mtd 05_idealgaspolat32
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Molecular Thermodynamics (CH3141)
The ideal gas II: Intramolecular degrees of freedom
N.A.M. (Klaas) Besseling
SandlerCh4
Molecular rotations
Molecular vibrations
Equipartition of energy
2
Summary of what we learned about the ideal gas so far:
The total partition function Qof an ideal gas can be build
up fromsinglemolecule partition functions q:
Q = exp !E
i
kT
"
#$%
&'i(
Q =1
N!qN
Nis the number of molecules
division byN! because of the indistinguishability of theN
atoms of the same kind.
For a onecomponent ideal gas (MaxwellBoltzmann statistics)
Divide and Rule! (independent modes or atoms)
q = exp ! !
n
kT
"
#$%
&'n(with

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hence, the singlemolecule partition function can be factorised:
q =qtrans
qint
Divide and rule further!
For the monoatomic ideal gas at nottoohigh T, qintis trivial and only qtransmatters,
 we can write:
q = qtrans
Translational and internal degrees of freedom are independent
(taking the electronic and nuclear partition functions equal to 1)
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There are 3 independent translational degrees of freedom,
henceztrans itself can be factorised:
qtrans
= qxqyqz with qx = exp !
!lx
kT
"#$
%&'lx
( etc.
. . . and further (independent translational modes)
where are the energy levels of a particle in a 1D box!lx
=
h2lx
2
8mLx
2
Because the energy levels are closely spaced ,and vary smoothly with lx, the sum can be replaced by an
integral, solving the integral yields
qx =
Lx
! where ! =
h
2!mkT
thermal wavelength
(!"

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Hence expressions for all desired thermodynamic propertiescan be obtained:
e.g. pressure (ideal gas law)
energy ( per degree of freedom)
heat capacity
entropy
1
2kT
5
Q =1
N!qN
=
1
N!
VN
!3N
We can use this to construct the total partition function:
for the monoatomic ideal gas
which directly yields an expression for the Helmholtz energy
because
A =!kTlnQ
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Intramolecular degrees of freedom
So far we discussed the monoatomic ideal gas at nottoohigh T,
for which only translational degrees of freedom are relevant.
Then thesinglemolecule partition function essentiallyequals the
translational partition function:
q =qtrans = exp !!ltrans
kT
"
#$
%
&'
ltrans
(

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Hence, we can write the singlemolecule partition function as a product
q =qtrans
qint
qtrans = exp !!ltrans
kT
"#$
%&'ltrans
( qint = exp !!iint
kT
"#$
%&'iint
(with
check this; show that indeed
exp !!
ltrans+ !
iint
kT
"#$
%&'iint
(ltrans
( = exp !!
ltrans
kT
"#$
%&'
exp !!
iint
kT
"#$
%&'iint
("
#$%
&'ltrans(
"
#$
%
&'
Oftenintramolecular degrees of freedomare relevant:
q = exp ! !i
kT
"#$
%&'
i
( = exp !!ltrans
+ !iint
kT
"#$
%&'iint
(ltrans
(
Internaland translationaldegrees of freedom are independent;
!Energies are additive
A =!kTlnQ =kTN lnN!lnqtrans! lnqint!1( )
=kTN ln N"3
eV
#$%
&'(! lnq
int
#
$%&
'(8
qint
!qrotqvibqeleqnuc
Internal molecular degrees of freedom
The internal molecular state is a combination of:
the rotational, vibrational, electronic, and nuclear states
Sandler4.1
this presumes that
rotational, vibrational, electronic, and nuclear states are independent
!!! This is only approximately true under certain conditions !!!
(details later)
Q =1
N!qN
=
1
N!qtransqint( )
N
q =qtrans
qint
lnqint
!ln qrot+ lnqvib + lnqele + lnqnuc
check this
check this

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what modification to the expression for chemical potential
that we have seen so far:
arise from the intramolecular degrees of freedom?
A =kTN ln N!3
eV
"#$
%&'(lnq
int
"#$
%&'
=kTln !!3( )
!kTNlnqint
=kT ln !!3( )"lnqint( ) =kTln !!3
qint
#$%
&'(
Term of Helmholtz energy:
!
!NV,T
!kTlnqint
with intramolecular
degrees of freedom:
!!"#
independent ofN
?
10
Often ok to assume that only
ground states(states with lowest energy levels) are relevant.
(ok at moderate temperatures, then )!! >>kT
Nuclear and electronic states
There may be more than one state with the same energy level.
the nr. of states with same energy is called
the degeneracyor multiplicityof that energy level
Then !!" = !#$ !
!!"%"
#$
"
#$
%
&'"=&
(
) *"!"%&!#$ !!
!"%&
#$
"
#$
%
&' ="!"%&
choosing the ground state level
to be the zero of energy

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!1enters qjust as a constant factor.
Hence it enters the Helmholtz energy only as
aconstant contribution per molecule:
Irrelevant for measurable thermodynamic properties
(if no chemical reactions occur!)
So it is often taken to be just 1 (even if it is not).
Then terms in the free energy are simply omitted
Similar arguments apply tonuclear states.
qint
!qrotqvibqeleqnuc !qrotqvib
!"!#!$%
lnqel,1
= ln1=0
( )
hence:
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Polyatomic molecules have vibrationaland rotationaldegrees
of freedom.
It is a reasonable approximation to assume that vibrational and
rotational degrees of freedom are independent:
qint
! qvib
qrot
This is not completely exact; it neglects centrifugal effects
Vibrational and rotational states

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qvib
!qvib
1
qvib
2
qvib
3
. . .
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Molecular vibrations
If vibration energies not too high,
molecular vibrations can be approximated as
harmonic oscillation (oscillations in a parabolic potential energy),
and as
separable in independent vibrational normal modes.
The number of modes depends on the molecular structure.
Hence
Sandler4.1c
diagram by Derek Kverno at http://www.cartage.org.lb/en/themes/sciences/Physics/MolecularPhysics/MolecularSpectra/MolecularSpectroscopy/MolecularSpectroscopy.htm
vibration modes of the CO2 molecule
(linear triatomic molecule!threemodes)A diatomicmolecule
has only onevibration
mode
more complex
molecules!more
vibrational modes
In the following we
focus on just one mode,
indicated by subscript vib
without a number.14

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Harmonic potential(= parabolic potential):
V(x) = 12 Kx2 ! F(x) = "
dV
dx= "Kx
x= displacement of oscillator from its minimumenergy state
K= spring constant
V= potential energy
F= restoring force
We have chosen
For mass min a parabolic potential
the vibrationfrequencyis
V(0) = 0
! =
1
2!
K
m
from Classical Mechanics:Christiaan Huygens 1673Horologium oscillatorium sive de motu pendulorum
more generally:
mto be replaced by a reduced mass
e.g. diatomic molecule: m1m
2 m
1+ m
2( )
(Hookes law)
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According to the
Schrdinger equation for
a harmonic potential energy well
!vib,j = j+12( )h!
(j = 0, 1, 2, . . . !)
equidistant energy levels: !!vib
=h"
ground state energy:(fromh
ttp://131.1
04.1
56.2
3/Lectures/CHEM_
207/vibrational_spectroscopy)
The temperature for which is!"vib= kT !vib =h" k
the energy levels are
!O 1000K( )
choosing the minimumpotential as the zero of energy:
!0 =
1
2h"!
!
!
h"
j = 0
j = 1
j = 2
j = 3
j =4
j = 5
j = 6

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If that is if T
then only the ground state is occupied,
the higher energy states are virtually unoccupied.
With the Boltzmann distribution law in mind,
we understand immediately, without any further calculations,
that
This is usually approximately true at room temperature since
(this is the lowTlimit)
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The partition function for one vibrational mode is
!!"#
= $%& !!
!
"#
"
#$
%
&'
!=0
(
) =
=exp ! h!
2kT
"#$
%&'
exp !h!
kT
"#$
%&'
"#$
%&'
j
j=0
(
)using (geometric series)xi
i=0
!
" =1 1#x( )
qvib =exp ! 12h! kT( )1! exp !h! kT( )
=
exp ! 12!vib T( )1! exp !!
vib T( )
= exp ! h!
2kT
"#$
%&'exp !
jh!
kT
"#$
%&'j=0
(
)
!"# ! !+ 1
2( )!!!"
"
#$$
%
&''
#=0
(
)
!!"#
!
$"
%vibrational temperature

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(energylevel spacing )
then ~ only the ground state occupied:
T ! h" k
qvib! exp " h!
2kT
#$%
&'( =exp "
"vib
2T
#$%
&'(
qvib =exp ! 12h! kT( )1! exp !h! kT( )
!!vib
=h" >>kT
Helmholtz energy: avib =!
kTln
qvib =
1
2h!
entropy:
Contributionsper moleculeto the:
energy:(mean energy of vibrational mode)
!!"#
= !"#
$%& '
"1 !=
1
2!"
heat capacity: !!"#
=
!!!"#
!"= 0
!!"#
= !!"#! !
!"#( ) ! = 0
Low temperature limit
(just the first term of partition func.)
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energylevel spacing : classical limit
qvib
!
kT
h!=
T
"vib
!">> #!
Helmholtz energy: avib
=!kTlnqvib =!kTlnkT
h!
Contributions per molecule to the
energy: !vib= k
d avib
T
d1 T= k
dln1 T
d1 T= k
1
1 T= kT
heat capacity: cvib =d!
vib
dT= k
with for x

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Graphical summaryof dependence of
"viband cvibon T
!
Cvib
k!
lowTlimit:
just the
groundstate energy
lowTlimit:
exac
tresul
t
highTlimit:
high
Tlimit
!vib
!vib
NB:
!vib "O(1000K)
!vib
=h"
eh" kT
!1+h"
2
!!
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Molecular rotations
Assuming that vibrations and rotations are independent:
qint
= qvib
qrot
implies that rotations are treated as if molecules are rigid rotors
(neglecting centrifugal effects)
Sandler4.1b
 monoatomic molecules: 0 rotational modes
 linear molecules: 2 rotational modes
 nonlinear molecules: 3 rotational modes

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Heteronucleic diatomic molecule (e.g. CO)
R R
mA
mB
=mAmB
mA+m
B
reduced mass
rotation of diatom AB with AB distance =R
equivalent withmovement of a mass on the surface of a sphere with radiusR
Hence in the classical limit the partition function
is the same as that for translational movement
of a particle with massin a 2D box of area :4!R2
~
2424
qrot =2!kT
h2!"#
$%&
2 2
4!R2
= moment of inertiaI= R2
qrot =8!
2IkT
h2
=
T
!rot
= rotational temperature!rot
=
h2
8!2Ik
= translational singleparticle partition function
of a particle with massin a 2D box of size
in the classical limit (check this)
4!R2
So we can use expression ~ translational motion

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If the classical limit does not apply
(low T, or smaller)
the actual values of the lowest energy levels are important.
These do not just depend on the size, but also on the shape
of a 2D box.
Then the correct rotational energy levels have to be used.
Makes things more complicated (see Sandler)
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kT ! "!
26
Homonuclear diatomic molecule (e.g. N2)
The two nuclei are indistinguishable
Rotational partition function as given above
needs to be devided by 2:
Generally:
where thesymmetry number ! =1 heteronuclear
2 homonuclear
"#$
qrot =8!
2IkT
"h2
=
T
"!r
Molecular symmetry
each orientation of a homonuclear = symmetric linear molecule
corresponds to
twoorientations of a heteronuclear = asymmetric linear molecule

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qrot =8!
2IkT
"h2
The contribution to the Helmoltz energy per molecule is
arot =!kTln qrot with
The contribution to the energy per molecule is
!!"#
=
$%!"#
&
$1 &= '
$!"1 &
$1 &= '&
That is per degree of freedom
(a diatomic molecule has two rotational degrees of freedom)
1
2kT
For nonlinear molecules: !rot =3
2kT
crot=
d!rot
dTThe contribution to the heat capacity per molecule is
rotations (classical limit applies)
translations (classical limit applies)
vibrations (classical limit may be used )
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Overview for molecular systems at room temperature:
!"

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The Law of Equipartition of Energies
The Equipartition Theorem
for translations and rotations and vibrations
the mean energy per molecule, per degree of freedom is always
1
2kT
For vibrations it is kTper molecule for each normal mode.
This is in the classical limit
for potential energy and
for kinetic energy.
1
2kT
1
2kT
In the classical limit(high temperatures),
Equipartition = division in equal parts
each degree of freedom contributes an equal amount to energy12kT
30
if the energy for a degree of freedom of a particle can be written
as Cy2
where Cis some constant coefficient
ydefines the state for a degree of freedom
ycan be,

in classical mechanical terms: a (angular) velocity
component. Then Cy2is a kinetic energy.
 a position coordinate in the case of a harmonic potential.
Then Cy2is a potential energy.
 in quantum mechanical terms: a quantum number, e.g. as for
translation (check e.g. in previous handouts)
Why so often this ?12kT
Because of the quadratic expressions for the energy:

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for each translational degree of freedom,
in classical mechanical terms: 12mv
x
2
for each rotational degree of freedom,
in classical mechanical terms: 12I!
2
I= moment of inertia, != angular velocity
for vibrational modes, kinetic energy:
+ potential energy; if vibration is harmonic: 12kx
2
(in quantum mechanical terms this is )! lx
2
1
2mv
x
2
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q = exp !Cy2
kT
"
#$%
&'dy
!(
(
) =!kT
C
the partition function
classical expression
y(treated as) continuous
the corresponding Helmholtz energy contribution
a =!kTln q =!kTln !kT
C=! 1
2kT ln
!k
C+ lnT
"
#
$%
&
'
with the GibbsHelmholtz relation:
! =d a T
d1 T=
1
2k
d ln1 T
d1 T=
1
2kT
This result does not depend on C,
only on the quadratic character of !y =Cy
2
ydefines microstate of some mode
if energy of the state can be written as !y = Cy
2

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Give a rough estimate of the heat capacity of a mole of CObased on the equipartition theorem
at ~ room temperature (assuming lowTlimit can be used for vibr)
at ~ 5000 K (assuming that it does not decompose)
vibrations (lowTlimit)
!"