5

Introduction

Often we are interested in the total energyinvolved when one mole of a substance changes its energy
state: for this we multiply by the Avogadro numbcr N= 6.02 x 10.
However, the spectroscopist measures the various characteristics of the absorbed or emitted radiation
during transitions betwccn energy states and often, rather loosely, uses frequency,
wavenumber as if they were energy units. Thus in referring to ´an energy of 10 cm thewavelength,
and
spectroscopist
means a separation between two energy states such that the associated radiation has a wavenumber value
of 10 cm.The first expression is so simple and convenient that it is
essential to become familiar with
wavenumber and frequency energy units if one is to understand the spectroscopist's language. Throughout
this book, we shalluse the symbol [ to
represent energy in cm.
It cannot be too fimly stressed at this point that the frequency of
radiation associated with an
change does not imply that the transition between energy levels occurs a certain number of timesenergy
each
second. Thus an electronic transition in an atom or molecule may absorb or emit radiation of
some 10° Hz, but the electronic transition does not itself occur 10 times per frequency
second. It may oCCur
once or many times and on each occurrence it will absorb or emit an energy quantum of the
appropriate
frequency.
1.3 REGIONS OF THE SPECTRUM
Figure 1.5illustrates in pictorial fashion the various, rather arbitrary, regions into which electromagnetic
radiation has been divided. The boundaries between the regions are by no means precise, although the
molecular processes associated with each region are quite different. Each succeeding chapter in this
book deals essentially with one of these processes.
In increasing frequency the regions are:
1. Radiofrequency region: 3x10°-3 x10° Hz; 10 m-1 cm wavelength. Nuclear magnetic resonance
(n.m.r.) and electron spin resonance (e.s.r.) spectroscopy. The energy change involved is that
arising from the reversal of.spin of a nucleus or electron, and is of the order 0.001-10 joules/
mole (Chapter 7).
2. Microwave region: 3x 10°-3 x 10 Hz; 1cm -100um wavelength. Rotational spectroscopy.
Separations between the rotational levels of molecules are of the order of hundreds of joule per
mole (Chapter 2).
3. Infra-red region: 3x 10-3 x10 Hz; 100 um -lum wavelength. Vibrational spectroscopy.
One of the most valuable spectroscopic regions for the chemist. Separations between levels are
some 10 joules/mole (Chapter 3).
4. Visible and ultra-violet regions: 3 x 10*-3 x 10° Hz; 1 um -10 nm wavelength. Electronic
spectroscopy. The separations between the energies of valence electrons are some hundreds of
kilojoules per mole (Chapters 5 and 6).
5. X-ray region: 3 x 10°-3 x 10 Hz; 10nm -100 pm wavelength. Energy changes involving the
inner electrons of an atom or a molecule, which may be of order ten thousand kilojoules (Chapter 5).
6. ray region: 3x 1o!8-3x 100 Hz; 100 pm-1 pm wavelength. Energy changes involve the rear
rangement of nuclear particles, having energies of 10- 10" joules per gram atom (Chapter 9).
Another type of spectroscopy, that was discovered by Raman and bearing his name is discussed in Chapter 4.
This, it will be seen, yields information similar to that obtained in the microwave and infra-red regions,
although the experimental method is such that observations are made in the visible region.
 6

Change of
Change of nuclear
Change of spin orientation Change of clectron distribution configuration

Visible and
N.m.r. E.s.r. Microwave K-rav Y-ray

Fundamentals

of
Wavenum Molecular
102 100 104 106 10°

Wavelength 10 m 100 cm 1çm 100 um 1pm 10 nm 100 pm SpectrosCopy

Frequency Hz3x 106 3 x 108 3x,1010 3x 1012 3x 1014 3 x,1016 3x 1018

Energy
joules/mole 10-3 10 10 10 103 10 09

Fig. 1.5| Regions of the electromagnetic spectrum.

produced
molecular,
radiation. In
2. 1. order
3.
plotted
are electric
microwave place emitted, have spectrum.
insee thatchargehydrogen) emission
Theradiation this The
orparticles,
that
change. The
linearly a dipole There by or
said given that if microwave a radiofrequency
infra-red and only
permanent
separation, the there
electronic
Consider
with toand field in the at and are
spectroscopy.
Directionof Direction of
the be direction the plus a carries consequent the change Shall
Fig, momentcomponent
Vertical rotation molecule the of
lower pure it several
a dipole of dipole 'microwave and appropriate follows be
small region: 1.6 radiation
rotation rotation electric region:
a which
the has changes some
half (sayminus permanent
carbon
net a upon possibilities:
region:
Here Rotation
in is upwards
of zero A that can
mechanism
positive the 'microwaveactive'. gives (cf. Fig. charges takes dipole frequency. the depicted
dipole. molecule their be
dioxide is it dipole Fig. spin We
of rise 1.6, in place,moment. net influenced
chargevibration, amoment polar a If to1.2). and change spin
the Ifpositive reversal may infor
0C molecule inactive'. there
plane the we Consequently such is Fig.Introduction
a interaction
on diatomic spectrum. Thus it consider H,
is placescenter
associated consider
the rathermeasured is charge as can by
28+
no
interaction seen of or
hydrogen the 1.4,
carbon an as the interact
molecule, Thisdipole, periodically,
to of Cl,, the electric between
there
0
example, than + paper) gravity the onand all wi th
in imposes AIl be
and rotation the the nucleus must
chloride, such with
a
a
particular molecules as exactly a
smallrotation, showing in
canfluctuates
of other other spin the tiny magnetic bethe
or
in a H,occur, and the of
which limitation or similar magnetic and
magnetic
a reversals
incident
some
negative which direction. the having the
molecule HC1 hand, net HCI,
Cl,, energyregularly. This electron
the fluctuation component (Fig.negative fieldsradiation electric
in in in
charges three must Time no
on a form which which produce field
dipole.The
interaction
permanent can must 1.5, to
atoms give
the
to charge, of associated or
on rise applicability bethe dipole notwhere there electromagnetic
one
be magnetic and
fluctuation is
absorbed an tiny
the are fluctuating move), we is atom absorption reversal the
to can
moment moment is
oxygens:
arranged a no
notice such said
charged with nuclear,
dipole take (the effect
of or we to the
of
 of these
atoms known One thereHowever, 'infra-red
inactive`
thus
1.8. dpole During
tretched the
atoms
motions further Here
are is
as a moment
Fig during thperiodic
e onethere and
very
moment
vibration mode
1.8Asymmetric Componento
moment
dipole Dipoie stretch1ng
vibrationC26 bond is
Asymmetric
vibrations much bending
does
another Fig.1.7 compresscd.
remains
showing the alteration
stretches of
exaggeratedcenter is vibration
mode' .
allowed stretching zero
is molecule
Symmetric both
the
stretching
fluctuotion
seldom
gravity
of
This, to the
while
in throughout C-0Fundamentals
known
in the
thisdipolevitbration with
C
more Figs. as other stretching
molecule
move.shown amplitude thebonds as of
vibrotion
in than 1.7, thMolecular
e
moment, called
the is
vibration whole
changing
dipole about 1.8, Note compressed,
in 'symmetric
of
and Fig. (see the much ofSpectroscopy
the C particularly Chapter and this
moment. 10 'a1.nti-symmetrical
9, of
corbon per 1.9; the exaggerated. the simultancously,
-0-o
cent in alsovibration is and carbon motion, stretch,
dioxide real 3
that for vice
of infra-red NormalStretched and
molecules,
the a Compressed
dioxide the
molecule the more versa. is this as
time
-Obond relative thusstretch,
particularmolFig.in
ecule
C
detailed
active. In As
length. infra-red the
the
motions depicted 1.7. is
displacement figure
discussion), vibration
neither of Plainly
alternately
active'
shows. in
of
the Fig.
is

ultraviolet, We
5. 4.
show 1.4 The Although
There
fully spectrum
electrical spectroscopy,
information
electronic |Fig.

in visible
visible
REPRESENTATION
SPECTRA
Fig. OF in is 1.9|
Chapter a
by
polarizabilityrather charges dipole
and 1.10 its andabout the Bending component
interaction
4.specialultra-violet moment Vertical Dipolevibration
moment Bending
infra-redhighly a change
appearance dipole
in the motion
of
1.10|Fig. the
strscture
of
requirement requirements 0C-0
Recorder Amplifier schematic
regions the molecule.
with of
region: or the 28+ 8

Schematic molecule non-appearance

the of carbon
of a
diagram undulatory
fora The
particular
The
the do Introduction
Source I Rotatable, mustconsequent dioxide
diagram spectrum; molecular excitation impose
of change electric molecule of molecule
Grating a
of recording certain some
a S, since motion change of
grating during alimitation
field valence (see and
it vibration
to of Chapter its
ectrometer. uses
spectrometer the be in associated
the
radiation.electron
a motion.
Raman electricfrequencies of
grating the 254
3). time

involves application dipole

suitable This
active'; dipole
(a
block will can fluctuation.
M:
bethis gives
M the give
offor of
reflective use discussed is movingrise infra-red
valuable
that
in
the the to of 9
a
 MICROWAVE SPECTROSCOPY
2
2.1 ROTATION OF MOLECULES
We saw in Chapter 1that spectroscopy in the microwave region is concerned with the study of rotating
molecules. The rotation of a three-dimensional body may be quite complex and it is convenient to
resolve it into rotational components about three mutually perpendicular directions through the center
of gravity-the principal axes of rotation. Thus abody has three principal moments of inertia, one about
each axis, usually designated ¡, l¡, and l
Molecules may be classified into groups according to the relative values of their three principal moments
of inertia-which, it will be seen, is tantamount to classifying them according to their shapes. We shall
describe this classification here before discussing the details of the rotational spectra arising from each
group.

1. Linear molecules: These, as the name implies, are molecules in which all the atoms are arranged
in a straight line, such as hydrogen chloride HClor carbon oxysulphide OCS illustrated below:
H CI
CS
The three directions of rotation may be taken as (a) about the bond axis,(b) end-over-end rotation
in the plane of the paper, and (c) end-over-end rotation at right angles to the plane. It is self-evident
that the moments of (b) and (c) are the same (that is Ig= l) while that of (a) is very small. As an
approximation we may say that I, =0, although it should be noted that this is only an approximation
(see Section 2.3.1).
Thus for linear molecules, we have
I=lç ,=0 (2.1)
2. Symmetric tops: Consider a molecule such as methyl fluoride, where the three hydrogen atoms
are bonded tetrahedrally to the carbon, as shown below:
H
H
H
 the Perhaps
Herzberg. into
34
above the 4. 3.
four Simple Asymmetric
moments no In
dipole top.
Spherical to There
symmetrical. =l¢>l haveWe
because
a asare As
description
Molecular should
it molecule the still in
rotational fact, as
rotational A oblate. main the
examples moment these simple are
of then identical
it case
be tops: then two involves
rotational
inertia spinning
is pointed
Spectra spectrum
tops: molecules example In the An
subdivisions of
classes owing When thismolecule
example and linear
adequate. are different: the
These case about s we
and inout H
water
is to is a rotation aximolecules,
observable. their are molecule I of Symmetric have
Molecular far that H,0 molecules, the
is of this since Fundamentols Molecularof
more only tetrahedral
molecule =21 tcalled he this l,=
one symmetry, axis of
H
and Spherical
has latter
= class three the l, the
rigorous can of 2l can center
typeprolatewhich tops: comparatively
Structure, (and vinyl toacademic all a end-over-endThe
H be of
which rotation three
termsoften
chloride tops: H
H is I, imagined gravity moment
moments symmetric
boron we =
vol.
H the interest I, may
1ç#l
H CI
II).
thandoes) CH,=CHCI. majority alone Ip=lc = methane
trichloride, mention: massive
as inertia
Spectroscopy
lies rotation
of
have
However,describe can in of top; a along
top,
produce this CH,inertia
. in,
been of whereasif, hydrogen
and itabout
) and
H substances chapter. We which, is
the C identical, as hence theCF now
for used in out
classification
moleçulesof no have if
the
abOve dipoleSince asmethyl
/g=le<l4, atoms not of
purposes then shown, the the
belong, it negligible,
(see, is name off bond plane
change they called fluoride
have is it this
of for can of axis of
this and a planar is axis.
example. thehowever.
all have spherical referredabove, (chosen the
book (2.4) three hence (2.3) (2.2) class. Such paper
no and
I,

byWe 2.3.1 rselyes

hecause shallWe to for w ngilar oans
ver,orf Ine equation: orThe
balancing, prinmolecule
ciple the f We
moment molecule rigid astart
2.3 point
complicated that have 2.2
with much cach molecule.momentum) that
bar Rigid consider out
unduly be seen
a
of MOLECULUES
DIATOMIC calculated ROTATIONAL
moleCuleSPECTRA
inertia
rotates (the this, of where concerned. that
Eqs. |Fig. its sysStems
with For
bond)Diatomic
the
treatment reasonable simple
rotational
(2.5) 2.1
end-over-end simplest this, but
cannot
about class for
| whose it its
and however, is any energy The
Cis joined rigid A can
of probably
molecules, have
energy,
(2.6): of approximations permitted
lengthMolecule
all
rotating molecule
= m,defined
Vlm+ r; by
diatomic about
linear
be
being isany
a directly limited along
mr =r(m mytmi
my>yl1 = rigid is mnolecule impossible arbitrary Spectroscopy
Microwave
)+ by point
a
molecules, content
the energy
solving
by
=m,y molecule
bar C =m,>ymr mathematics to with
extended may certain
C, amount
length
of in withoutvalues--the all
=
treated
the turn, lead.merely the other
M,(ro center shown to Schrödinger definite of
r, discussingthe
symmetrical involved
to forms
=r, as gross rotational
r) (from
(2.6))Eq. of in accept so-called
+ztwo gravity: Fig. values of
approximations.
masses, the equation is molecular
2.1. straightforward dependingenergy
m
this and linear results rotational
m, Masses
is unsymmetrical for (ie..
and of energy,
defined molecule th e
m m, existing We levels-mayenergy in onany
and system the
by shall for arbitrary
quantized: is
the m, in
solutions tedious, shape
are molecules. most not
represented
moment,
joined concern and value
(2.7) (2.6) (2.5) detail, while
and size this 35
or
diss. rotational
Erom .h where centimetes.
where wavenumber.
between and equal. thisIn
rotationalnumber:
Equation
Sobetween ByWhere
rigid detines Replacing Therefore,
Equation B6
0. it the
weFa is is diatomic
B, c, useful by The
expression, use the we
energy these(2.10)energy noits
have(2.12\
constant C, ue the (2.11) the
quantity
restriction of have
moment Eq.
bave
2BE,
rotational velocity to
energies, v means molecule the written (2.8)
and =0 we used is consider levels. expressed levels Schrödinger
usually AElhc = arbitrary, J, ish of in
aand can but of which
to inertia Eq.
ating explicitly constant, ight, In to Planck's are =
shownotationthe
we abbreviated ¬,energies
= the cm, or, the the integral E,g11
can given
m,m,/(m, (2.7):
would h rotational conveniently
equation
the is more molecule. and constant,
allowed
ecule
allowed
say
the
given
is ¬,
BJJ+
=
here
expressed
the
of values
take
itintegral the
by
m,),
+ Fundamentals
moment
(2.13) of
is expreSsed
to
8z'lc
h
particularly, is J(J+ it
hat radiationenergies
region thisarises expression: may
then B= by -JJ+1)cm and inand
energy
tne 1) in restriction values I joules,1) be terms u of
has
moiecule
conventional. inerda
8r'
ofIgc
h
cm,
in spectra
these
emitted joules; in in
directly
the
is
shown called
is and , Molecular
ts levels cm from of
units. the moment the
owest out that
mmaically is
lB-
We
cm J=0,
since ,
s
.
usualy
We
ar
corresponding
e
or we,
which
of
zero where
the
atomic the
reduced m
+
m
SpectrosCopy
gular not 1, J=0, absorbed
write: however, effectivelyupwards,the of J=
rotational
rotaung mgnt 2, the solution inertia, 0, masses
...) discussed 1,
tum, unit 1,
is 2, mass
equally 2, as
al of ...) frequency, rotational
are allows to called theither
e . . energy and of
all. a the
as wavenumber consequence ininterested the the
I
For in well terms Schrödinger
only Or levels system.
bond
WeJ=Fig, have V= I,
of certain
may l.2.2. used
the is wavenumber. AElh in since allowed length. Equation
inuePlainly differences
of
tational reciprocal
I changes equation
discrete quantum both
(2.13) (2.12) (2.11) Hz, (2.10)
and to (2.,9) (2.9) (2.8)
to for a or are the

while Thus
nabsorption 1, In immediately: now Intherefore,
and, cotation
raise der
We
st t h priorce
poirenngt actiofce, callimitculatae withE,
Totatoras
selection quaton
transitions a, eredransitionspectrum.
LactSiIeighbor, we
general, other tonoW
opically rather the
a
deriving
similarstepwise
a a would
the
moleculewords,
it
to discuss
occurs),
the J=0
need
the is
not rap1dly a
of of f
the course. theto
rule, shows either can
This
spectrum VyJH to reached
JH=2B(J VJ=l+J=2=Ej=2-Ej=l
in have EJ=| J
sequence occur this raise state the to bond,
sophisticated
which
that, above pattern, shown lowering is
raising =
an
-EJ-0= State. I consider rotatingrotational
we increasing
there
and
froma consisting B[J'+ B(J the absorption
raised
is
VJ=0-J=2=cm-2B
l spectrum.
can (the normal at
and
changes J for
we of or
of + molecule
+ =6B-2B Plainly ground
let diatomic
the
Comes
on forbimadyden. application transitions
this below: we wvoul atd 1)(J+2)
3J from 28incident differences energy
particular have the the l)
cm line -0= temperatures.
If molecule a values
J
foImulate byby molecule, s
ve footresultrotational
of +2-(J+ from = the the
rotational we molecule point the
rule: one made lines 4B J= wil 2Bradiation
energy imagine
J=0’J=2’J=4...
Such
unit-all
it we of
have level
the Fi g
of
. in at
anenergy
BJ(J the cm appear l
cm! between
molecule
is which
and, Spect
at MicroWave

A= for a need the not, only 2B. Ji) +1) state to absorbed state, the disrupted: is there in
resul t,
the Schrödinger assumpion for identical
2.3. 4B. the
at be greater
centrifugal thprinciple.
e may
+l to in the
oth er
rigid it only instance, its
J J=2 2B absorbed molecule
6B....results to which levels
but than have.
is
transitons immedate emission stateJ+ level cm. (2.14)
wil
consider (2.15)
(2.17) diatomicalc led
(2.16) this the
wave that cm-!in be tono to In
coD an by If in
a In a
the

Fig.
absorption Fig2.2
2 3
2.3| 0

tfrom
hem.molecule
spectrum between Allowed
antdhe energy
whiarisesch levels the molecule.
diatomic
rotational
allowed
of energies The
6B
more
ofa
tronsitions
rigid energy. of
a
rigid
diatomic 02B 6B 12B 20B 30B 42B
we
see
 same ero of prime and Hence. ofand (cf.where Rewriting moment Tosuch J=0molecule
Or
rotation do Eqs. that will with Is OfThus
want2.3.2
38
energy We carbon conclude homonuclearcoune.
4J=2, the Eq. a not there
(2.10) radiation. not. tq.
-
Precisely E
requirement transition stateoccur Remember,
all, levels. absolute(2.9)) we spectrum of
than
Intensities
now to and express Eq. inertia this for
requires
ortwo
are only (2.16)
to under
the
a 3, Does.for oxygen. and. rotation (2.11) Thusthere if
similar
good elc., (2.13) should reasons the gives
mass section.
ofand a
transition consider
brieflythe
is knowing the carbon
nomal great
we molecules
also, wil
molecule the
Oximation, was
plainly a velocity as hence oocur.about
calculations respectively, of I= deal s ee f or whole
forbiddeninstance, of 19.921 the = h/8TBc. spectroscopic that this. that
we be
no
J=l’ the 14.56954 8rx monoxide the shall the of rotation
Spectral hydrogen of there suchdipole spectrumis
knowledge 46.483 03 x 68 relative light bond bond energy th e
Firstlv, asvmmetric Fundamentals
a 6.626
10-%4 x apply energy as
show J=2? ro=0.J131 x as 2.997 we will
are molecule
in relative 26.561I19.921 atomic cm in × Vo- length.
as axis, to th e
component
about
HCl
atom 104' have 3.84235 Eq. conditions.
be be to
cqually
likelyto thal other We ofLines s,since
93 B=1.92118 cm
=3.84235 noand raisedlevels moment (heteronuclear)the and be of
the = 1027 36
68 to kg
10' x x Gilliam(2.16) dipole bond expected Moleculor
hc
menlloned
words, have
intensities weights they would CO
1.2799 x be cm. change
probablly relalive nm 10S4 and1.673 m to may Thus from ofaxis wil
B B change
the
more
(or
x 26.56]
(H=1.0080) is = al."et an
Hence,
inertia
the be was showduring from
Spectroscopy
probabilities =11.383 in 27.9907 >x 2B observed diatomic
be extremely
ransilion
occur. above or of I.J31 10 43 cm.
have J
cm-! and said =0 rejected is wil such
o1 less the 36 X
from veryrotational athe this
m? 10*
xHowever, to to
spectrum hence be(and the small rotation, molecule. a
all
calculatiornschance spectral B measured
Ä) 65 10
Clhlanges
probabilily kg, 1o
10-47 Eq. widely in spectrum
of x
kg. be nonot all J=l Section
of lines
transition t 102 we -kg m (2.16) spectrum. linear) aboutspectrum,
The can C=the the inrotating. state, spaced: and
wih which making be
of
kg moment first order t h e
12.0000,0
reduced henceno
2.I: observed,
AJ1or between Eq. calculate molecules and
bond while
the line Secondiy,
=all show (2.16);
these
to such this we
]
transition of determine (J
mass means so, can
is thut the the inertia = = transitions interaction N, since
çhanges
alm for 15.9994, 0) are applying now
. is
masses even and
various then is in in that
Jthis see 0, if
= u the the if the it
is 0 a a

numbersnolecule
probabilities ihere that This
probability a
existence
Therotator, Awisimilar
th and e, where, moleculus Heremolecules
T=300K)thshows The
lhwhere,
at hence,and momentum. Where second first docs
ollowing ue we we wil
P, energy defining increasing see
how we ofmoving
/ we way, must
knowfactor in be not
like is of N; in molecules
the may twfactor
o the
that
varies this each aredifferent mean,
E, level
Rearrangement equations there remember,
relative that identical,
governing molecule
fromsingle in
is moment approachor is and J tw o state, level.
more also graphs are the
zed. expression with numbers however,
J=
nvention, of forrequired-the
energy
with
almost population J;
the
rotational
the
will
thecarry
inertia, the the larger of for isc NN,number population to1
of Fi g . example, the line J=2,
of that
of
these energyproblem states as = velocity
out
molecules
we Eq. o B. exp(-0.019)
2.4 many in = energy
in intensities
transitions
the all
take P=JJ the exp(-EJkT) any in
SpectrOScopy
J=0 Microwave
(2. gives the and which possibility
in have spectral
an
h/2r 10) rotational E=lo terms molecules 2x taking J= of higher intheof assemblage state,
2El can 2 angular been light the in
have = 6.63 state 1 a levels will
between each lines
+),=thJU+
e
as
2 JU = P=2EI be
rewritten: frequency momentumangular i
of calculated,
ts
exactly
degeneracy
of
in
0.98 1.38
10*x3x x
x typical in statelowest
is cms exp = be
isdirectly level. will wil|
say,

fundamental +D
P=lo
the
J=
10x value [-BhcJ(J given level the
when
is
the
molecules, to
of
move be
the Boltzmann variousbegin
(in showing l of by is equally
momentum.
of same in state, 300
10"x1x2 proportional
unit ) radians a
B= B
is1)/kT] +
zero, to

of
units rotator energy. the
the at 2
cm,cm.A in distribution
levels. with, J=
such intense.
energy equilibrium, since I
angular per more and asthe is
are: In and J to In
second), states.
the rapid = the fact,there in
same
Although
room very 0, normal a
momentum. case so, (cf. initialsince fore
and Degeneracy decrease as in
calculation
temperature
simple as
of Section
if different that
P the the we numbers the sample,gas the
Thus the J=0. of
diatomic of have intrins1C intrinsic a
angular is N,/ In (2.18) 1.7.2).
total
single
we(2.19) the (say o
39
se NÍ a of
 nerate.enfoldFigure samethe 40
threeAll betom only J= We momentum taken it
proportionalThroughout
oriented oftake For has Fi2.g4.
in
three 1.can angular
egeneracy,2.6(a) the Here see integral
up direction to
rotational
rotational directions be
integral
plane
in
and P the
momentum vector along
to the
only implications= values drawn The
of /1 the as
above
(b) the Boltzmann
spectively. energy: directionsthree the or x2 of may
such magnitude well taking 0.1 0.2 0.3 0.4 0.5 0.7 0.8 0.9
shows zero the axis as 0.6
different paper): units= units.
that take derivation,
components rotational magnitude.
about values
the the of its up populations
situationIn
are,
J +1, V2, this component is of Fundamentals
general,
=
directions of 0, quantumlimited thwhich e P ofB 2
level
thus is
course, and, most momentum. The
has =
andalong a 5 of
for as rotation been andRotational 3
-1. easily along
number by direction the of
it (Fig. Fig. 10
may asSociated
=2 J Thus
reference quantum
a printed rotational
cm 4 Molecular
degenerate.
threefold 2.5(a) by given a occurs
The quantum
readily (P= 2.5(b) (in
the means case
thisJ), numbèr of and 5
angular showS, reference the bold in energy T= OB=5cm-1 D Spectroscopy
V6 with direction to mechanical and B=
be of angular 300 number
seen and ) (d)) it face
the a of is levels 10
momentumwith diagram. a direction the K 7
J= vector different
usually type in J cm-!
hat same (here angular law momentum Eq. of
each (P 3 respect to (2.18).
Fig. --08
angular assumed of is which 2.2.
In zero directions drawn show
energy 2v3 vector length Fig. momentum The
to vector that
or may
th )
wimomentum the
to
V2(= 2.5, an as diagram 10
level reference in be integral be an it
fivefold thisfrOm we vector whicharrow is is
is stated:
conventionally a
1.4 1) show vector-i.e., has
(2J instanoe may an
and
directin top multinle of been
+and can
1)-fold heno to the only angularlength
sey. bot. hawe caee

n theWhen is energy the Fig.

ion number |Fig.2.6
against
this We 2.5
E, see |
is wil ofthat,
aplotted plainly molecule The
rises degenerate
althoughrespectively. The three Reference
direction
five +1
to be +1 with
degenerate
a and 0
J
levels the
hmum
at seven (a)
= (a)
e J 1.
the molecular P=2
st Popul ation avai lable degenerate orientations Spectroscopy
Microwave
andpoints
gral then fall increases population
on «
rotational of
minishes.J a(2/+ the
value curve +1 +2 +3 rotational
l)rapidly each inorientations
-3 -2

exp(-E,theof/kT)
to
entiation type with level (b) angular
decreases
J. fora
shown The momentum
molecule
total
of in
Eq. Fig. exponentially
relative with
(d) (c) (b)
vector
(2.20) 2.7, =2
J
population an
at for
indicating
shows andJ a
(Eq.
=3.
that (2.20) (2.18),
the that
 much Considering
prime B> change thereexceptWhen 2.3.3intensitieshence We Fig.2.7
increase 42
have
ggerated, B. isfor a it
This henceand in no
total particular its is
seen
change carbon appreciableatomic Effect whileplain The The
mass that diogram
relative
the that
the monoxidea massatom of
intensity
line d(2J+ 1) cxp[-BJ(J
3.0
+1) hckT]
4.0 5.0 6.0f
wil and ransitions populotions, 1.0
2.0
relative decrease change Isotopic intensities has
0
be hence in
reflected -the a been
molecule will
lowering inas resulting in in
an
the internuclear thSubstitution
e between be are including
drawn MaximumFundamentals
example, a
2
in B moment
the 1
of value. substance is maximum directly forRotational
3
the
rotational we
replaced levels t
degeneracy
same
he population: of
"Clevels If distance of proportional 4
we see inertia with
conditions quantum
Molecular
t hat is by or
designate
energy identical of
on isotope-an
its near very J=,
with
going isotopic
and on
the low torotational
the
number, 6 V2hcB Spectroscopy
respectlevels the chemicallyB as
value J or the Fig. 7 © O
from value
populations
very .J B=B=
2.4.
tothe of substitution.
C for energy 5
10 cm
2
thosemoleculeclÛo the element with given high
molecule molecule. cm
levels 9
of to identical the by J of
l2c wit!clo values
Eq.
h original.
There the ofa 10
lainly, and a (2.21).rotational diatomic
will
Fi2.g8. there is,
In in
however, every have
as
we isparticular, molecule.
shown shows mass a smalllevels; (2.21)
have way
a

between
for by
clarity, the
where of et Fig. diagram
Substitution.where Observation
CB°O al., the
as 2.8 and
u the already | lines
is prime was. the at
ofmolecule The 6 J (2B) the
Taking the at this effect 2
4 5 transitions
foot
reduced refers 3.673stated, 0
than
dcreased such of
the of
to found
37 isotopic
aS 12CO that Fig.
mass mass, 2B' due
the cm. Carbon 2B

heavier 1.92118cm-B= the to of2.8.

of ! separation
The first monoxide.
substitution the the the
oxygen and B' B 4B 4B' SpectrOScopy
heavierlighter
Microwave
spectrum
.046 the 8IC molecule. values
rotational
to h has
internuclear of molecule one
+= be
on 6B
6B
15.99948c'I'c_IL1.046
m' B led (2B), theof
15.9994ml5.9994 We absorption to
determined
and
the
have the energy RR heavier
areAgain
distance
and B= evaluation
15.9994 x 12 that
12
X immediately from"C°o of
levels 13CO shown the
species
+15.9994 1.83669 10
of is
these of
and dashed. effect
carbon-12
considered to rotational
precise 12R'
CIm wil
cm
figures be has
12B show
at been
to 3.842atomic
unchanged
be are: spectrum much
a
35
smaller
12.00, weights.
cm, exaggerated
of
weby a separatton
have while diatomic
isotopic Gilliam 43
that
 44
Fundamentals of Molecular Spectroscopy
1
from which m', the atomic weight of carbon-13, is found to be 13.0007. This is within 0.02 per cent of
the best value obtained in other ways.
CO molecules in
It is noteworthy that the data quoted above were obtained by Gilliam et al. from
Thus, besides allowing an
natural abundance (i.e., about one per cent of ordinary carbon monoxide).
studies can give directly an estimate of
extremnely precise determination of atomic weights, microwave
intensities.
the abundance of isotopes by comparison of absorption
 2.7 MICROWAVE OVEN
One area where microwave radiation has become very familiar in recent years is the kitchen, in the
nearly as
not nearly
shape of the microwave oven. While obviously not sophisticated as a spectrometer, its mode
as sophisticated
of operation depends entirely upon the absorption by the food of the microwave radiation in which it
is bathed.n fact. it is the water molecules only 'which absorb the radiation and so become raised into
high rotational states-the biological molecules in food are far too large to be able to rotate. As with
many other excited states, the excess rotational energy of the water molecules is re-emitted as heat and
the food becomes cooked,)
The efficiency of the oven lies in the fact that this heatingis internal)In a conventional hot oven a piece
of meat or acake is heated from the outside, and it must be left to cook until its center has been raised to
asufficiently high temperature. In mnicrowave heating,however, water molecules throughout the whole
bulk of the food are simultaneously excited and 'heated',so cooking times are drastically reduced.
The effect of such concentrated microwave radiation on the human body, unfortunately, is similar
whatever is exposed to the radiation is rapidly heated and cooked from the inside! It is essential, therefore,
to ensure that the door seal on a microwave is in good condition, so that no radiation is allowed to leak
out.