
Ideas and Examples
13
Energy
I
Electronic
Vibrational Rotational
Figure
1.7
Schematic diagram o the energy levels o a diatomic molecule showing on ly the
two lowest electronic levels, which are widely separated in energy. Associated
with each is a stack o more closely spaced vibrational levels, which in turn
embrace a series o rotational levels. Under the Simple Harmonic oscillator
app rox imation, the vibrational levels of each electronic level are equally spaced,
but the rotational levels predicted by the rigid rotor model are not. T his diagram
gives an impression
of
the relative sizes
o
the electronic, vibrational and rota-
tional energies but o nly a very f e w o f t h e lower energy levels can be shown here
without loss o clarity . Rea l molecules have fa r more levels. Spectroscopic
transitions can be excited between the rotational levels alone, between the
rotational levels in diferent vibrational states and between the sub-levels
o
the
electronic energy sta tes. T he three ty pes occur with very diflerent energies and
are observed in quite di fer ent regions of the electromagnetic spectrum.
Th e gaps between electronic energy levels are en orm ous in co mp arison,
yet electronic energy makes no contribution to the to tal energy of most
systems a t room temperature.
A
com m on er ror is to believe that since the
energies of molecular electronic levels may be high then the electronic
energy of the system must be high. If the molecule existed in one of the
higher levels the energy would indeed be high. But it does not at low
temperatures, where the molecule is in its lowest electronic level. F o r a
gas at room tempe rature the translational energy is greatest in magn itude
followed by the rotation al energy and then the v ibrational one. The o rder
is exactly reversed from th at
of
the energy gaps:
E(translationa1)
>
E(rotationa1)
>
E(vibrationa1)
>>
E(e1ectronic))
(1.10)
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14
Chapter 1
1.5 SUMMARY
Atom s an d m olecules may possess several different con tribution s t o their
total energy b ut each one ca n have only certain discrete quantised values.
Th e energies associated w ith different m ode s of mo tion differ in m agn i-
tude, with the g ap s between the energy levels for translation being sm aller
than those for rotation that, in turn, are smaller than those for vibration.
The gaps between the electronic energy levels of molecules normally
vastly exceed all of these. This causes translational motion to occur at
lower tem peratures than rotation al mo tion, which occurs a t lower tem-
peratures t ha n vibration (and much lower than electronic excitation). Yet
we remain u naw are of quantisation in every day life, an d we have som e
indication already that this is inherently because systems behave classi-
cally at high en ough tempe ratures. W e need to sha rpen this concept and
decide w hat a ‘high enough ’ tem pera ture is, an d then we sho uld be able to
predict the behaviour
of
systems from a knowledge of the quantised
energy levels of the at om s an d m olecules from w hich they are m ade .
1.6
FURTHER
IMPLICATIONS FROM SPECTROSCOPY
We are so familiar today with spectra that we tend to miss a very
remarkable fact about them. An atomic spectroscopic transition in ab-
sorptio n, for example, occurs when an ato m in a specific energy level
accepts energy from the radiation an d jum ps to ano ther specific level, in
accordance with the Bohr condition and under various selection rules
th at limit the transitions th at a re possible. These have been discovered by
experiment, and can be rationalised using quantum mechanics.
So
the
spec trum of a single at om und ergo ing a single transition is a single line at
one specific frequency. But w hen we talk abo ut the spec trum of a n ato m
(a loose term ) we imm ediately think of a w hole family of transition s a t
different frequencies (Figu re
1.5).
Since the electron in on e ato m ca n only
m ake one jum p between energy levels a t a time it follows that wha t we see
is th e result of a large number of individual atom s simultaneously absorb -
ing energy and jump ing t o a whole range of possible q ua ntu m states. Th at
is, instead of seeing the sp ectrum of a single ato m , we a re observing the
spectra of a very large number of individual atoms simultaneously.
We
say that spectroscopy is an
ensemble phenomenon,
meaning precisely tha t
w hat we observe is the result of wh at
is
hap pen ing in the large collection
of atoms.
But w hat w ould h ap pen if we were clever eno ugh t o observe a single
atom over
a
long period of time, rath er tha n instantaneously? Following
the initial abs orptio n of energy from the light beam , the a tom enters a
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Some Basic Ideas and Exam ples
15
higher energy level. Let us now postulate that an efficient mechanism
exists (it does ) for returnin g it t o the lowest level, where it m ight a bs or b a
different frequency from the incident radiation and attain a second,
different, higher level. It w ould then retu rn an d the process be repeated
over a whole cycle of possible transitions covering the total range of
frequencies. We conclud e th at over infinite time the a to m w ould perform
all the possible transitions allowed to it and the spectrum of the single
atom would be identical in appearance to the ensemble one. This has
been pu t to d irect experimental test in recent years, alth ou gh in mo lecular
rather than atomic spectroscopy, and found to be correct. It opens the
possibility of calculating ensem ble behavio ur of a collection of molecules
from the b ehaviou r over time of a single one, a concept close to the basis
of using ensembles in statistical therm ody nam ics (see later).
A noth er unexpected asp ect of spectroscopy lies in the Bo hr cond ition.
We tend to think of atoms or molecules absorbing energy from an
incident light beam and jumping to higher energy states, but Equation
(1.8) does n ot dictate a direction for the energy change to occur. T h a t is,
whilst an atom in a lower energy state might jump to a higher energy
state, one in th at s tate might em it energy unde r the influence of the light
beam , and fall back t o the lower state. In this case the beam would exit
more intense than it arrived. These processes are known as stimulated
absorption and stimulated emission respectively. Einstein w as th e first t o
consider this a nd showed by a simple kinetic argum ent (it is now mo re
satisfyingly don e using qu an tum mechanics) tha t the abso lute probab ility
of an upward or a downward transition caused by light
of
the correct
frequency is exactly the sam e. It follows that if there a re m olecules in b oth
energy levels then the intensity of a spectroscopic line depends on the
difference between the nu mb er of a tom s th at ab so rb energy from th e light
beam an d those that emit energy to it, a n d therefore on the
diflerence
in
the populations of the two energy levels. This is further considered in
Section
4.4.
So not only can spectroscopy measure the energy gaps in
ato m s and m olecules, bu t it indicates this difference in popu lations, to o.
Since all atoms and molecules at thermal equilibrium with their sur-
roundings are found experimentally
to
exhibit absorption spectra, we
conclude tha t
at
thermal equilibrium the lower energy states are the mo re
highly populated. Samples in which the a tom s are d eliberately excited to
higher levels, for example, by an electric discharge through them, have
their upper levels overpopulated and exhibit emission spectra. This is
how streets are lighted, using the emission spectrum of sodium. We
should always remember that systems are not necessarily at thermal
equilibrium; indeed, equilibrium can be disturbed or avoided in many
ways
of
increasing technical im portance. F o r example, lasers depen d on
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16 Chapter
1
deliberately prod ucin g o verp opu lations of higher energy states.
1.7
N TURE OF QU NTISED SYSTEMS
Th e quan tised world is a strange a nd un-instinctive one, complicated by
the fact that polyatomic molecules possess millions of discrete energy
levels. But we can understand wh at q uan tisation implies by study ing it a t
its simplest and we initially consider a system that has just two energy
levels available to it. Th a t is, one in w hich the energy can not vary w ithout
restriction, as in the classical physical world, but in which the atoms,
molecules, nuclei or whatever t ha t com prise the system can individually
ad op t on e of jus t tw o possible energy states. Such systems actually exist.
F o r example, the nucleus of the hydrogen atom , the pro ton, is magnetic
and can interact with an applied magnetic field to affect its energy.
Experiment shows that it is a q ua ntu m species whose magnetic mo me nt
can adopt either of two orientations with respect to the field direction,
rather t ha n the one tha t a classical com pass needle would. In o ne orienta-
tion the m agnetic m om ent lies alon g the direction of the applied field, an d
its energy is lowered, whilst in the oth er it o ppo ses it, an d its energy is
increased. These a re simple experimental facts an d they imply th at appli-
catio n of the field creates a tw o-level system [F igu re 1.8(i)]. Th is is
exploited in Nuclear M agnetic Resonance (N M R ) spectroscopy, in which
transitions are excited between the two levels. A
different example is
found in the halogen atoms that behave as though they were two-level
systems a t low tempe rature.
Let us be clear what is implied by the existence of the two levels. We
define the energy of the low er to be
0,
an d that of the upper on e
E
If we
have one particle (an atom ,
a
nucleus o r w hatever) in its lowest level its
energy is 0, whereas if, som ehow , we pu t it in to the upp er s tate its energy
is E It is crucial to o u r understanding of qu antised systems that there is no
other possibility. Th e particle ca nno t, for example, have a n energy
of c/2
or 1.28. This seem s at od ds w ith everyday life in which, up to som e limit,
systems seem to be ab le to possess any energy. F o r example, we can h eat a
kettle to any temperature below the boiling point of water. We need
somehow to reconcile this difference with the classical world since we
know that a t atomic o r molecular level
all
energies are quan tised.
Th e secret once mo re lies in the fact th at in the exp eriments we usually
perform we do not study individual particles but rather collections of
them in which they may be dis tributed between the two energy states. W e
sta rt by simply add ing a seco nd [Figure 1.8(ii)]. N ow bo th m ay be in the
sam e energy level, to give a tot al energy of 0 o r 2c or on e may be in one
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Some Basic Ideas and Examples
17
Magnetic Field
- I
0
(ii)
Figure 1.8
(i)
A two-level system results when hydrogen nuclei are placed inside a magnetic
field. Some protons align with their magnetic moments along the applied j e l d ,
and some against it, resulting in two difSerent energy states. At thermal equilib-
rium there are more nuclei in the lower energy state than in the upper one.
Nuclear Magnetic Resonance Spectroscopy consists in causing a transition
between the two.
(ii)
If just two particles enter
a
two-level system their total
energies can be
0 , 2 ~r E,
but i ft h e y enter the levels with equal probability then
the latter can be obtained in two way s, unlike 0 and 2~ which can be obtained in
jus t one.
level and the other in the other, giving a total of
E .
Now consider the
average energy of the two. F o r this latter case this is &/2,
so
tha t we now
have an energy tha t is not one of the quan tised values. Th e total energy is
simply the n um ber
of
particles times the average value
(2
x
~ / 2 ) , nd in
this case has a value equal to th at
of
one
of
the qua ntised levels. But this is
rarely true as we increase the num ber
of
particles in the system. Co nside r
one in which there are
( m +
n) particles divided between th e tw o energy
states, with
m
in the lower energy one. N ow the total energy is
nE,
which
can take a range
of
values determ ined by n, an d the average energy
(1.11)
which is determined by the values of m and n. W hereas the energies of the
individual particles have discrete, quantised, magnitudes, the average
energy, and the total energy, have no such restrictions and can take a
wide range
of
values. This lies at the heart of why systems containing
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18 Chapter 1
species with q uantised energy levels nevertheless display classical behav-
iour und er m ost c onditions. These energies clearly depend on how the
particles are distributed over the two energy states, tha t is on the num bers
m
and
n.
This simple example leads to a further important insight. If we can
distinguish between the particles,
i.e.
know which is which, then the
situations with
0
o r 2~ in total energy can each be obtained in just one
way. But an energy of E is obta ined if either is in the low er level an d th e
oth er in the upp er o ne, in tw o w ays. If b oth levels are equ ally likely to be
populated this energy is twice as likely to occur as either of the others.
Generalising, this implies that som e population distributions a nd some
energy values ar e m ore likely to occur tha n others. This is
a
conclusion of
mo men tous im portance. However, we stress tha t it depend s on the likeli-
hood
of
pop ulating each level being inherently equal.
We shall now consider systems containing
a
large number of, e.g.
molecules, which may exist in any of a large number of energy levels,
before returnin g t o the two-level system.
1.7.1
Boltzmann Distribution
Within chemistry we habitually deal with systems that contain a very
large numb er ( N )of molecules an d we consider how these are distributed
between their numerous energy levels when the systems are in thermal
equilibrium with their surroundings at temperature T. For example,
1
mole of gas at 1 bar pressure contains N ,
(6.022
x molecules. W e
need the q ua nt um analo gue of the M axwell distribution of energies. It is
given by the Boltzmann distribution, which we shall state and use here
before deriving it in Chapter
2.
It is a statistical law that applies to a
constant number
of
independent non-interacting molecules in
a
fixed
volume, an d is subject to the to tal energy
of
the system being consta nt
(the system is isolated), an d the several ways
of
obta ining this energy by
distributing the molecules between the quantised levels being equally
likely. That is, it does not matter which particular molecules are in
specific energy states provided tha t the total energy is con stant. The
distributio n is
(1.12)
where ni s the number of molecules in a level of energy ii and
g i
s the
degeneracy
of
that level (the number
of
states
of
equal energy, for
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Some Basic I d e m and Example s
19
example, the three p o rbitals
of
a
H
ato m have the same energy and
so for
this
g i
=
3).
The denominator includes summation over all the energy
levels of the molecules. This equation results from statistical theory
applied t o a very large num ber of molecules, und er which condition s on e
particular distribution of the molecules between the quantised levels
becomes
so
mu ch m ore likely than the rest that it alone need be consider-
ed. This is the ultimate extension of ou r conclusion concerning just two
levels.
O n first encounter, the B oltzman n d istribution looks formidable, es-
pecially because it apparently involves summation over the millions of
energy levels present in po lyatom ic molecules. But we now return to the
two-level system to discover the circumstances where this is only an
ap pa ren t difficulty.
1.7.2 Two-level Systems
Consider a two-level system in which there are no particles (atoms,
molecules or wh atever) in the lower level an d n l in the upper on e so that
the total number N
=
(no+ nl). At therm al equilibrium the Boltzmann
distribution tells us that these are related th roug h
Rea rrangem ent yields
(1.13)
(1.14)
In the simplest case the degeneracy of each state is 1, e.g. for protons
inside a magnetic field. H ere the ratio of the po pulatio ns dep end s directly
an d solely on the value
of
the dimensionless expon ent (@T) ha t varies as
the temp erature is change d, being a fixed characteristic of the system.
The denom inator,
kT,
is kno wn as the 'thermal energy'
of
the system. This
energy is always freely available to us in systems a t therm al equ ilibrium
with their su rroundings a nd, indeed, we canno t avoid it without decreas-
ing the temp erature to
0
K.
Th is gives us a simple physical picture. The
therm al energy is wh at a system possesses by virtue
of
the motion
of
the
particles th at com prise it, an d we see th at it is closely related, for exam ple,
to the mean thermal energy due to translation +kT, above. But the
distribution tells us th at in q uantised systems we must c om pare
kT
to
rather than
2
imes it. If
kT
is mu ch less than
E
we do n ot have the energy
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20 Chapter 1
to raise the particle from its low er energy level to the higher on e, but as
the temp erature is increased it becomes possible t o d o
so.
So much for the basic picture; now let us investigate the distribution
semi-quantitatively. At low temperatures
kT
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Some
Busic
Ideas
and
Examples
21
T T T
0)
(ii) (iii)
Figure 1 9 (i) I n a two-level sys tem all the population is in the lower level at absolute zero
but t he number in the upper level
(n , )
grows as the tem perature is increased. I t
does not grow indefinitely, however, but reaches an asymptotic value at high
temperature.
I f
the levels are singly-degenerate, half of the total number
o
molecules is then in each level.
(ii)
Th e total energy
of
the sys tem is the product
o
the population o the upper level times its energy (see te xt ) and
so
it varies with
temperature exactly as does n l . Th e energy also reaches an asym ptotic value at
high temperature. (iii) Variation o C,with temperature is given by the difleren-
tial with respect t o temperature
of
the energy curve,
(ii)
It starts
from
zero , goes
through a ma xim um at the point o inflexion of the energy curve, and returns to
zero a t high temperature.
This is a simple multiple
of
y1
so
that the variation of energy with
temperature has exactly the same form as the variation of n , [Figure
1.9(ii)]. This is astonishing. It show s th at as th e tem peratu re is increased
E
does not increase continually but again tends to a n asym ptote. T o check
this we must again turn to a calculation a nd m easurement
of
C,, which is
obtained over the temperature range simply by differentiating Figure
1.9(ii). [ C v = (aE/aT),]. Th is is shown in F igu re 1.9(iii). It predicts, what
wou ld be very stran ge in classical physics, th at the heat cap acity increases
through a maximum and then falls
to
zero as the temperature is in-
creased, a n d is wh at is observed expe rimentally.
This precise behaviour is unique to the two-level system. But the
argum ents we have used are no t. Th us the B oltzmann distribution in any
system, in which an y particle m ay exist in an y on e
of
a number of discrete
energy levels, always contains exponential terms in which quantised
energies are c om pared with kT, an d it is the values of these exp onentials
th at largely determ ine level pop ulation s a nd all the physical prope rties of
the sample. That is, they all depend upon the ratio ( /kT).his simple
realisation gives prob ably the most im po rtan t insight in to physical chem -
istry.
An exam ple is seen if we extend th e arg um ent used abo ve to calculate
the energy of the two-level system t o on e with m any levels. W e realise tha t
the particles are distribu ted am ong st the levels each
of
which has its own
characteristic qu antised energy, ii nd we must sum the energies
of
them
all. It follows tha t
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22
E = E n i s ,
1
Chapter
I
(1.19)
This seemingly obvious statem ent h as astounding implications when we
realise what we have done. It says that if we know the values of the
qua ntised energies, which we can determ ine from spectroscopy o r calcu-
late using qua ntu m m echanics, then we can calculate a thermody nam ic
property. It establishes a direct relationship between the properties of
individual atoms and molecules and the thermodynamic properties of
samples made u p of large numb ers of them, an d it is one of the funda m en-
tal equa tions of statistical thermody nam ics. W e shall return t o it later.
1.7.3
Two-level Systems
with
Degeneracies Greater than Unity;
Halogen Atoms
At roo m tempe rature the electrons in ato m s are found exclusively in their
lowest orbitals. Th is is because the highe r orbitals a re greatly separa ted in
energy from them
so
that (&/kT)
>
1.Th is is true of the halogens, bu t w ith
these the lowest level is split into tw o by spin-orbit coupling, which
is
smallest in fluorine and largest in iodine (Figure
1.10).
Such coupling
results because motion of the electrically charged electron around the
nucleus in a p -orbital causes a m agnetic field there th at is experienced by
the electron itself. Ho wev er, the electron possesses spin angu lar m om en-
tum tha t causes it to have a q uite separate magnetic mo me nt. As with the
proton in an external magnetic field the quantised electron magnetic
moment can adopt just two orientations inside the field due to orbital
mo tion, an d two energy levels result. Experiment shows tha t the energy
sepa ration between them is very low com pared with the energies sepa rat-
ing the orbitals
so
that the halogens behave as if they were two-level
systems a t n orma l temperatures.
2p3 2
0
g = 4
Figure 1.10 The lowest electronic energy level of the halogen atoms is split into two by
spin-orbit coupling, the interaction between th e magnetic moments due jirs tly
to the orbital motion
of
the electron in its p-orbital and secondly due to its
intrinsic spin. Inpuorine the splitting in energy is of the order
of
k T a t room
temperature and both levels are populated. The next lowest electronic level is
comparaticely very high in energy (energy
>>
k T ) a n d is completely un-
populated at room tem perature. A t thi s temperature the atoms behaue as though
they are two-level system s, but the t wo levels have difer en t degeneracies.
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Some Basic
Ideas and Examples
23
We take F as an example. Its electron configuration is ls22s22p5 o
tha t it has a single unpaired electron in a p-orbital. The states that result
from spin-orbit coupling can be calculated simply using Term Sym bols
(Appendix 1.2) tha t show the gro und state to split into
2P1,2
nd
2 P 3 , 2
components. The subscript indicates the total angular m omentum qu an-
tum num ber of the state, J , and according to H und's rule the state with
the highest J , 4, ies lowest in energy w hen a n electron shell is over half
full. How ever, the Term Sym bol has an oth er crucial piece of inform ation
encoded in it since the degeneracy of a sta te is (2 J
+ l ) ,
an d we have seen
that the degeneracy enters the Boltzmann distribution. For the upper
state g,
=
2 whilst for the lower o ne
go =
4 (Fig ure 1.10)so that
(1.20)
Th e expo nential term changes with tempe rature exactly as before, tend-
ing to unity as
T
-
00.
In conseq uence the asym ptotic value of the ratio is
no longer 1 but 0.5. Th at is, at high tempe ratures one-third of the ato m s
are in the upper state compared with the half obtained when the levels
have equal degeneracies. Had the state with
J
=
been the upper one
then tw o-thirds of the atom s would have been in the upper state at the
higher temperatures. Simply put, a t high temp eratures, a state of degener-
acy g can hold
g
times m ore atom s than one of degeneracy one -t h e states
behave a s thoug h they were buckets. Deg eneracy has a significant effect
on level populations.
Th rou gh the direct relationship between energy an d heat c apacity it is
clear tha t the halogen ato m s have C, values tha t reflect their ab ility to
accept electronic energy within these split ground state levels besides
possessing trans lationa l energy.
It rem ains to pu t in som e values to see how significant this is. No tably,
the exponent always appears as a ratio so, provided that we express
num erator a nd deno mina tor in the sam e units, i t does not ma tter what
the un its are. Spectroscopists measure
E
using experimen tally convenient
reciprocal wavelength units, den oted an d usually quo ted in cm-' (1
cm
=
m). These are directly related to energy th roug h the relations
E
= hv
and c = v / l o r c =
vV,
where
v ,
1 and c are respectively the
frequency, wave length and velocity of the light. It follows th at
E
=
h c b .
But rather than calculating this each time it is convenient to calculate
kT/hc in cm-I an d to use the measurem ent un its. (A discussion o n units is
provided in Appendix 2.1, Ch apter 2). F o r F,
=
401 cm-l, whilst kT/hc
a t 298 K (room temperature)
=
207.2
ern- ,
so
that
E/kT
=
1.935, and
e - ~ / k T= 0.144. T o work ou t the electronic contributio n to the energy
of
1
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24 Chapter
1
mole of F atoms at this temperature we first return to the Boltzmann
distribution to ob tain the num ber of atom s in the upper state:
g
l e
l k T
2
x
0.144
= 0.067
1
N A
g o e WkT + y lecE ikT 4
+
2
x
0.144
(1.21)
where, as usual, we have defined the lower state t o have zero energy. Th e
total electronic energy per mole at this temperature is
E = n , & N , =
0.067
x
401N, cm-' mol-', wh ich is 328
J
mol-l. Th us the
presence of the low-lying sta te increases the tot al energy of the
system from the pure translation value of 12.47 x 298
=
3716 J rno1-I
(recall
C
=
12.47
J
mo1-I K-' for tran slation) by roughly 9 % . This
increases rapidly with tem pera ture.
The electronic heat capacity is given by
C,
= (dE/dT) = &(dn,/dT),
where the latter is obtained by differentiating the previous eq ua tion .
1.7.4
A
Molecular Example:N O
Gas
NO
is the only simple diatomic m olecule th at contains a single unpaired
electron and it is in a
n*
orbital, implying that it possesses one unit of
orbital angular momentum about the bond axis, yielding a magnetic
mo m ent. Magn etic interaction with the electron spin magnetic mo me nt
once mo re results in spin-orbit coupling (Figure 1.11) and the gro und
state is split into two, giving
2111.,2
and 2113,2 tates, with now the former
the lower in energy. In diatomic molecules, as opposed to atoms, the
degeneracies cannot be assessed from thesesymbols, but each is doubly
degenerate
(go
=
g1 =
2). Th e higher sta te lies 121 cm-' abo ve the lower
so
that a t room temperature
e /kT
.58 and
(1.22)
showing that over one-third of the molecules are in it at room temp era-
ture. As before, we could wo rk ou t wh at this implies as a mo lar contribu-
tion to the tota l energy of the system but it is obviously appreciable, an d
so
is the effect on the heat capacity. But, as with all light diatomic
molecules, there is n o co ntribu tion to the heat capacity a t this tempera-
ture from vibratio nal m otion, since the first excited vibration al level is too
far removed in energy from the g roun d state to be occupied.
Clearly, it is simple t o calculate the e lectronic energy an d hea t capacity
of F and
NO.
But our calculations need not be restricted to these
properties. A sample
of NO
a t room tempe rature is found to be magnetic
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+ - -
t P
b Ps
-
- -
(ii)
Figure
1 11
I n
N O
there is an unpaired electron in a n-orbital w ith orbital angular momen-
tum (with quantum number = 1) about the bond a xis; the vector representing
this is therefore drawn along this axis. (i) Relative to this the spin angular
mome ntum vector o the electron s =4 can lie either parallel to it, to give an
overall momentum of ,or antiparallel to it 4, eading to I IS l 2nd 211112 tates.
(ii)
These motions
o
the negatively charged electron produce magnetic mo-
men ts, also along th e axi s but in the opposite direction to the angular momen-
tum vectors, and experiment shows that the magnetic moment due to spin
motion is (almost) equal to that due t o orbital motion. In the for me r state th e
moments add to make the state magnetic but in the latter the moments are
opposed and the state is not magnetic. I n the molecule, as opposed t o the atom ,
each state is doubly degenerate g =
2)
and, with the next lowest molecular
orbital well removed in energy from the lowest, N O behaves as a two-level
system e xac tly as shown in Figure 1.9.
(actually param agnetic). At first sight this seems unsurprising in a mo l-
ecule tha t co ntains an unpaired electron, since electrons are m agnetic, but
we m ust rememb er th at in spin-orbit cou pling we have discovered the
influence of a second m agn etic field within th e m olecule, due to orb ital
m otion. W e have therefore to consider the resultant m agnetic field of the
two ra ther tha n just tha t of electron spin. By qua ntu m laws these can lie
only parallel (the 21-1312 tate) or antiparallel (the 2111j2tate) to each
oth er alon g the molecular axis. In the former case the m agnetic mo me nts
re-enforce each other, and in the latter they are opposed. It was dis-
covered experimentally that the m agnetic mom ent d ue t o the spin m otion
is
almo st exactly equal (to 0.11 ) to tha t due to orbital motion so that the
two cancel in the
2111,2
tate, m akin g it essentially non-m agnetic. Since
this is the lower energy sta te all the molecules w ould be in it a t sufficiently
low temperature and the sample would not be appreciably magnetic.
Th at a roo m tem perature sam ple is magnetic results from therma l popu-
lation of the up per state.
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26
Chapter 1
A
measure of the magnetism of a bulk sample is its susceptibility
x,
Greek chi) which is again straightforward to calculate. Calling the mag-
netic m om ent of
a
molecule in the 2113 2tate
p,
it is given by
and n,,z per mo le is obtained from the B oltzmann distribution as before
(but remember
go = g1
here). Th e tem peratu re dep endence of the suscep-
tibility ha s exactly the sa m e m athem atical form as d oes the pop ula tion of
the upper state, an d th e energy of the system. It therefore increases from
(near) zero at
0 K
an d rises to an asym ptotic value. We see how powerful
some rathe r s traightforward ideas are in calculating the physical pro per-
ties of collections of ato m s and molecules.
APPENDIX 1.1
THE
EQUIPARTITION INTEGRAL
From Equations
(1.3)
and
(1.4),
Since the stan dard integral
(1.24)
(1.25)
we find th at th e terms in m cancel and
mu2
= SkT
(1.26)
This result obviously holds for an y ‘squared term’ whose energy distribu -
tion
is
given by the Maxwell equa tion.
APPENDIX 1.2 TERM SYMBO LS
A Term Symbol provides a straightforward means for assessing what
states of an ato m exist as a result of spin-orbit coupling from kn owledg e
of the electronic structure
of
the atom . In general an atom possesses many
electrons and coupling occurs between their spin and orbital motions
according to definite rules. For light atoms (those with low atomic
numbers) Russell-Saunders coupling decrees that the spin and orbital
angular m om enta sum according t o the rules:
s csi
I
(1.27)
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Some
Basic Ideas
and
Examples
27
L = p i
I
(1.28)
where
S
is the vector sum of all the individual spin vectors
of
the
electrons,
si
nd likewise for
L ;
the qua ntum unit of angular mom entum
(h/2n) is omitted by con vention. Th e total angular mo me ntum resulting
from coupling between the magnetic moments due to these resultant
m om enta is then given by
J = L + S
(1.29)
A further convention is to write all these quantities in terms of the
qua ntum numbers (scalars) rather than actual values
of
the angular
mom enta, despite having to remember tha t vector addition is involved.
Th e result is expressed in the T erm Sym bol
2 s + lLJ (1.30)
Orbital angular momentum quantum numbers of individual electrons
are given letter symbo ls. Th us
s,
p, d and refer to values of
0,
1 , 2 an d
3
respectively. W hen the vector ad dition has tak en place capital
S,
P,
D
and
F symbols are used for the correspo nding tota l
L
values.
In filled electron shells individual si values cancel, as do the corre-
sponding
i
ones, an d we need consider only the u npa ired electrons. In the
F atom , with one unp aired electron in a 2p-orbital,
S = s i =
and L=
i =
1, which is a
P
state. By the general laws of qua ntu m mechanics the
two vec tors can lie only parallel o r antipara llel to each other, yielding jus t
two possible values of
J ,
4and
.
Their Term Sym bols are 2P3,2nd 2Pli2
respectively, since (2 s + 1)= 2. They are described as ‘doublet P three
halves’ an d ‘doublet P half’ states. Their energy separation depends on
the spin-orbit cou pling con stant, which differs between different halogen
atoms . Fr om general qua ntu m m echanical principles, the degeneracy of
each state is given by ( 2 J + 1).
W ith diatom ic molecules a sim ilar con ven tion is used, except that the
total angular momenta are summarised in Greek alphabet symbols,
C
(sigma), Il (pi), an d
A
(delta) corresponding to S, P and D in atoms. The
basic symbo l becomes
2c
‘A where s analogous to
S
and A (lambda)
to
L,
an d spin-orbit coupling ma y again occur between the two m om en-
ta . Fo r NO with its single unp aired e lectron in a
n*
orbital (A = 1, a ll
state) the resultant states are, consequently, and
2113/2.
ut these
both have degeneracies
of
2, no t wh at we would expect for atom s. They
ar e referred t o a s ‘doublet Pi half’ an d ‘doublet Pi three halves’ states.
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Some Basic Ideas and Examples
29
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CHAPTER 2
Partition Functions
Chapter 1established that atom s a nd molecules possess quantised energy
levels and that the energy gaps between them can be measured directly
using optical spectroscopy . If the energy of the lowest state is know n th en
this allows absolute values for the energies of the systems to be ascer-
tained. In all cases but on e, the energy du e to v ibration in a m olecule (see
below), the lowest energy is defined to be ze ro. In addition , assuming the
Boltzmann distribution for the populations of the energy levels in systems
a t thermal equilibrium with their surrounding s allowed us to calculate
the physical properties of some two-level systems. We could take the
agreem ent between results
so
calculated a n d experiment as evidence that
the distribu tion is correc t. But th e theoretical d erivation
of
the d istribu-
tion gives insight into the conditions under which it is valid, and we
return to this below. Meanwhile we continue to demonstrate that the
simple ap pro ach we have used w ith two-level systems can be generalised
to ones con taining an y num bers of levels.
2.1 MOLECULAR PARTITION FUNCTION
Th e energy
of
any quantised system can be ob tained, as stated above, by
summ ing over the popu lations of the individual s tates multiplied
by
the
energies of those states:
This is, though, an impractical formula for estimating E for systems,
including molecules, in which the number
of
energy levels may be very
large indeed. First, it involves a su m m ation over term s in each of these
and, second, we appear to need to know the energies of all of states.
How ever, we have alread y seen this is no t necessarily the case. Since the
populations (n ,)depend on e-&iikT, ny terms in the
expansion of
the
sum
for which ii>> kT have near-zero values of
ni
can be neglected. This
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Partition Functions
31
limits the nu mb er tha t needs be considered and has far-reaching conse-
quences.
A common approximation is to assume that the various modes of
energy that a molecule m ay possess are indepe ndent,
so
th at for each level
‘total = Eelectronic ‘vibrational + ‘rotational + ‘translational
(2.2)
This is the sam e assum ption m ade in interpreting molecular spectra. It is
a very good approximation but it is one and there are many cases in
which it can be seen to be (slightly) inaccurate.
A
simple exam ple is tha t a
change in the v ibrational energy of a n an harm onic oscillator (e.g. a real
diatom ic molecule) involves a ju m p to a h igher vibratio nal level in which
the ave rage internuclear distance differs from th at in the lowest one, and
therefore causes a change in the moment of inertia and the rotational
energy of the m olecule, too. H ow ever, the app roxim ation is sufficiently
good for us to be able to understand the properties of molecules and to
calculate their thermodynamic properties, often within measurement
accuracy. If we require their exact values we have no alternative to
me asuring the energy levels experimentally an d e ntering them into the
un-approx imated energy equation, without assuming tha t the individual
contributions are indepe ndent.
Summ ing over the know n energy levels is the mo st convenient way of
obtainin