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Calculation of absorption coefficient for the ν2 and ν3 bands of sulfur dioxide
.L Quant. Spectrosc. Rodiat. Transfer, Vol. 17, pp. 493-499. Pergamon Press 1977. Printed in Great Britain
CALCULATION OF ABSORPTION COEFFICIENT FOR T H E v2 A N D v3 B A N D S O F S U L F U R D I O X I D E M. OSUMI Research and Development Center, Sanyo Electric Co. Ltd., Hirakata, Osaka, Japan
and T . KUNITOMO Department of Engineering Science, Faculty of Engineering, Kyoto University, Kyoto, Japan
(Received 16 July 1976) Abstract--Absorption coefficients Sld of sulfur dioxide were calculated in the v2 and v3 bands for a temperature range from 300 to 1200°K. Calculations were carried out for such vibrational transitions that their integrated absorption coefficients a at T°K are greater than 0.1% of the total integrated absorption coefficients a °~' observed at 300°K. Rotational lines with intensities greater than 10-~ cm -2 atm-' STP were included in the calculation. The line intensities were averaged over small wavenumber intervals of Ato = 10 cm-' throughout the bands to obtain the absorption coefficients Sld. They were compared with the experimental values of other investigators and good agreement was obtained.
E F Go J
K_,K,
O
S T To W £
d h k
NOTATION total energy, cm-' rotational energy, cm ' vibrational energy above the ground state, cm -I rotational quantum number index to distinguish rotational level total partition function line intensity, cm -2 atm-' temperature, °K reference temperature, °K asymmetry coefficient of rotational energy speed of light, cm sec-' line spacing, cm ' Planck's constant, erg sec Boltzmann's constant, erg°K -' vibrational quantum number associated with the vibrational mode v~
Greek symbols a r I#-[ v, to ~oo o~, to*
integrated absorption coefficient, cm -: atm -~ asymmetry parameter rotational matrix element normal mode of vibration line position, cm -z band center, cm-' frequency of normal vibrations, cm-' band center of the fundamental band, cm -'
Subscripts i is a quantity associated with the ith mode of vibration v is a quantity which takes the effect of vibration into consideration
Superscripts ",' denote the values in the initial and final states involved in a transition, respectively iso denotes the value for the isotopic species $3402
1. I N T R O D U C T I O N
THE AnSORPTIONcoefficient SId is a parameter of importance, comparable to a line shape parameter, and governs gaseous absorption and emission of radiation in the infrared region. It is a thermophysical property of interest in determining the concentration of combustion or exhaust gases, in estimating the amount of the gas in the atmosphere or in studying environmental or
geophysical heat-transfer problems. QSRT Vol. 17, No. 4--D
493
494
M. OSUMIand T. KUNITOMO
The i.r. spectra of the sulfur dioxide molecule consist of a great number of rotational lines; its principal bands are at 8.7 gm (v,), 7.4 gm (v~) and 19.3 gm (v2). The authors "~ reported previously the absorption coefficients for the u, band by calculation from normal to high temperature and obtained results in good agreement with experimental values. This paper presents calculated absorption coefficients of the v2 and u3 bands of sulfur dioxide from 300 to 1200°K. Using also the results presented in previous paper for the v, band,"' we have now described the main absorption bands of sulfur dioxide. 2. THEORY 2.1 Vibrational and rotational energies The vibrational energy in the state (v, v:v3) above the ground state for the species $3~O2 can be expressed in the form ~2) Go(v~ v2v3) = 1151.74v~ + 522.44v2 + 1369.07v3 - 3.99v, 2 - 3.00v2: - 5.17v32
- 2 . 0 5 v , v 2 - 13.71v,vs+O.39v2v3+... (cm-').
(1)
The rotational energy in the state JK ,K,, for the species S~202, is Fv(JK ,K,; K) = (1.1587 - 0.00019v, + 0.01943 v2 - O.Ol092v~)J(J + 1)
+(0.8644 + 0.00123 v, + 0.01996vz - 0.00491 v3) WK ~K,(x) (cm-')
(2)
where = -(0.8165 + 0.00149Vl + 0.01936v2 - 0.00982 v~)/(0.8644 + 0.0013 v, + 0.01996v2 - 0.00491v3); (3) here v,, v2 and v~ are vibrational quantum numbers and J is rotational quantum number. (2~The asymmetry coefficient W,¢ ,,¢,(x) cannot be expressed as a simple function of the rotational quantum numbers. It was determined by using the same program as in the previous paper. "'~ The total energy is then obtained as the sum of the vibrational energy and the rotational energy, i.e. E(vt v2v3; JK ,r,) = Go(V, v2vs) + F~(Jr_,r, ; r).
(4)
The energy for the species S~40~ can be calculated by using the vibrational and rotational constants for the $3'O~. "'~) 2.2 Rotational line intensities and their wavenumbers The wavenumber of the line due to the transition (v,v~v~,J,¢ ......... ,K,) ~ (v,v2v~,J,¢_,,¢,)' ' " ' is t t I. ! - E ( v l v.2. .v. 3. .,. to = E(vlv2v3, JK_,K,)
j
tv
-
~,K,)
! . vv . = too + Fo,(Jg_,r,, ~ v ) - Fo,~JK_,r,,
KV~).
(5)
The line intensity, S, for the transition is obtained in terms of the integrated absorption coefficient a °b~ at 300°K, viz. S=a
ob~ To to . ,, (- hcE"/kT) T-~oo I'v' + l)l/'tl2 exp Q~Q,
1 - exp ( - hcto/kT) 1-exp(-hcto*/kT)'
(6)
where too is the band center; to* is that of the fundamental band; Q is the total partition function; Igl 2 is the square of rotational matrix element, which has been calculated for the t,= and t'3 bands; ¢L4' v'[ is the vibrational quantum number of the lower state associated with the vibrational transition. 3. CALCULATION 3.1 Calculation criteria The wavenumbers and the line intensities were calculated for the allowed transitions from
Calculation of absorption coefficient for the v2 and v3 bands of sulfur dioxide
495
eqns (5) and (6), respectively, to obtain the absorption coefficient. Weak lines with intensities less t h a n 10 - 6 c m -2 atm-' STP were neglected since they do not contribute appreciably to the bands. Calculations were performed from 300 to 1200°K. Hot bands with band intensities greater than 0.1% of a °bs at 300°K were included in the calculations. The number of transitions that must be included increase rapidly as the temperature is raised. 3.1.1 The v2 band. Since the v2 band consists of subbands due to the transitions &v2 = 1, calculations were carried out for these transitions. The value of 120 cm 2 atm-' stp ~5~was used for a °b" at 300°K. The numbers of vibrational transitions considered were 7, 17, 43 and 69 for 300, 600, 900 and 1200°K, respectively. Table 1 shows the band centers and the integrated absorption coefficients for the main transitions at each temperature. It can be seen that, with the increase in temperature, the number of hot bands must be increased to obtain the required total band intensity a °bs for the band. The selection rule for rotational transitions for the v2 band ~ are AJ=0,+I
(J"=O~J'=O);
AK_,=+I,+3
.... ;
AK,=+I,+3
....
(7)
since the oscillating dipole moment changes along the axis of the intermediate moment of inertia. The most intense line occurs for the transitions AK_, = + 1 and zlK~ = + 1. Rotational lines between 410 and 630 cm -' were calculated by using the selection rules for the v2 band. The region was divided into small intervals of &~o = 10 cm-' to calculate the absorption coefficients, following the procedure used in the previous paper, "~ i.e.
(8)
SIal = ~, (Sld)j, J
where (S/d)j is the jth subband of the v2 band. Table 1. The band centers and the integrated absorption coefficients for principal transition of the ~,2band v 2 Band (VlV2V 3) -- (VlV2+l v 3) ~(cm-2atm-iSTP) (VlV2V3)
~o
300°K
600°K
900°K
1200°K
517.29
96.51
53.06
27.71
15.17
010
511.29
15.92
30.16
23.78
15.99
020
505.29
2.02
13.04
15.45
12.73
030
499.29
0.24
5.08
9.00
9.07
040
493.29
1.88
4.97
6.10
050
487.29
0.68
2.65
3.97
060
481.29
1.39
2.53
00~
$3202
070
475.29
0.72
1.59
080
469.29
0.37
0.98
090
463.29
0.19
0.61
i00
515.24
Ii0
0.38
3.33
4.37
3.79
509.24
1.90
3.76
4.00
120
503.24
0.83
130
497.24
140 150
2.45
3.19
1.43
2.28
491.24
0.79
1.54
485.24
0.42
1.00
160
479.24
0.22
0.64
200
513.19
0.21
0.70
0.95
210
507.19
0.12
0.60
1.01
220
501.19
0.39
0.81
012
503.47
001
513.38
0.30
0.61
2.00
3.10
011
2.93
507.38
1.15
2.68
3.10
021
501.38
0.50
1.75
2.48
031
495,38
0.20
1.03
1,77
041
489.38
0.57
1,20
051
483.38
0.30
0.78
i01
511.33
0.50
0.74
111
505.33
0.43
0.79
121
499.33
0.28
0.63
1.20
0.69
000
$3402
517.35
0.14
0.13
4.23
2.22
496
M, OSUMIand T. KUNITOMO
3.1.2 The v~ band. The transitions At:3 = 1 were considered for the ~'3 band. The observed absorption coefficient of 853.3 cm -2 atm -~ STP ~6) was used for calculation of the line intensities. The numbers of transitions calculated were 6, 17, 47 and 68 for temperatures 300, 600, 900 and 1200°K, respectively. Table 2 presents the band centers and the integrated absorption coefficients of the main transitions at each temperature. The rotational selection rule for the ~'3 band differs from that for the ~,z and z,2 bands since the dipole moment changes along the axis of the least moment of inertia, viz. AJ=0,-+I;
AK_, = 0, -+2, -+4 . . . . ;
AK_,=-+I,-+3 . . . . .
(9)
The wavenumber region considered extends from 1200 cm -1 to 1400 cm-'. 3.2 Results and discussions 3.2.1 The v2 band. The mean line spacings for the effective rotational lines of the p2 band are shown in Fig. 1. They are given for the entire ~,2 band and for its main subbands. Their order of magnitude is 10-2 and it is found that there are enough rotational lines in the small intervals to calculate the absorption coefficients S/d. Figure 2 presents the mean absorption coefficients Sld at 300°K both for the entire v2 band and for its main subbands. The fundamental band dominates at 30&K. The line spacings of the 1,2 band are shown in Fig. 3 for all of the temperatures considered. Figure 4 shows the mean absorption coefficients Sld at 300, 600, 900 and 1200°K. As the temperature is raised, the absorption coefficient around the band center decreases while that in the wing region increases. The band centers for the transitions (VlV2V3)--->(v~v2+ l v3) are obtained from eqn (1) as
(1o)
a,o = 517.29 - 2.05v, - 6.0v2 + 0.39v3 Table 2. The band centers and the integrated absorptioncoefficients for principal transition of ~3 Band (VlV2V3) -- (VlV2V3+l) ~(cm-2atm-ISTP) (VlV2V3) 000
$3202
~o
300°K
600°K
900=K
1200=K
1361.75
747.9
510.7
310.6
187.8
147.2
135.3
100.6
010
1357.84
62.39
020
1353.93
5.36
43.06
59.54
54.28
030
1350.02
0.47
12.78
26.45
29.50
040
1346.11
3.85
11.86
16.14
050
1342.20
1.18
5.37
8.90
060
1338.29
100
1348.04
2.95
2.46
4.94
31.89
48.63
46.54
110
1344.13
9.24
21.26
24.99
120
1340.22
2.72
9.38
13.52
130
1336.31
4.18
7.36
140
1332.40
1.88
4.27
200
1334.33
7.71
11.64
210
1330.42
002
1341.07
012 022
2.03
3.38
6.27
11.94
21.32
1337.16
5.27
11.53
1333.25
2.35
6.27
102
1327.36
1.95
5.46
001
1351.41
69.74
72.62
2.25
2.16
38.66
011
1347.58
11.25
30.58
39.08
021
1343.59
3.32
13.54
21.18
031
1339.68
0.99
6.05
11.56
061
1335.77
I01
1337.70
111 121 201 003 000
2.73
6.36
11.16
18.29
1333.79
4.91
9.87
1329.88
2.18
5.36
1323.99
1.81
4.65
2.50
1330.73 $3402
1362.22
33.6
22.55
1.85
5.63
13.50
8.14
~,
band
Calculation of absorption coefficient for the v2 and v3 bands of sulfur dioxide 0,1
E"
v \ 0.01
300 °K ))2 band
~__~ (o0o)-(oI0) (010)-(020)
(020) - (030)
0.001
I
s3%2
(ooo>-(OlO}
t
400
I I I 500 550 WAVENUMBER (cm -1 )
450
1
I
I
I 600
Fig. 1. Mean line spacings for the v2 band and for its main subbands at 300°K,
)I0 II- I i t ~ I [ i "~2 band = (000)-(010) - - o - - (010)-(02.0) •
,,
(020)-(030)
--~
(000)- (010)
~ ~ , 300°K
i i i [
i
53402
% ¥~J "o
0.1I
01 ~ t 40O
45O
500 550 WAVENUMBER (cm "1)
600
Fig. 2. Absorption coefficients S[d for the u2 band and for its main subbands at 300~K.
0.~, i i , i
i
i i
i i i
))z band
I 0.01 ~
"o
0.00' ...9oo
00001 400
[
.
i
450
50O 550 WAVENUMBER (cm -1)
600
Fig. 3. Mean line spacings d f o r the entire ~'2 band.
49"/
498
M. OSUMf and T. KUNITOMO 10,
i
I
i
i
}
~
i
1
.
.
.
l
i
")2 band
o.,i7 F. oo
, I
oo1-,
400
{
450
.
500
.
WAVENUMBER
550 (¢rn-1)
600
4.Absorption coefficients S/d for the entire v2 band.
Fig.
and they are located at lower wavenumbers than the band center of the fundamental unless the ~3 vibrational mode is at a high level. Therefore, the band center of the v2 band moves toward lower wavenumbers as the temperature is raised. 3.2.2 The p3 band. The mean line spacings for the v~ band at each temperature are shown in Fig. 5. As the temperature is raised, more rotational lines are needed to describe the band. The absorption coefficients for the entire v3 band at 300, 600, 900 and I200°K are presented in Fig. 6. The characteristic feature of the v3 band compared to the v2 band is that the band width of the v2 band is narrow. This result is due to the fact that the p2 band contour spreads out since the P, Q and R branches for v2 subbands for the different K_, and AK l are located at lower and higher wavenumbers than the wavenumber of the band center of the subband. At the same time, the v3 band broadens less since the P, Q and R branches for the subband of the strongest transition AK_I = 0 overlap those for different K : I . The result of the present calculation at 300°K is compared with the absorption coefficients for the v3 band measured by CHANand TIEN(7) in Fig. 7. The dotted line in the figure shows that the results of the calculation corrected for the slit width &o = 30cm -t is consistent with the
0.1i ,
i ] l l l r
V3
i i i i
ii
ba~
0.01 l
600
300oK
/
~ ' ~ ~
0.001
013001
I
l
IIIllllll
l
1300
i i
1350 1400 WAVENUMSER (cm-1)
Fig. 5. Mean line spacings d for the entire v3 band.
Calculation of absorption coefficient for the v2 and ~3 bands of sulfur dioxide i 1 ~
,
1
1
499
,
~3 ~nd I
10
.
--
/
600
i
L
i
/
300"K
i
i
i
i
1
~
i
i
i
J
1300 1350 1400 WAVENUMf3ER (cmq )
l
Fig. 6. Absorption coefficients Sld for the entire ~3 band. ~00
~
,
,
_ _
10
~
~
1
i
,
i
,
band 300° K present~3 calculation "( Chartet al. experiment
i
] / / ~ i
1
i
,
,
'
,
7~
/\
!
I ]
I "o
I
o.1
0.01
I
,-/-
I
i
[
IJ/
II
1300
l
I
i
t
I
I
I
l,,
1350 1400 WAVENUMBER(cm-1)
Fig. 7. Comparison of SId calculated for the u3 band with the experimental result of Chan and Tien at 300°K.
experiment of Chart and Tien. Comparisons for the v2 band and for the p~ band at high temperatures could not be carried out since there are no experimental data.
1. 2. 3. 4. 5. 6. 7.
REFERENCES M. OSUMIand T. KUNITOMO,JQSRT 15, 1055 (1975). G. HERZBERG,Infrared and Raman Spectra--ll. Van Nostrand, New York (1968). G. KING, R. M. HAINERand P. C. CROSS, J. Chem. Phys. I1, 27 (1943). P. C. CRoss, R. M. HAINERand G. W. KING, J. Chem. Phys. 12, 210 (1944). J. MORCILLOand J. HERRANZ,J. Pulds inst. quire, ks. 10, 162 (1956). D. F. EFFERS,Jr. and E. D. SCHMm,J. Phys. Chem. 64, 279 (1960). S. H. CB^N and C. L. TIEN, J. Heat Trans. ASME Ser. C93, 172 (1971).
CALCULATION OF ABSORPTION COEFFICIENT FOR T H E v2 A N D v3 B A N D S O F S U L F U R D I O X I D E M. OSUMI Research and Development Center, Sanyo Electric Co. Ltd., Hirakata, Osaka, Japan
and T . KUNITOMO Department of Engineering Science, Faculty of Engineering, Kyoto University, Kyoto, Japan
(Received 16 July 1976) Abstract--Absorption coefficients Sld of sulfur dioxide were calculated in the v2 and v3 bands for a temperature range from 300 to 1200°K. Calculations were carried out for such vibrational transitions that their integrated absorption coefficients a at T°K are greater than 0.1% of the total integrated absorption coefficients a °~' observed at 300°K. Rotational lines with intensities greater than 10-~ cm -2 atm-' STP were included in the calculation. The line intensities were averaged over small wavenumber intervals of Ato = 10 cm-' throughout the bands to obtain the absorption coefficients Sld. They were compared with the experimental values of other investigators and good agreement was obtained.
E F Go J
K_,K,
O
S T To W £
d h k
NOTATION total energy, cm-' rotational energy, cm ' vibrational energy above the ground state, cm -I rotational quantum number index to distinguish rotational level total partition function line intensity, cm -2 atm-' temperature, °K reference temperature, °K asymmetry coefficient of rotational energy speed of light, cm sec-' line spacing, cm ' Planck's constant, erg sec Boltzmann's constant, erg°K -' vibrational quantum number associated with the vibrational mode v~
Greek symbols a r I#-[ v, to ~oo o~, to*
integrated absorption coefficient, cm -: atm -~ asymmetry parameter rotational matrix element normal mode of vibration line position, cm -z band center, cm-' frequency of normal vibrations, cm-' band center of the fundamental band, cm -'
Subscripts i is a quantity associated with the ith mode of vibration v is a quantity which takes the effect of vibration into consideration
Superscripts ",' denote the values in the initial and final states involved in a transition, respectively iso denotes the value for the isotopic species $3402
1. I N T R O D U C T I O N
THE AnSORPTIONcoefficient SId is a parameter of importance, comparable to a line shape parameter, and governs gaseous absorption and emission of radiation in the infrared region. It is a thermophysical property of interest in determining the concentration of combustion or exhaust gases, in estimating the amount of the gas in the atmosphere or in studying environmental or
geophysical heat-transfer problems. QSRT Vol. 17, No. 4--D
493
494
M. OSUMIand T. KUNITOMO
The i.r. spectra of the sulfur dioxide molecule consist of a great number of rotational lines; its principal bands are at 8.7 gm (v,), 7.4 gm (v~) and 19.3 gm (v2). The authors "~ reported previously the absorption coefficients for the u, band by calculation from normal to high temperature and obtained results in good agreement with experimental values. This paper presents calculated absorption coefficients of the v2 and u3 bands of sulfur dioxide from 300 to 1200°K. Using also the results presented in previous paper for the v, band,"' we have now described the main absorption bands of sulfur dioxide. 2. THEORY 2.1 Vibrational and rotational energies The vibrational energy in the state (v, v:v3) above the ground state for the species $3~O2 can be expressed in the form ~2) Go(v~ v2v3) = 1151.74v~ + 522.44v2 + 1369.07v3 - 3.99v, 2 - 3.00v2: - 5.17v32
- 2 . 0 5 v , v 2 - 13.71v,vs+O.39v2v3+... (cm-').
(1)
The rotational energy in the state JK ,K,, for the species S~202, is Fv(JK ,K,; K) = (1.1587 - 0.00019v, + 0.01943 v2 - O.Ol092v~)J(J + 1)
+(0.8644 + 0.00123 v, + 0.01996vz - 0.00491 v3) WK ~K,(x) (cm-')
(2)
where = -(0.8165 + 0.00149Vl + 0.01936v2 - 0.00982 v~)/(0.8644 + 0.0013 v, + 0.01996v2 - 0.00491v3); (3) here v,, v2 and v~ are vibrational quantum numbers and J is rotational quantum number. (2~The asymmetry coefficient W,¢ ,,¢,(x) cannot be expressed as a simple function of the rotational quantum numbers. It was determined by using the same program as in the previous paper. "'~ The total energy is then obtained as the sum of the vibrational energy and the rotational energy, i.e. E(vt v2v3; JK ,r,) = Go(V, v2vs) + F~(Jr_,r, ; r).
(4)
The energy for the species S~40~ can be calculated by using the vibrational and rotational constants for the $3'O~. "'~) 2.2 Rotational line intensities and their wavenumbers The wavenumber of the line due to the transition (v,v~v~,J,¢ ......... ,K,) ~ (v,v2v~,J,¢_,,¢,)' ' " ' is t t I. ! - E ( v l v.2. .v. 3. .,. to = E(vlv2v3, JK_,K,)
j
tv
-
~,K,)
! . vv . = too + Fo,(Jg_,r,, ~ v ) - Fo,~JK_,r,,
KV~).
(5)
The line intensity, S, for the transition is obtained in terms of the integrated absorption coefficient a °b~ at 300°K, viz. S=a
ob~ To to . ,, (- hcE"/kT) T-~oo I'v' + l)l/'tl2 exp Q~Q,
1 - exp ( - hcto/kT) 1-exp(-hcto*/kT)'
(6)
where too is the band center; to* is that of the fundamental band; Q is the total partition function; Igl 2 is the square of rotational matrix element, which has been calculated for the t,= and t'3 bands; ¢L4' v'[ is the vibrational quantum number of the lower state associated with the vibrational transition. 3. CALCULATION 3.1 Calculation criteria The wavenumbers and the line intensities were calculated for the allowed transitions from
Calculation of absorption coefficient for the v2 and v3 bands of sulfur dioxide
495
eqns (5) and (6), respectively, to obtain the absorption coefficient. Weak lines with intensities less t h a n 10 - 6 c m -2 atm-' STP were neglected since they do not contribute appreciably to the bands. Calculations were performed from 300 to 1200°K. Hot bands with band intensities greater than 0.1% of a °bs at 300°K were included in the calculations. The number of transitions that must be included increase rapidly as the temperature is raised. 3.1.1 The v2 band. Since the v2 band consists of subbands due to the transitions &v2 = 1, calculations were carried out for these transitions. The value of 120 cm 2 atm-' stp ~5~was used for a °b" at 300°K. The numbers of vibrational transitions considered were 7, 17, 43 and 69 for 300, 600, 900 and 1200°K, respectively. Table 1 shows the band centers and the integrated absorption coefficients for the main transitions at each temperature. It can be seen that, with the increase in temperature, the number of hot bands must be increased to obtain the required total band intensity a °bs for the band. The selection rule for rotational transitions for the v2 band ~ are AJ=0,+I
(J"=O~J'=O);
AK_,=+I,+3
.... ;
AK,=+I,+3
....
(7)
since the oscillating dipole moment changes along the axis of the intermediate moment of inertia. The most intense line occurs for the transitions AK_, = + 1 and zlK~ = + 1. Rotational lines between 410 and 630 cm -' were calculated by using the selection rules for the v2 band. The region was divided into small intervals of &~o = 10 cm-' to calculate the absorption coefficients, following the procedure used in the previous paper, "~ i.e.
(8)
SIal = ~, (Sld)j, J
where (S/d)j is the jth subband of the v2 band. Table 1. The band centers and the integrated absorption coefficients for principal transition of the ~,2band v 2 Band (VlV2V 3) -- (VlV2+l v 3) ~(cm-2atm-iSTP) (VlV2V3)
~o
300°K
600°K
900°K
1200°K
517.29
96.51
53.06
27.71
15.17
010
511.29
15.92
30.16
23.78
15.99
020
505.29
2.02
13.04
15.45
12.73
030
499.29
0.24
5.08
9.00
9.07
040
493.29
1.88
4.97
6.10
050
487.29
0.68
2.65
3.97
060
481.29
1.39
2.53
00~
$3202
070
475.29
0.72
1.59
080
469.29
0.37
0.98
090
463.29
0.19
0.61
i00
515.24
Ii0
0.38
3.33
4.37
3.79
509.24
1.90
3.76
4.00
120
503.24
0.83
130
497.24
140 150
2.45
3.19
1.43
2.28
491.24
0.79
1.54
485.24
0.42
1.00
160
479.24
0.22
0.64
200
513.19
0.21
0.70
0.95
210
507.19
0.12
0.60
1.01
220
501.19
0.39
0.81
012
503.47
001
513.38
0.30
0.61
2.00
3.10
011
2.93
507.38
1.15
2.68
3.10
021
501.38
0.50
1.75
2.48
031
495,38
0.20
1.03
1,77
041
489.38
0.57
1,20
051
483.38
0.30
0.78
i01
511.33
0.50
0.74
111
505.33
0.43
0.79
121
499.33
0.28
0.63
1.20
0.69
000
$3402
517.35
0.14
0.13
4.23
2.22
496
M, OSUMIand T. KUNITOMO
3.1.2 The v~ band. The transitions At:3 = 1 were considered for the ~'3 band. The observed absorption coefficient of 853.3 cm -2 atm -~ STP ~6) was used for calculation of the line intensities. The numbers of transitions calculated were 6, 17, 47 and 68 for temperatures 300, 600, 900 and 1200°K, respectively. Table 2 presents the band centers and the integrated absorption coefficients of the main transitions at each temperature. The rotational selection rule for the ~'3 band differs from that for the ~,z and z,2 bands since the dipole moment changes along the axis of the least moment of inertia, viz. AJ=0,-+I;
AK_, = 0, -+2, -+4 . . . . ;
AK_,=-+I,-+3 . . . . .
(9)
The wavenumber region considered extends from 1200 cm -1 to 1400 cm-'. 3.2 Results and discussions 3.2.1 The v2 band. The mean line spacings for the effective rotational lines of the p2 band are shown in Fig. 1. They are given for the entire ~,2 band and for its main subbands. Their order of magnitude is 10-2 and it is found that there are enough rotational lines in the small intervals to calculate the absorption coefficients S/d. Figure 2 presents the mean absorption coefficients Sld at 300°K both for the entire v2 band and for its main subbands. The fundamental band dominates at 30&K. The line spacings of the 1,2 band are shown in Fig. 3 for all of the temperatures considered. Figure 4 shows the mean absorption coefficients Sld at 300, 600, 900 and 1200°K. As the temperature is raised, the absorption coefficient around the band center decreases while that in the wing region increases. The band centers for the transitions (VlV2V3)--->(v~v2+ l v3) are obtained from eqn (1) as
(1o)
a,o = 517.29 - 2.05v, - 6.0v2 + 0.39v3 Table 2. The band centers and the integrated absorptioncoefficients for principal transition of ~3 Band (VlV2V3) -- (VlV2V3+l) ~(cm-2atm-ISTP) (VlV2V3) 000
$3202
~o
300°K
600°K
900=K
1200=K
1361.75
747.9
510.7
310.6
187.8
147.2
135.3
100.6
010
1357.84
62.39
020
1353.93
5.36
43.06
59.54
54.28
030
1350.02
0.47
12.78
26.45
29.50
040
1346.11
3.85
11.86
16.14
050
1342.20
1.18
5.37
8.90
060
1338.29
100
1348.04
2.95
2.46
4.94
31.89
48.63
46.54
110
1344.13
9.24
21.26
24.99
120
1340.22
2.72
9.38
13.52
130
1336.31
4.18
7.36
140
1332.40
1.88
4.27
200
1334.33
7.71
11.64
210
1330.42
002
1341.07
012 022
2.03
3.38
6.27
11.94
21.32
1337.16
5.27
11.53
1333.25
2.35
6.27
102
1327.36
1.95
5.46
001
1351.41
69.74
72.62
2.25
2.16
38.66
011
1347.58
11.25
30.58
39.08
021
1343.59
3.32
13.54
21.18
031
1339.68
0.99
6.05
11.56
061
1335.77
I01
1337.70
111 121 201 003 000
2.73
6.36
11.16
18.29
1333.79
4.91
9.87
1329.88
2.18
5.36
1323.99
1.81
4.65
2.50
1330.73 $3402
1362.22
33.6
22.55
1.85
5.63
13.50
8.14
~,
band
Calculation of absorption coefficient for the v2 and v3 bands of sulfur dioxide 0,1
E"
v \ 0.01
300 °K ))2 band
~__~ (o0o)-(oI0) (010)-(020)
(020) - (030)
0.001
I
s3%2
(ooo>-(OlO}
t
400
I I I 500 550 WAVENUMBER (cm -1 )
450
1
I
I
I 600
Fig. 1. Mean line spacings for the v2 band and for its main subbands at 300°K,
)I0 II- I i t ~ I [ i "~2 band = (000)-(010) - - o - - (010)-(02.0) •
,,
(020)-(030)
--~
(000)- (010)
~ ~ , 300°K
i i i [
i
53402
% ¥~J "o
0.1I
01 ~ t 40O
45O
500 550 WAVENUMBER (cm "1)
600
Fig. 2. Absorption coefficients S[d for the u2 band and for its main subbands at 300~K.
0.~, i i , i
i
i i
i i i
))z band
I 0.01 ~
"o
0.00' ...9oo
00001 400
[
.
i
450
50O 550 WAVENUMBER (cm -1)
600
Fig. 3. Mean line spacings d f o r the entire ~'2 band.
49"/
498
M. OSUMf and T. KUNITOMO 10,
i
I
i
i
}
~
i
1
.
.
.
l
i
")2 band
o.,i7 F. oo
, I
oo1-,
400
{
450
.
500
.
WAVENUMBER
550 (¢rn-1)
600
4.Absorption coefficients S/d for the entire v2 band.
Fig.
and they are located at lower wavenumbers than the band center of the fundamental unless the ~3 vibrational mode is at a high level. Therefore, the band center of the v2 band moves toward lower wavenumbers as the temperature is raised. 3.2.2 The p3 band. The mean line spacings for the v~ band at each temperature are shown in Fig. 5. As the temperature is raised, more rotational lines are needed to describe the band. The absorption coefficients for the entire v3 band at 300, 600, 900 and I200°K are presented in Fig. 6. The characteristic feature of the v3 band compared to the v2 band is that the band width of the v2 band is narrow. This result is due to the fact that the p2 band contour spreads out since the P, Q and R branches for v2 subbands for the different K_, and AK l are located at lower and higher wavenumbers than the wavenumber of the band center of the subband. At the same time, the v3 band broadens less since the P, Q and R branches for the subband of the strongest transition AK_I = 0 overlap those for different K : I . The result of the present calculation at 300°K is compared with the absorption coefficients for the v3 band measured by CHANand TIEN(7) in Fig. 7. The dotted line in the figure shows that the results of the calculation corrected for the slit width &o = 30cm -t is consistent with the
0.1i ,
i ] l l l r
V3
i i i i
ii
ba~
0.01 l
600
300oK
/
~ ' ~ ~
0.001
013001
I
l
IIIllllll
l
1300
i i
1350 1400 WAVENUMSER (cm-1)
Fig. 5. Mean line spacings d for the entire v3 band.
Calculation of absorption coefficient for the v2 and ~3 bands of sulfur dioxide i 1 ~
,
1
1
499
,
~3 ~nd I
10
.
--
/
600
i
L
i
/
300"K
i
i
i
i
1
~
i
i
i
J
1300 1350 1400 WAVENUMf3ER (cmq )
l
Fig. 6. Absorption coefficients Sld for the entire ~3 band. ~00
~
,
,
_ _
10
~
~
1
i
,
i
,
band 300° K present~3 calculation "( Chartet al. experiment
i
] / / ~ i
1
i
,
,
'
,
7~
/\
!
I ]
I "o
I
o.1
0.01
I
,-/-
I
i
[
IJ/
II
1300
l
I
i
t
I
I
I
l,,
1350 1400 WAVENUMBER(cm-1)
Fig. 7. Comparison of SId calculated for the u3 band with the experimental result of Chan and Tien at 300°K.
experiment of Chart and Tien. Comparisons for the v2 band and for the p~ band at high temperatures could not be carried out since there are no experimental data.
1. 2. 3. 4. 5. 6. 7.
REFERENCES M. OSUMIand T. KUNITOMO,JQSRT 15, 1055 (1975). G. HERZBERG,Infrared and Raman Spectra--ll. Van Nostrand, New York (1968). G. KING, R. M. HAINERand P. C. CROSS, J. Chem. Phys. I1, 27 (1943). P. C. CRoss, R. M. HAINERand G. W. KING, J. Chem. Phys. 12, 210 (1944). J. MORCILLOand J. HERRANZ,J. Pulds inst. quire, ks. 10, 162 (1956). D. F. EFFERS,Jr. and E. D. SCHMm,J. Phys. Chem. 64, 279 (1960). S. H. CB^N and C. L. TIEN, J. Heat Trans. ASME Ser. C93, 172 (1971).