1 ideal diatomic gas: internal degrees of freedom polyatomic species can store energy in a variety...
TRANSCRIPT
1
Ideal diatomic gas: internal degrees of freedom
• Polyatomic species can store energy in a variety of ways:
– translational motion– rotational motion– vibrational motion– electronic excitation
Each of these modes has its own manifold
of energy states, how do we cope?
2
Internal modes: separability of energies
• Assume molecular modes are separable – treat each mode independent of all others– i.e. translational independent of vibrational,
rotational, electronic, etc, etc
Entirely true for translational modesVibrational modes are independent of:– rotational modes under the rigid rotor
assumption– electronic modes under the Born-
Oppenheimer approximation
3
Internal modes: separability of energies
Thus, a molecule that is moving at high speed is not forced to vibrate rapidly or rotate very fast.
An isolated molecule which has an excess of any one energy mode cannot divest itself of this surplus except at collision with another molecule.
The number of collisions needed to equilibrate modes varies from a few (ten or so) for rotation, to many (hundreds) for vibration.
4
Internal modes: separability of energies
Thus, the total energy of a molecule j:
jel
jvib
jrot
jtrs
jtot
5
Weak coupling: factorising the energy modes
• Admits there is some energy interchange – in order to establish and maintain thermal
equilibrium
• But allows us to assess each energy mode as if it were the only form of energy present in the molecule
• Molecular partition function can be formulated separately for each energy mode (degree of freedom)
• Decide later how individual partition functions should be combined together to form the overall molecular partition function
6
Weak coupling: factorising the energy modes
• Imagine an assembly of N particles that can store energy in just two weakly coupled modes and
• Each mode has its own manifold of energy states
and associated quantum numbers
• A given particle can have:- -mode energy associated with quantum number k
- -mode energy associated with quantum number r
rktot
7
Weak coupling: factorising the energy modes
The overall partition function, qtot:
statesall
totrieq
)(
expanding we would get:
...
)()()()(
)()()()(
)()()()(
)()()()(
33231303
32221202
31211101
30201000
eeee
eeee
eeee
eeeeqtot
8
Weak coupling: factorising the energy modes
but e(a+b) = ea.eb, therefore:
...
......
......
......
221202
211101
201000
eeeeee
eeeeee
eeeeeeqtot
each term in every row has a common factor of
in the first row, in the second, and so on. Extracting these factors row by row:
0e 1e
9
Weak coupling: factorising the energy modes
...
...)(
...)(
...)(
...)(
32103
32102
32101
32100
eeeee
eeeee
eeeee
eeeeeqtot
the terms in parentheses in each row are identical and form the summation:
statesall
je
10
Weak coupling: factorising the energy modes
statesallstatesall
statesalltot
jj
j
exe
eeeeeq
...3210
If energy modes are separable then we can factorise the partition function and write:
qxqqtot
11
Factorising translational energy modes
jztrs
jytrs
jxtrs
jtottrs ,,,,
which allows us to write:
ztrsytrsxtrstrs
stateszallstatesyallstatesxalltrs
statesallstatesalltrs
qxqxqq
exexeq
eeq
ztrsytrsxtrs
ztrsytrsxtrstottrs
,,,
)(
,,,
,,,,
Total translational energy of molecule j:
12
Factorising internal energy modes
elvibrottrstot qqqqq ...using identical arguments the canonical partition function can be expressed:
elvibrottrstot QQQQQ ...
Total translational energy of molecule j:
but how do we obtain the canonical from the molecular partition function Qtot from qtot? How
does indistinguishability exert its influence?
13
Factorising internal energy modes
When are particles distinguishable (having distinct configurations, and when are they indistinguishable?
• Localised particles (unique addresses) are always distinguishable
• Particles that are not localised are indistinguishable– Swapping translational energy states between such
particles does not create distinct new configurations
• However, localisation within a molecule can also confer distinguishability
14
Factorising internal energy modes
When molecules i and j, each in distinct rotational and vibrational states, swap these internal states with each other a new configuration is created and both configurations have to be counted into the final sum of states for the whole system. By being identified specifically with individual molecules, the internal states are recognised as being intrinsically distinguishable.
Translational states are intrinsically indistinguishable.
15
Canonical partition function, Q
This conclusion assumes weak coupling. If particles enjoy strong coupling (e.g. in liquids and solutions) the argument becomes very complicated!
Nelvibrottrstot
Nel
Nvib
Nrot
Ntrs
tot
qqqqN
Q
qqqN
...!
1!
and thus:
16
Ideal diatomic gas: Rotational partition function
Assume rigid rotor for which we can write successive rotational energy levels, J, in terms of the rotational quantum number, J.
1
12
2
2
)1(
)1(8
)1(8
cmJBJ
cmJJIc
h
hc
E
JJI
hE
JJ
J
joules
where I is the moment of inertia of the molecule, is the reduced mass, and B the rotational constant.
17
Ideal diatomic gas: Rotational partition function
Another expression results from using the characteristic rotational temperature, r,
)()1(8 2
2
jouleskJJE
hcBkk
hcB
Ik
h
rJ
rr
• 1st energy increment = 2kr
• 2nd energy increment = 4kr
18
Ideal diatomic gas: Rotational partition function
Rotational energy levels are degenerate and each level has a degeneracy gJ = (2J+1). So:
TJJkTJrot
rJ eJegq /)1(/ )12(
If no atoms in the atom are too light (i.e. if the moment of inertia is not too small) and if the temperature is not too low (close to 0 K), allowing appreciable numbers of rotational states to be occupied, the rotational energy levels lie sufficiently close to one another to write:
19
Ideal diatomic gas: Rotational partition function
2
2
0
/)1(
8
)12(
h
IkTTq
eJq
rrot
dJTJJrot
r
• This equation works well for heteronuclear diatomic molecules.
• For homonuclear diatomics this equation overcounts the rotational states by a factor of two.
20
Ideal diatomic gas: Rotational partition function• When a symmetrical linear molecule rotates
through 180o it produces a configuration which is indistinguishable from the one from which it started. – all homonuclear diatomics
– symmetrical linear molecules (e.g. CO2, C2H2)
• Include all molecules using a symmetry factor
rrot
Tq
= 2 for homonuclear diatomics, = 1 for heteronuclear diatomics = 2 for H2O, = 3 for NH3, = 12 for CH4 and C6H6
21
Rotational properties of molecules at 300 K
r/K T/r qrot
H2 88 2 3.4 1.7
CH4 15 12 20 1.7
HCl 9.4 1 32 32
HI 7.5 1 40 40
N2 2.9 2 100 50
CO 2.8 1 110 110
CO2 0.56 2 540 270
I2 0.054 2 5600 2800
22
Rotational canonical partition function
Nrotrot qQ
relates the canonical partition function to the molecular partition function. Consequently, for the rotational canonical partition function we have:
NNN
rrot h
IkT
hcB
TTQ
2
28
23
Rotational Energy
2
28lnlnln
h
IkNTNQrot
this can differentiated wrt temperature, since the second term is a constant with no T dependence
molecules)diatomic(forNkTU
TT
NkTT
QkTU
rot
V
rotrot
ln
ln 22
24
Rotational heat capacity
molecules)(linearRC
RTU
mrot
mrot
,
,
this equation applies equally to all linear molecules which have only two degrees of freedom in rotation. Recast for one mole of substance and taking the T derivative yields the molar rotational heat capacity, Crot, m. Thus, when N = NA, the molar rotational energy is Urot,m
molecules)diatomic(forNkTU rot
25
Rotational entropy
53.106//ln/ 12 KTmkgIRSrot
Srot is dependent on (reduced) mass (I = r2), and there is also a constant in the final term, leading to:
2
2
2
2
8lnln1
8ln
lnlnln
h
kITNkS
h
IkTk
T
NkT
QkT
UQk
T
QkTS
rot
N
rot
Vrot
26
Rotational entropy
282 1010 trsrot qbutq
Typically, qrot at room T is of the order of hundreds for diatomics such as CO and Cl2. Compare this with the almost immeasurably larger value that the translational partition function reaches.
27
Extension to polyatomic molecules
2
1
,,,
zryrxrrot
TTTq
• In the most general case, that of a non-linear polyatomic molecule, there are three independent moments of inertia.
• Qrot must take account of these three moments – Achieved by recognising three independent characteristic
rotational temperatures r, x, r, y, r, z corresponding to the three principal moments of inertia Ix, Iy, Iz
• With resulting partition function:
28
Conclusions
• Rotational energy levels, although more widely spaced than translational energy levels, are still close enough at most temperatures to allow us to use the continuum approximation and to replace the summation of qrot with an integration.
• Providing proper regard is then paid to rotational indistinguishability, by considering symmetry, rotational thermodynamic functions can be calculated.
29
Ideal diatomic gas: Vibrational partition function
Vibrational modes have energy level spacings that are larger by at least an order of magnitude than those in rotational modes, which in turn, are 25—30 orders of magnitude larger than translational modes.– cannot be simplified using the continuum
approximation– do not undergo appreciable excitation at room
Temp.
– at 300 K Qvib ≈ 1 for light molecules
30
The diatomic SHO modelWe start by modelling a diatomic molecule on a simple ball and spring basis with two atoms, mass m1 and m2, joined by a spring which has a force constant k.The classical vibrational frequency, oscis given by:
Hzk
osc
2
1
There is a quantum restriction on the available energies:
...),2,1,0(2
1
voscvib hv
31
The diatomic SHO model
The value is know as the zero point energy
• Vibrational energy levels in diatomic molecules are always non-degenerate.
• Degeneracy has to be considered for polyatomic species
– Linear: 3N-5 normal modes of vibration– Non-linear: 3N-6 normal modes of vibration
osch2
1
32
Vibrational partition function, qvib
• Set 0 = 0, the ground vibrational state as the reference zero for vibrational energy.
• Measure all other energies relative to reference ignoring the zero-point energy.
– in calculating values of some vibrational thermodynamic functions (e.g. the vibrational contribution to the internal energy, U) the sum of the individual zero-point energies of all normal modes present must be added
33
Vibrational partition function, qvib
The assumption (0 = 0) allows us to write:
Thvib
hhhvib
vib
vib
eeq
eeeeq
hhhh
/
32
4321
1
1
1
1
...1
...,4,3,2,
Under this assumption, qvib may be written as:
a simple geometric series which yields qvib in closed form:
where vib = h/k = characteristic vibrational temperature
34
Vibrational partition function, qvib
• Unlike the situation for rotation, vib, can be identified with an actual separation between quantised energy levels.
• To a very good approximation, since the anharmonicity correction can be neglected for low quantum numbers, the characteristic temperature is characteristic of the gap between the lowest and first excited vibrational states, and with exactly twice the zero-point energy, .
osch2
1
35
Ideal diatomic gas: Vibrational partition function
Vibrational energy level spacings are much larger than those for rotation, so typical vibrational temperatures in diatomic molecules are of the order of hundreds to thousands of kelvins rather than the tens of hundreds characteristic of rotation.
Species vib/Kqvib
(@ 300 K)
H2 5987 1.000
HD 5226 1.000
D2 4307 1.000
N2 3352 1.000
CO 3084 1.000
Cl2 798 1.075
I2 307 1.556
36
Vibrational partition function, qvib
• Light diatomic molecules have:
– high force constants – low reduced masses
• Thus:– vibrational frequencies (osc) and characteristic vibrational
temperatures (vib) are high
– just one vibrational state (the ground state) accessible at room T
• the vibrational partition function qvib ≈ 1
kosc 2
1
37
Vibrational partition function, qvib
• Heavy diatomic molecules have:– rather loose vibrations – Lower characteristic temperature
• Thus:– appreciable vibrational excitation resulting in:
• population of the first (and to a slight extent higher) excited vibrational energy state
• qvib > 1
38
Vibrational partition function, qvib
• Situation in polyatomic species is similar complicated only by the existence of 3N-5 or 3N-6 normal modes of vibration.
• Some of these normal modes are degenerate
(1), (2), (3), … denoting individual normal modes 1, 2, 3, …etc.
Species vib/K∏(qvib)
(@ 300 K)
CO2 3360 1.091
1890
954(2)
NH3 4880(2) 1.001
4780
2330(2)
1360
CHCl3 4330 2.650
1745(2)
1090(2)
938
523
374(2)
...)3()2()1()( xqxqxqqq vibvibvibnvib
totvib
39
Vibrational partition function, qvib
As with diatomics, only the heavier species show values of qvib appreciably different from unity.
Typically, vib is of the order of ~3000 K in many molecules. Consequently, at 300 K we have:
in contrast with qrot (≈ 10) and qtrs (≈ 1030)
For most molecules only the ground state is accessible for vibration
11
110
eqvib
40
High T limiting behaviour of qvib
At high temperature the equation gives a linear dependence of qvib with temperature.
If we expand , we get:
Tvib vibeq /1
1
Tvibe /1
vibvibvib
T
Tq
...)/(11
1High T limit
41
T dependence of vibrational partition function
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.41.0
1.2
1.4
1.6
1.8
2.0
qvi
b
Reduced Temperature T/
As T increases, the linear dependence of qvib upon T becomes increasingly obvious
42
The canonical partition function, Qvib
Using we can find the first
differential of lnQ with respect to
temperature to give:
VT
QkTU
ln2
N
TNvibvib vibeqQ
/1
1
1
ln/
2
T
vib
Vvib vibe
Nk
T
QkTU
43
The vibrational energy, Uvib
This is not nearly as simple as:
)1( /, T
vibmvib vibe
RU
RTU
kTU
mrot
mtrs
,
, 2
3
linear molecules
44
The vibrational energy, Uvib
This does reduce to the simple form at equipartition (at very high temperatures) to:
)1( /, T
vibmvib vibe
RU
Re
RU
RTU
mvib
mvib
7
1
)1(
300010,
,
Normally, at room T:
(equipartition)
45
The zero-point energy• So far we have chosen the zero-point energy
(1/2h) as the zero reference of our energy scale
• Thus we must add 1/2h to each term in the energy ladder
• For each particle we must add this same amount– Thus, for N particles we must add U(0)vib, m = 1/2Nh
hNe
R
Ue
RU
ATvib
mvibTvib
mvib
vib
vib
2
1
)1(
)0()1(
/
,/,
46
Vibrational heat capacity, Cvib
The vibrational heat capacity can be found using:
2/
/2,
, )1(
T
Tvib
V
mvibmvib vib
vib
e
e
TR
T
UC
The Einstein Equation
This equation can be written in a more compact form as:
T
RC vibEmvib
F,
47
Vibrational heat capacity, Cvib
FE with the argument vib/T is the Einstein function
The Einstein function
Tu
e
eu vibu
u
E
2
2
)1(F
48
The Einstein heat capacity
0.1 1 100.0
0.5
1.0
FE
Reduced temperature T/
High Tlow T
49
The Einstein function
• The Einstein function has applications beyond normal modes of vibration in gas molecules.
• It has an important place in the understanding of lattice vibrations on the thermal behaviour of solids
• It is central to one of the earliest models for the heat capacity of solids
50
The vibrational entropy, Svib
vibvibvib
vibvibvibvibvib
QkT
UUT
AA
T
UUS
ln)0(
)0()0(
Nvibvib qQ • We know and N = NA for one mole, thus:
TT
vibmvib
vibvibAvib
vib
vibe
e
T
R
S
qRqkNQ
//
, 1ln)1(
/
lnlnln
51
Variation of vibrational entropy with reduced temperature
0.1 1 100.0
0.5
1.0
1.5
2.0
2.5
3.0
Svi
b/R
Reduced temperature T/
TT0
52
Electronic partition function• Characteristic electronic temperatures, el,
are of the order of several tens of thousands of kelvins.
• Excited electronic states remain unpopulated unless the temperature reaches several thousands of kelvins.
• Only the first (ground state) term of the electronic partition function need ever be considered at temperatures in the range from ambient to moderately high.
53
Electronic partition function
It is tempting to decide that qel will not be a significant factor. Once we assign 0 = 0, we might conclude that:
1)(00/,
i
kTel termshighereeq iel
To do so would be unwise!One must consider degeneracy of the
ground electronic state.
54
Electronic partition function
The correct expression to use in place of the previous expression is of course:
00
0/
)(0, gtermshigheregegqi
kTiel
iel
Most molecules and stable ions have non-degenerate ground states. A notable exception is molecular oxygen, O2, which has a ground state degeneracy of 3.
55
Electronic partition functionAtoms frequently have ground states that are degenerate. Degeneracy of electronic states determined by the value of the total angular momentum quantum number, J.Taking the symbol as the general term in the Russell—Saunders spin-orbit coupling approximation, we denote the spectroscopic state of the ground state of an atom as:
spectroscopic atom ground state = (2S+1)J
56
Electronic partition functionspectroscopic atom ground state = (2S+1)J
where S is the total spin angular momentum quantum number which gives rise to the term multiplicity (2S+1). The degeneracy, g0, of the electronic ground states in atoms is related to J through:
g0 = 2J+1 (atoms)
57
Electronic partition functionFor diatomic molecules the term symbols are made up in much the same way as for atoms.
• Total orbital angular momentum about the inter-nuclear axis. Determines the term symbol used for the molecule ( etc. corresponding to S, P, D, etc. in atoms).As with atoms, the term multiplicity (2S+1) is added as a superscript to denote the multiplicity of the molecular term.
58
Electronic partition functionIn the case of molecules it is this term multiplicity that represents the degeneracy of the electronic state.For diatomic molecules we have:
spectroscopic molecular ground state = (2S+1)
for which the ground-state degeneracy is:g0 = 2S + 1 (molecules)
59
Electronic partition function
SpeciesTerm
Symbol gn el/K
Li 2S1/2 g0 = 2
C 3P0 g0 = 1
N 4S3/2 g0 = 4
O 3P2 g0 = 5
F 2P3/2 g0 = 42P1/2 g1 = 2 590
NO 21/2 g0 = 223/2 g1 = 2 178
O23-
g g0 = 31g g1 = 1 11650
60
Electronic partition functionWhere the energy gap between the ground and the first excited electronic state is large the electronic partition function simply takes the value g0.
When the ground-state to first excited state gap is not negligible compared with kT (el/T is not very much less than unity) it is necessary to consider the first excited state.The electronic partition function becomes:T
eleleggq /
10
61
Electronic partition function
For F atom at 1000 K we have:
109.524 1000/590/10 eeggq T
elel
674.322 1000/178/10 eeggq T
elel
For NO molecule at 1000 K we have: