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High-temperature spectra of the pure rotational band of H2O
J. Quanr. Spcctrosc. R&at.
Tramfer. Vol. 5. pp. 697-714.
HIGH-TEMPERATURE
Persamon
Press Ltd., 1965. Printed ia Great Britain
SPECTRA OF THE PURE ROTATIONAL BAND OF I&O*
C. B. LUDWIG, C. C. FERNSO, W. MALKMUS and F. P. BOYNTON Space Science Laboratory, General Dynamics/Convair, San Diego, California (Received 23 March 1965) Abstract-The short wavelength wing of the pure rotation emission spectrum of water vapor (between 10 and 22 cc) has been measured at temperatures between 500” and 2200°K and at a nearly constant
optical depth of 2-4 x lo-( g/cmz. Apparent line positions and relative intensities agree with those determined from previous work. An approximate calculation of the local mean spectral absorption coefficient has been performed. This calculation is the weighted mean value of results obtained using energy levels of the prolate spheroidal top and of the “most asymmetric” top. A comparison of the results of the calculation (applying the statistical band model using broadening p&meters obtained in vibrationrotation bands) with the experimental data indicates that the calculation is acceptable for the long wavelength contribution to the total emissivity of hot water vapor. An estimate of the integrated band intensity has been obtained. INTRODUCTION
IN THE couRsE of a recent investigation (I) of the high-temperature spectral emissivities of the 6.3-p fundamental band of water vapor, it was found that the short wavelength wing (X c 22 p) of the rotational band grows very rapidly with temperature. At high temperatures, the rotational band overlaps the long wavelength wing of the 6.3-p band. Previous investigations@-‘) have shown that the rotational band is quite strong, as a consequence of the large dipole moment and rotational constants of the water molecule. Earlier estimates(8) of the contribution of this band to the total emissivity of water vapor, while relatively accurate at high temperatures and moderately thick optical paths, appear to be too low at other conditions. The rotational band will contribute significantly to the total radiant emission from hot water vapor under conditions of moderately high temperature (below 12OO”K), high pressures, and/or low to moderate optical depths (below 1000 cm-atm).(Q) The observed emission between 10 and 22 p reported here is attributed to the rotational band only. The possible vibration-rota~on bands in this region are difference bands of negligible intensity, since the corresponding summation bands are very weak (of the order of 1 atm-l *cm-2 at 273°K); moreover, the difference bands are further reduced by the Boltzmann factor which is of the order of O-1 at 2000°K. In general, far-infrared measurements of hot gases in static systems are quite difficult because of the lack 07 suitable window materials. Measurements of flow systems, such as the “supersonic burner” which was previously employed(l) in this laboratory in studies of emission from the vibration-rotation bands of Ha0 and COa, are not subject to * This work is supported by Project Defender, ARPA, under ONR Contract Nonr 3902(00), and Air Force Cambridge Research Laboratories, A0 363, Contract AF19(628)-4360. 697
698
C. B. LUDWIG, C. C. FERRISO,W. MALKMUSand F. P. BOYNTON
window limitations. The spectral emissivities reported here were measured between 10 and 22 p at temperatures between 550” and 2200X, using the supersonic burner technique. These measurements were not carried to longer wavelengths, since the emitted energy becomes too low for accurate measurements under the present experimental conditions. To obtain an estimate of the total band emission, values of the spectral emissivities were calculated by an approximate formulation and compared with the measured values. The comparison appears rather good, considering the approximate nature of the calculations. An estimate of the integrated band intensity was also obtained. The results of the calculations of the absorption coefficient for the rotational band of H,O are given for temperatures between 300” and 3000°K; spectral emissivities derived from the calculation are compared with our own experimental data and with those of NELSON. EXPERIMENTAL
RESULTS
A detailed description of the experimental technique has been presented elsewhere(l) and need not be repeated here. The spectral emission measurements were performed with a Perkin-Elmer Model 99 double-pass monochromator, using a KBr prism. A constant mechanical slit width of 0.5 mm was maintained; the resulting spectral slit width was 8 cm-l at 12 p and 1.2 cm-l at 22 p. Some spectral structure is resolved, especially at the longer wavelengths. Representative spectral emissivities are shown in Figs. l-6 where the temperatures are 590”, 850”, 1040”, 1640”, 1830”, and 2200”K, respectively.
T = 590°K
WAVELENGTH
FIG. 1.
pressure
[MICRONS)
Hz0 at 590°K from 9 to 22 p. Total = 3.12 cm; pl = O-69 cm-atm. Slitwidth = O-14 p at 10 p and = 0.065 at 20 p.
Spectral emissivities of the rotational band of = 1 atm; pathlength
High-temperature spectra of the ‘pure rotational band of HnO
699
T = 850°K 0.5
WAVELENGTH
(MICRONS
1
FIG. 2. Spectral emissivities of the rotational band of HpO at 850°K from 9 to 22 c. Total pressure = 1 atm; pathlength = 3.12 cm; pl = O-92 cm-atm. Slitwidth = 0.14 p at 10 p and = 0.065 at 20 p. 0~6,.,~,,,,,,,,,,,,,.
,
,
,
,
T = 1040°K
c 5 g
04
-
0.4
-
. 03
-
0.2
-
0.1
-
5
,
IO
II
12
13
14
.,
,
15
r 16
,
r 17
.,.,
., I6
., 19
20
.J 21
22
WAVELENGTHtMICRONS)
FIG. 3. Spectral emissivities of the rotational band of H1O at 1040°K from 9 to 22 p. Total pressure = 1 atm; pathlength = 3.12 cm; pl = l-10 cm-atm. Slitwidth = 0.14 c at 10 p and = O-065 at 20 B.
loo
C. B. LUDWIG,C. C. FERIUXI,W. MALICMUS and F. P. B~YNTON
T=
1640’K
WAVELENGTH
(MICRONS)
FIG. 4. Spectral emissivities of the rotational band of Ha0 at 1640°K from 9 to 22 p. Total pressure = 1 atm; pathlex@h = 3.12 cm; pl = l-68 cm-atm. Slitwidth = 0.14 p at 10 p and = O-065 at 20 CL.
T = 1830’K 0.6
0.5
g 2 w a
o’4
Id
0.3
0.2
0.1
01 ‘1 ,““‘.‘. 9 10 I,
IP
13
14
““‘*
15
16
WAVELENGTH
17
18
19
20
21
22
(MICRONS)
FIG. 5. Spectral emissivities of the rotational band of Ha0 at 1830°K from 9 to 22 p. Total pressure = 1 atm; pathlength = 3.12 cm; pl = l-91 cmeatm. Slitwidth = 0.14 p at 10 p and = 0.065 at 20 p.
High-temperature
spectra of the pure rotational
T
701
band of Ha0
= 2200°K
WAVELENGTH
iMICRONS)
FIG. 6. Spectral emissivities of the rotational band of Hz0 at 2200°K from 9 to 22 cc. Total pressure = 1 atm; pathlength = 3.12 cm; pl = 2.34 cm-atm. Slitwidth = 0.14 p at 10 p and = 0.065 at 20 p.
OPTlCAL DEPTH PATH LENGTH
0.5-
-2200
OK
-.-.-,640
w
u ~0.30 = 3.12
1: 5X f
I.5 w
(CM -ATM
km
CM
--,040 .K . . . . . . . . . . 540-K
0.4 -
: 5 z
0.3-
z
0.2 -
0.1 -
)i
IMICRONS)
FIG. 7. Spectral emissivities of the 6*3-~ fundamental and portions of the rotational of H1O at 540”, 1040”, 1640”, and 2200°K.
band
702
C. B. LUDWIG, C. C. FERRISO,W. MALKMUS
and F. P. BOYNTON
The appreciable increase with temperature of the spectral emissivities in this wavelength region is readily apparent in Fig. 7, where several spectra at different temperatures (540”, 1040”, 1640”, and 2200°K) are superimposed. This increase originates from the so-called “hot lines” which come from transitions within higher vibrational states, in addition to the transitions between higher rotational levels in the vibrational ground state. A detailed analysis of high-resolution high-temperature spectra will result in the identification of many specific “hot” transitions, not heretofore observed. However, the low resolution of the present data between 600 and 100 cm-l gave only the band contour and no single lines could be distinguished, Below about 600 cm-l the resolution was adequate to identify individual lines. We have chosen a spectrum at 850°K for analysis in greater detail to compare with previous work. Readings at approximately every 0.2 cm-l were taken in the spectral range from 445 cm-l to 520 cm-l; these data are shown in Fig. 8. Transitions allowed by the symmetry of the molecule give rise to hundreds of lines in the spectral region investigated here. However, only those transitions in which [AT~= 0 or 2 in the ground and higher vibrational states can be expected to give rise to strong lines in this region.
450
460
470
480 WAVENUMBER
490
FIG. 8. Spectral emissivities of the rotational band of Ha0 520 cm-l. Most
of the lines shown in
500
510
520
(CM-‘)
at 850°K
between 440 and
Fig. 8 were previously observed and identified by TAYLOR, cell with water vapor heated to 500°C. These identified lines are indicated as triangles in Fig. 8, together with their relative intensity at 773°K. It is seen that the agreement is good. The low resolution in our measurement as well as the uncertainty in the wavenumber calibration (& 0.5 cm-l) did not permit a more complete analysis. However, we have indicated a few additional lines (dotted triangles) which seem to appear in our spectrum. The relative intensities of these additional lines were estimated by using the method of Ref. 3. BENEDICT and STRONG,(~) who used an absorption
703
High-temperature spectra of the pure rotational band of H.QO APPROXIMATE
ABSORPTION
COEFFICIENT
CALCULATIONS
To estimate the spectral absorption coefficients of the long wave-length regions of the rotational band of water vapor and to obtain integrated band intensities, we have calculated the local mean spectral absorption coefficient by an approximate technique. The intensity of a group of transitions from states of a given value of J is assigned as the weighted mean of the intensities of a most asymmetric top and a prolate spheroidal top. The weighting factor is the asymmetry parameter K, defined by K=
2B-A-C A-C
(1)
*
The results of these calculations are shown to give good agreement with experimental data in the wings of the band where the weak line approximation is valid. The local mean spectral absorption coefficient is given approximately by P = S;+l[d,
(2)
where S;+l is the sum of the integrated line strengths for all transitions from a state of given J to a state in which J’ = J+ 1, and d is the average spacing between consecutive groups of lines. The “Q-branch” (which falls at very long wavelengths) is thus neglected. The value of d is given approximately by w/(J+ 1). The sum of the integrated line strengths ,S;“l is obtained fromol)
where (DFSare the direction cosines between the space fixed F and rotating g axes,(ll) w is the average wavenumber of the transitions, QR = rotational partition function, p0 = permanent dipole moment (1.87 debye for H,O), and N
1
-=
-=--
PO
To
P
kT
PO T
from the ideal gas law, where p. = Loschmidt number and p. = 1 atm. It is convenient to introduce a quantity P* =
P/[l-exp(
-%)I,
(5)
from which the explicit dependence on w has been removed. We have then p(
J>
=
81r” PO& To [ 3hc
-][(J+o QR T
~m+-E(J94/kTl m
1
IQ~A~],
(6)
FMM’
thus separating P* into two terms, the first of which is independent of J. Water is an asymmetric top, and therefore no explicit expression may be written for the matrix elements and the energy levels as functions of J, T, and the rotational constants A, B, and C. The levels of a symmetric top are explicitly given, and the levels of slightly asymmetric tops may be calculated approximately by replacing the nearly equal rotational 5
C. B. LUDWIO, C. C. FERRISO,W. MALKMUS and F. P. BOYNTON
‘lo4
constants by their arithmetic or geometric mean. This technique has been appliedo2) to the calculation of the absorption coefficient of H20, using the prolate spheroidal expression. On the other hand, we have found that the energy levels of the “most asymmetric” molecule may be represented by a simple approximate formula. The values of S*/d for a given class of transitions J-+7’ are not greatly different for the two cases, and we assume that a reasonable value is given by the weighted mean. Symmetric top model. The expression for the mean absorption symmetric top is as followso3) &(J)
=
1 hc $p,, To ABC T (1~)~‘~kT ( A -(BC)1’2
)“2
coefficient of a nearly
(exp[ -f$BC)l/2~(~+
1)] (7)
-
exp
- g(BC)‘!‘(J+
l)(J+2)]]
x (J+ 1)2G[ (A -(BC)‘,‘)
l/2 hc (=)
112
(JS l)].
A similar, but slightly different, expression has previously been given,(12) and a discussion of the differences between that expression and the one used here is given in Ref. 13. The function G(z) in Eq. (7) is defined as G(z) = (w)lj2( 1 - -$) for which the power series representation
erf(z) + i exp( -z2),
(8)
is found to be
( - l)n z2n+l n= IJ n!(2n + 1)(2n f 3) *
G(z) = 4 i
(9)
This series is absolutely convergent, and, hence, valid for all z; however, for convenience in computation, the following expressions, namely, G(z) FZfz-
4 2 -23 + -25 15 35
- &z7
(for z < 1.45)
(10)
and G(Z) z (,)112( 1 - --&)
(for z 2 l-45)
(11)
are found to approximate G(z) to within 3 per cent for all z. The closest representation of the energy levels of the Hz0 molecule by the symmetric top model is obtained by choosing the A axis as the top axis and selecting an effective value for the remaining constant, such as (BC)l12, as has been done previously.(12) A difficulty arises in the case of the rotational spectrum in that strict application of the selection rules (AJ = 1, AK = 0) yields the expression w = 2(BC)1’2(Jf
1)
(12)
for the frequency as a function of quantum number. The permanent dipole moment, however, lies along the B axis, which must be considered the top axis in applying the selection rule AK = 0. On the basis of this argument, the effective rotational constant (BC)1’2 in Eq. (12) would be better replaced by (AC)‘12 or 4 (A + C).
High-temperature spectra of the pure rotational band of H20
705
For an asymmetric top, a reasonable value for the (average) frequency is given by the difference of the average term values for each value of J. This assumption gives for the average frequency w = 2(A+;+C)(J+l).
(13)
This relationship is consistent with the observed frequencies of identified lines. We note that for Ha0 the values of + (A + B + C), 4 (A + C), and (AC)lj2 do not differ greatly. We will use Eq. (7) for evaluating F&(J), but will use Eq. (13) as the relationship between o and J. In effect, this procedure retains the characteristic structure of the symmetric top energy levels but releases the restriction on the frequency which results from the symmetric top selection rule. Most asymmetric top model. The development for the most asymmetric top model proceeds as follows: The energy levels of a rigid asymmetric rotor are given by’14) E(J,T,A,B,C)
A+C = hc ---J(J+ 2
1) +-
A-C
2
&(J,T,K)
1 ,
(14)
where E, is a complicated function of J, r, and K, and for a given Jassumes 2J+ 1 different values. For the most asymmetric rotor (K = 0), it is found that the energy levels calculated by CROSS, HAINER and KING ( w h’rch are odd functions of 7) can be reasonably well approximated by E,(J,T,O)
z JT.
(15)
The approximation is especially good at high values of 1~1,as one can see from Fig. 9, where E, is plotted against T for J = 12. It is assumed that the relationship of Eq. (15) also applies to higher J’s. We assume a simplified pattern of transitions, namely, that only those transitions for which IAT] = 0 or 2 are considered and that the total intensity is divided equally among these three groups. This assumption permits explicit evaluation of the summations in Eq. (3). In particular, we .&id
=
exp[(hc/kT)&(A-C)J(J+l)]-exp[-(hclkT)&(A-C).72] exp[(hc/k
kT 2 z -__hc A_C
T)3( A - C)Jj - 1
(16)
1 J+4{ ex p[(hc/kT)B(A-C)(J+a)2l-exp[-(hc/kT)b(A-C)(J+~)2i).
If the rotational partition function is calculated using Eq. (16), and replacing the sum over J by an integral, and using J(J+ l)=(J+ Q2, we obtain QRA = (“)1’2(~)3’2~AB~c~~,2~
(17)
C. B. LUDWIQ, C.
706
C. FERRISO,W.
I, II -12 -10 -6
I
-6
-4
I -2
MALKMUSand F. P. BOYNTON
II 0 T
2
I 4
I
6
I 6
I IO
I 12
14
FIG. 9. Plot of &( K = 0) versus 7 for J = 12.
where B* = $[B+(LIC)~‘~].
(18)
The development was carried through in powers of the parameter e = (A -C)/2B,
(19)
which for the most asymmetric rotor may have values from 0 to 1. The limiting case of e = 0 is that of the spherical top. Because of the approximations involved in setting up the model, the development was carried out only to the second order in e. To this approximation, we note that the rotational partition function becomes Q,,
=
(TT)~~~(~)~‘~--&-[~ +ge2].
(20)
The expression for S$+l becomes 2B(J+ 1)[*(2.7+ 1)(25+3)]
x[
1 -exp(
-%)I(
1 ++?(l
exp
-~B.J2)+$2[~B(J++)2]2).
-FTBJ(J+
1
1)
(21)
We note that the second term in the brace is nearly independent of J. Since the population is a maximum when (k/H’) BJ2 ~1, the second term has a value of approximately (23/36) e2= #e2, which will cancel the second order term in QRA[Eq. (20)].
High-temperature spectra of the pure rotational band of Ha0
707
The third term in the brace remains as a significant function of J. When Eqs. (20) and (21) are combined, we obtain ,;+I
= [ffp$;]-$(;)3’z2B(J+
x exp
+2J+
I)][ 1 -exp(
--$BJ(J+
1)(2J+3)
-$)I
(22) or, to the same (second-order) s;+l
accuracy in e,
= [ffp~~]-$~(~)3’z2B(~+
x exp
I$(~J+
I)]( 1 -exp[
-sBJ(J+
sinh[(hclkT)B(A
1)(2~+3)
-$I)
-C)(J+8)21
(23)
’
‘-(hc/kT)+(A-C)(J+Q2
Since for the most asymmetric top B = Q (A + B-I- C), we have ,;+l
= 2B(J+
1)
(24)
and d = 2B.
We find, then, that the mean absorption p(J)
(25)
coefficient is given by
= S;+l/d
l)3exp[ --$BJ(J+
= [ gP$]$($r’2$J+
-$2B(J+
I)]
1) I)
x
sinh[(hc/k T)#(A - C)(J-t
4 )2]
(he/k T)+(A - C)(J++)2
’
(26)
In applying the results of the most asymmetric top model, we define an “equivalent most asymmetric top” by the conditions A’B’C’
and
A’+B’+C’ A’+C’-2B’
= ABC = A+B+C = 0.
(27)
708
C. B. LUDWIG, C. C. FERRISO,W. MALKMUS and F. P. BOYNTON
From the valueso5) of the equilibrium rotational constants for HaO, A = 27.33 cm-l B = 14.58 cm-l
C = we obtain for the “equivalent
9.50cm-l,
most asymmetric top” A’ = 25.68 cm-l B’ = 17.14cm-l
C’ =
8.60cm-I.
When the two expressions, Eq. (7) and Eq. (26), are evaluated using the values of A’, B’, and C’ for HaO, it is found that the resulting values of pj& and PAS are not greatly different for a given value of J, when P* is large. In the wing of the band, they differ by
FIG. 10.
Plot of j$,
(dashed line) and FX, (solid line) versus J forj12OO”K.
an increasingly large factor, as can be seen in Fig. 10. For high values of J, P&(J) from Eq. (7) is proportional to J2 exp
and P&(J) from Eq. (27) is proportional Jexp[
- ( BQ~~“&J~]
to -G$J2],
High- temperature spectra of the pure rotational band of Ha0
709
Since C < (BC)lj2, it is evident that P& > P;r, for sufficiently high J. In other words, the lowest levels of the asymmetric rotor with rotational constants, A, B, and C of water are lower than the lowest levels of the symmetric top having rotational constants A and (BC)l12. Therefore, the asymmetric top has a greater total population of the energy levels of a given J, when J is large. We use for P* of water the value given by the weighted mean of P& and P&, p&o = (1 +‘%o)& where
~~~~
=
-0.436.
The local mean absorption P
Ha0
=
coefficient is then given by
P&o(l-ew[ --g-J)3
which is plotted in Fig. 11 as a function of temperature
WAVENUMBER
FIG. 11.
and wavenumber.
(CM-‘)
Plot of the local mean absorption coefficient &9 between 300”and 3000°K.
COMPARISON
(28)
-‘Qrgo%
OF CALCULATION
versus wavenumber
AND EXPERIMENT
In Figs. 12 through 15, the calculated and measured emissivities are compared for temperatures of 540”, 1040”, 1640”, and 2200°K. The dots represent the actual observed maxima and minima of the spectral structure. The solid lines were calculated using the relation r = 1 -exp( -&I), (30) where pl = optical depth and P is the local mean absorption coefficient in the weightedaverage asymmetric top expression, Eq. (29). Since the spectral emissivities calculated according to Eq. (30) are appropriate only to highly pressure-broadened gases, and since it is obvious that our experimental spectra are not highly pressure-broadened, we should expect Eq. (30) to provide an overestimate
C. B. LUDWIG, C. C. FERRISO,W. MALKMUS and F. P. BOYNTON
710
of the emissivity. In order to account for the spectral structure, statistical band model with exponential intensity distribution
the equation for the
E = 1 - exp[ - PpZ/( 1 + Ppr/4Z)l9
(31)
was used for the emissivity. The parameter
a = ~(yzpT[c(~)1’2+eg+(l +&]
(32)
is the ratio of collision-broadened line half-width to average line spacing, y0 is the line width at one atm and 300°K averaged over the band, c is the mole fraction of water vapor in the sample, CT* is the ratio of the optical cross-section for “inert” gases to that for water vapor, and CJthe ratio for non-resonant to resonant dipoles. Effective band-averaged values of 6 have been determined at various temperatures for the 6+3- and 2.7-p fundamental bands of water vapor. 06) These calculations are uncertain within a factor of 5 to 7. We have therefore calculated the emissivities by Eq. (31) in the upper and lower limits of d. The results are shown in Figs. 12 through 15 as dashed lines. The agreement with the measured values becomes better, especially at the longer wavelengths. I.0
I
,
,
SPECTRAL
,
,
,
EMISSIVITIES
I
,
OFTHE
,
,
Hz0
WAVENUMBER
,
,
I
ROTATIONAL
(CM-’
,
1
BAND AT
,
,
,
I
540°K
)
FIG. 12. Comparison of theory with present experiment, where the dots represent the maxima and minima of the experimental spectra; the solid line represents “thin gas” calculation, the dashed lines represent “thick gas*’ calculation with z = 0.085 (upper) and 0.0106 (lower). T = 540”K, pT = 1 atm, pl = 0.66 cm.atm.
The rigid rotor approach does not consider the actual spread of the lines in a group defined by a given J (which would tend to augment the emission at higher wavenumbers) nor the effect of centrifugal stretching (which would tend to decrease it). Thus any close agreement with experiment in the far wing of the band must be considered to some extent fortuitous. However, for the purpose of extrapolating to lower frequencies to determine total emissivities, the use of the calculated values is considered quite reasonable.
High-temperature
1.0
I
,
I
SPECTRAL
,
spectra of the pure rotational
I.,
1
EMISSIVITIES
,
I
OFTHE
(
H2b
I
,
I
ROTATIONAL
WAVENUMBER
(CM-’
711
band of Ha0
,
,
BAND
,
AT
I
,
’
,
1040°K
)
FIG. 13. Comparison of theory with present experiment, where the dots represent the maxima and minima of the experimental spectra; the solid line represents “thin gas” calculation, the dashed lines represent “thick gas” calculation with ii = 0.10 (upper) and 0.023 (lower). T = 1040°K,pT = 1 atm,pl = 1.1 cmeatm.
I.0
I
,
,
SPECTRAL
02
,
I
,
EMISSIVITIES
I
,
OF THE
I
,
Hz0
I
,
ROTATIONAL
I
,
BAND
I
,
AT
I
,
,
1640°K
. .
300
400
500
600
700
600
WAVENUMBER
900
1000
1100
l
-
L
*
1200
1300
(CM-‘)
FIG. 14. Comparison of theory with present experiment, where the dots represent the maxima and minhna of the experimental spectra; the solid line represents “thin gas” calculation, the dashed lines represent “thick gas” calculation with rT = 0.26 (upper) and 0.065 (lower). T = 164O”K, pT = 1 atm, pl = l-68 cmsatm.
712
C. I.0
,
B. LUDWIG,C.C.FERRISO,W.MALKMUS and F.P.BOYNTON
,
,
,
SPECTRAL
I
,
I
EMISSIVITIES
,
,
OF THE
,
I
Hz0
,
I
,
ROTATIONAL
1
BAND
,
AT
,
2200
,
,
OK
0.8-
300
500
400
700
600
WAVENUMBER
1100
1000
900
800
1200
13oc
(CM-‘)
FIG.15. Comparison of theorywith present experiment, where the dots represent the
maxima and minima of the experimental spectra; the solid line represents “thin gas” calculation, the dashed lines represent “thick gas” calculation with ii = 0.62 (upper) and 0.177 (lower). T = 2200”K,pT = 1 atm,pZ = 2.34 cmeatm.
As a further comparison, we show in Figs. 16 through 18 a similar comparison of our calculations with the data of NELSON(lo) at 555”, 832”, and 1111°K for p = 1 atm. The agreement appears to be quite good, particularly at the longer wavelengths. At the shorter wavelengths, there is a deviation of the calculated values from the measured data which we cannot explain, even on the basis of deviation from rigidity. Similar agreement is found with Nelson’s data for 2 atm. I -
E
’
I
SPECTRAL
’
I
’
EMISSIVITIES
I
’
I
OF THE
’ Hz0
I
’
I
ROTATIONAL
’
I
I
I’
BAND AT 555*K
-
-
0.5 -
0
’ 400
*
” 500
E WAVENUMBER
(CM-‘1
’s(lo) experiment, where the dashed-pointFIG.16. Comparison of theory with NELSON curve represents the experimental data, the solid line reprcscnts “thin gas” calculatbn, the dashed lines represent “thick gas” calculation, with Z = 0.281 (upper) and 0.0351 (lower). T = 555”K,p~ = 1 atm,pl = 38.6 cmsatm.
High-temperature spectra of the pure rotational band of Ha0 I
‘I(
I
_ SPECTRAL
‘II
EMISSIVITIES
1’1,
OF THE
Hz0
WAVENUMBER
I ‘I’ll ROTATIONAL BAND
AT 1lfl.K
713
-
(CM-‘)
Fro. 17. Comparison of theory with NELSON’S@)experiment, where the dashed-pointcurve represents the experimental data, the solid line represents “thin gas” calculation, the dashed lines represent “thick gas” calculation with ii = O-241 (upper) and 0.0482 (lower). T = 832”K,pT = 1 atm,pZ = 38.6 cm-atm. I
-
’
SPECTRAL
1’1’1
II
0
EMISSIVITIES
OF THE
I Hz0
n
1’1’1’
ROTATIONAL
I 1000 WAVENUMBER
BAND
I
I 1100
AT B32’K
I
I 1200
-
I
(CM-‘)
FIG. 18. Comparison of theory with NELWN’S(~~)experiment, where the dashed-pointcurve represents the experimental data, the solid line represents “thin gas” calculation, the dashed lines represent “thick gas” calculation with Z = O-295 (upper) and 0.059 (lower). T= llll”K,p, = 1 atm,pl= 38.6cmsatm. TOTAL
ROTATIONAL
BAND
INTENSITY
The total intensity of the pure rotational band system of a symmetric top is given by’ls’ a(T)
=
16= 3P,,B 2 3hckT ’
(33)
714
C. B. LUDWIG, C. C. FERRISO,W. MALTS
and F. P. BOYNT~N
In particular, Eq. (33) holds for a spherical top. If the expression for S,J+l, Eq. (23) is summed over J, retaining terms to the second power in e, we obtain the most asymmetric top 16+; a( T) w-B
3hckT
From the quoted values for the rotational
l+zez. [
1
(34)
constants, we find that e2 = 0.248 and
a(273”K) = 1350 [l-362] = 1840cmb2
-atmel.
Approximately the same value is obtained by a numerical summation of Eq. (7), the modified symmetric top model; an analogous series expansion has been developed for this case, but it is not as rapidly convergent. The effect of centrifugal stretching has not been considered here. An approximate treatment indicates that the overall effect of centrifugal stretching on the band intensity is quite small.(13) The experimental emissivity results (corrected to 273°K) appear consistent with this band intensity value, although the data are not sufficiently extensive to determine an experimental value. authors wish to acknowledge their indebtedness to assistancein obtaining and reducing the experimental data.
Acknowledgement-The
Mr. C. N. ABEYTA for
REFERENCES 1. C. B. LUD‘WIG, C. C. FERRISOand C. N. ABEYTA,JQSRT5,281(1965).
2. H. M. RANDALL,D. M. DENNISON,N. GINSBURGand L. R. WEBER,Phys. Reu. 52,160 (1937). 3. J. H. TAYLOR,W. S. BENEDICT and J. STRONG,J. Chem. Phys. 20,528 and 1884 (1952). 4. W. S. BENEDICT,H. H. CLAASSEN and J. H. SHAW, J. Res. Nat. Bur. Stand., Wash. 49,91 (1952). 5. C. H. PALMER,J. Opt. Sot. Amer. 47,1024 and 1029 (1957). 6. W. S. BENEDICT.M&n. Sot. Rov. Sci. LiPae. Suecial Vol. 2. 18 (1957). 7. N. G. YAROSLA~SKYand A. E. STANEVIC,, bp;. Spectrosc. 5,380 (1959). 8. S. S. PENNER, Quantitative Molecular Spectroscopy and Gas Emissivities, p. 322. Addison-Wesley, Massachusetts (1959). 9. C. C. FERRISO,C. B. LUDWIG and F. P. BOYNTON,submitted to Znt. J. Heat Mass Transfer. 10. K. E. NELSON, Experimental determination of the baud absorptivities of water vapor at elevated pressures and temperatures, M.S. Thesis, University of California (1959). 11. P. C. CROSS,R. M. HAINFZR and G. W. KING, J. Chem. Phys. 12,210 (1944). 12. S. S. PENNER,ibid. pp. 327-329. 13. W. MALKMUS,Intensities of pure rotational band systems, JQSRTS, 621-631(1965). 14. P. C. CROSS,R. M. HAINERand G. W. KING, J. Chem. Phys. l&27 (1943). 15. G. HERZBERG,Infraredand Raman Spectra, p. 488, Van Nostmnd, New York (1962). 16. C. C. FERRISO,C. B. LUDWIG and A. THOMSON,Empirical infrared absorption coefficients of Hz0 from 300” to 3000”K, to be published.
Tramfer. Vol. 5. pp. 697-714.
HIGH-TEMPERATURE
Persamon
Press Ltd., 1965. Printed ia Great Britain
SPECTRA OF THE PURE ROTATIONAL BAND OF I&O*
C. B. LUDWIG, C. C. FERNSO, W. MALKMUS and F. P. BOYNTON Space Science Laboratory, General Dynamics/Convair, San Diego, California (Received 23 March 1965) Abstract-The short wavelength wing of the pure rotation emission spectrum of water vapor (between 10 and 22 cc) has been measured at temperatures between 500” and 2200°K and at a nearly constant
optical depth of 2-4 x lo-( g/cmz. Apparent line positions and relative intensities agree with those determined from previous work. An approximate calculation of the local mean spectral absorption coefficient has been performed. This calculation is the weighted mean value of results obtained using energy levels of the prolate spheroidal top and of the “most asymmetric” top. A comparison of the results of the calculation (applying the statistical band model using broadening p&meters obtained in vibrationrotation bands) with the experimental data indicates that the calculation is acceptable for the long wavelength contribution to the total emissivity of hot water vapor. An estimate of the integrated band intensity has been obtained. INTRODUCTION
IN THE couRsE of a recent investigation (I) of the high-temperature spectral emissivities of the 6.3-p fundamental band of water vapor, it was found that the short wavelength wing (X c 22 p) of the rotational band grows very rapidly with temperature. At high temperatures, the rotational band overlaps the long wavelength wing of the 6.3-p band. Previous investigations@-‘) have shown that the rotational band is quite strong, as a consequence of the large dipole moment and rotational constants of the water molecule. Earlier estimates(8) of the contribution of this band to the total emissivity of water vapor, while relatively accurate at high temperatures and moderately thick optical paths, appear to be too low at other conditions. The rotational band will contribute significantly to the total radiant emission from hot water vapor under conditions of moderately high temperature (below 12OO”K), high pressures, and/or low to moderate optical depths (below 1000 cm-atm).(Q) The observed emission between 10 and 22 p reported here is attributed to the rotational band only. The possible vibration-rota~on bands in this region are difference bands of negligible intensity, since the corresponding summation bands are very weak (of the order of 1 atm-l *cm-2 at 273°K); moreover, the difference bands are further reduced by the Boltzmann factor which is of the order of O-1 at 2000°K. In general, far-infrared measurements of hot gases in static systems are quite difficult because of the lack 07 suitable window materials. Measurements of flow systems, such as the “supersonic burner” which was previously employed(l) in this laboratory in studies of emission from the vibration-rotation bands of Ha0 and COa, are not subject to * This work is supported by Project Defender, ARPA, under ONR Contract Nonr 3902(00), and Air Force Cambridge Research Laboratories, A0 363, Contract AF19(628)-4360. 697
698
C. B. LUDWIG, C. C. FERRISO,W. MALKMUSand F. P. BOYNTON
window limitations. The spectral emissivities reported here were measured between 10 and 22 p at temperatures between 550” and 2200X, using the supersonic burner technique. These measurements were not carried to longer wavelengths, since the emitted energy becomes too low for accurate measurements under the present experimental conditions. To obtain an estimate of the total band emission, values of the spectral emissivities were calculated by an approximate formulation and compared with the measured values. The comparison appears rather good, considering the approximate nature of the calculations. An estimate of the integrated band intensity was also obtained. The results of the calculations of the absorption coefficient for the rotational band of H,O are given for temperatures between 300” and 3000°K; spectral emissivities derived from the calculation are compared with our own experimental data and with those of NELSON. EXPERIMENTAL
RESULTS
A detailed description of the experimental technique has been presented elsewhere(l) and need not be repeated here. The spectral emission measurements were performed with a Perkin-Elmer Model 99 double-pass monochromator, using a KBr prism. A constant mechanical slit width of 0.5 mm was maintained; the resulting spectral slit width was 8 cm-l at 12 p and 1.2 cm-l at 22 p. Some spectral structure is resolved, especially at the longer wavelengths. Representative spectral emissivities are shown in Figs. l-6 where the temperatures are 590”, 850”, 1040”, 1640”, 1830”, and 2200”K, respectively.
T = 590°K
WAVELENGTH
FIG. 1.
pressure
[MICRONS)
Hz0 at 590°K from 9 to 22 p. Total = 3.12 cm; pl = O-69 cm-atm. Slitwidth = O-14 p at 10 p and = 0.065 at 20 p.
Spectral emissivities of the rotational band of = 1 atm; pathlength
High-temperature spectra of the ‘pure rotational band of HnO
699
T = 850°K 0.5
WAVELENGTH
(MICRONS
1
FIG. 2. Spectral emissivities of the rotational band of HpO at 850°K from 9 to 22 c. Total pressure = 1 atm; pathlength = 3.12 cm; pl = O-92 cm-atm. Slitwidth = 0.14 p at 10 p and = 0.065 at 20 p. 0~6,.,~,,,,,,,,,,,,,.
,
,
,
,
T = 1040°K
c 5 g
04
-
0.4
-
. 03
-
0.2
-
0.1
-
5
,
IO
II
12
13
14
.,
,
15
r 16
,
r 17
.,.,
., I6
., 19
20
.J 21
22
WAVELENGTHtMICRONS)
FIG. 3. Spectral emissivities of the rotational band of H1O at 1040°K from 9 to 22 p. Total pressure = 1 atm; pathlength = 3.12 cm; pl = l-10 cm-atm. Slitwidth = 0.14 c at 10 p and = O-065 at 20 B.
loo
C. B. LUDWIG,C. C. FERIUXI,W. MALICMUS and F. P. B~YNTON
T=
1640’K
WAVELENGTH
(MICRONS)
FIG. 4. Spectral emissivities of the rotational band of Ha0 at 1640°K from 9 to 22 p. Total pressure = 1 atm; pathlex@h = 3.12 cm; pl = l-68 cm-atm. Slitwidth = 0.14 p at 10 p and = O-065 at 20 CL.
T = 1830’K 0.6
0.5
g 2 w a
o’4
Id
0.3
0.2
0.1
01 ‘1 ,““‘.‘. 9 10 I,
IP
13
14
““‘*
15
16
WAVELENGTH
17
18
19
20
21
22
(MICRONS)
FIG. 5. Spectral emissivities of the rotational band of Ha0 at 1830°K from 9 to 22 p. Total pressure = 1 atm; pathlength = 3.12 cm; pl = l-91 cmeatm. Slitwidth = 0.14 p at 10 p and = 0.065 at 20 p.
High-temperature
spectra of the pure rotational
T
701
band of Ha0
= 2200°K
WAVELENGTH
iMICRONS)
FIG. 6. Spectral emissivities of the rotational band of Hz0 at 2200°K from 9 to 22 cc. Total pressure = 1 atm; pathlength = 3.12 cm; pl = 2.34 cm-atm. Slitwidth = 0.14 p at 10 p and = 0.065 at 20 p.
OPTlCAL DEPTH PATH LENGTH
0.5-
-2200
OK
-.-.-,640
w
u ~0.30 = 3.12
1: 5X f
I.5 w
(CM -ATM
km
CM
--,040 .K . . . . . . . . . . 540-K
0.4 -
: 5 z
0.3-
z
0.2 -
0.1 -
)i
IMICRONS)
FIG. 7. Spectral emissivities of the 6*3-~ fundamental and portions of the rotational of H1O at 540”, 1040”, 1640”, and 2200°K.
band
702
C. B. LUDWIG, C. C. FERRISO,W. MALKMUS
and F. P. BOYNTON
The appreciable increase with temperature of the spectral emissivities in this wavelength region is readily apparent in Fig. 7, where several spectra at different temperatures (540”, 1040”, 1640”, and 2200°K) are superimposed. This increase originates from the so-called “hot lines” which come from transitions within higher vibrational states, in addition to the transitions between higher rotational levels in the vibrational ground state. A detailed analysis of high-resolution high-temperature spectra will result in the identification of many specific “hot” transitions, not heretofore observed. However, the low resolution of the present data between 600 and 100 cm-l gave only the band contour and no single lines could be distinguished, Below about 600 cm-l the resolution was adequate to identify individual lines. We have chosen a spectrum at 850°K for analysis in greater detail to compare with previous work. Readings at approximately every 0.2 cm-l were taken in the spectral range from 445 cm-l to 520 cm-l; these data are shown in Fig. 8. Transitions allowed by the symmetry of the molecule give rise to hundreds of lines in the spectral region investigated here. However, only those transitions in which [AT~= 0 or 2 in the ground and higher vibrational states can be expected to give rise to strong lines in this region.
450
460
470
480 WAVENUMBER
490
FIG. 8. Spectral emissivities of the rotational band of Ha0 520 cm-l. Most
of the lines shown in
500
510
520
(CM-‘)
at 850°K
between 440 and
Fig. 8 were previously observed and identified by TAYLOR, cell with water vapor heated to 500°C. These identified lines are indicated as triangles in Fig. 8, together with their relative intensity at 773°K. It is seen that the agreement is good. The low resolution in our measurement as well as the uncertainty in the wavenumber calibration (& 0.5 cm-l) did not permit a more complete analysis. However, we have indicated a few additional lines (dotted triangles) which seem to appear in our spectrum. The relative intensities of these additional lines were estimated by using the method of Ref. 3. BENEDICT and STRONG,(~) who used an absorption
703
High-temperature spectra of the pure rotational band of H.QO APPROXIMATE
ABSORPTION
COEFFICIENT
CALCULATIONS
To estimate the spectral absorption coefficients of the long wave-length regions of the rotational band of water vapor and to obtain integrated band intensities, we have calculated the local mean spectral absorption coefficient by an approximate technique. The intensity of a group of transitions from states of a given value of J is assigned as the weighted mean of the intensities of a most asymmetric top and a prolate spheroidal top. The weighting factor is the asymmetry parameter K, defined by K=
2B-A-C A-C
(1)
*
The results of these calculations are shown to give good agreement with experimental data in the wings of the band where the weak line approximation is valid. The local mean spectral absorption coefficient is given approximately by P = S;+l[d,
(2)
where S;+l is the sum of the integrated line strengths for all transitions from a state of given J to a state in which J’ = J+ 1, and d is the average spacing between consecutive groups of lines. The “Q-branch” (which falls at very long wavelengths) is thus neglected. The value of d is given approximately by w/(J+ 1). The sum of the integrated line strengths ,S;“l is obtained fromol)
where (DFSare the direction cosines between the space fixed F and rotating g axes,(ll) w is the average wavenumber of the transitions, QR = rotational partition function, p0 = permanent dipole moment (1.87 debye for H,O), and N
1
-=
-=--
PO
To
P
kT
PO T
from the ideal gas law, where p. = Loschmidt number and p. = 1 atm. It is convenient to introduce a quantity P* =
P/[l-exp(
-%)I,
(5)
from which the explicit dependence on w has been removed. We have then p(
J>
=
81r” PO& To [ 3hc
-][(J+o QR T
~m+-E(J94/kTl m
1
IQ~A~],
(6)
FMM’
thus separating P* into two terms, the first of which is independent of J. Water is an asymmetric top, and therefore no explicit expression may be written for the matrix elements and the energy levels as functions of J, T, and the rotational constants A, B, and C. The levels of a symmetric top are explicitly given, and the levels of slightly asymmetric tops may be calculated approximately by replacing the nearly equal rotational 5
C. B. LUDWIO, C. C. FERRISO,W. MALKMUS and F. P. BOYNTON
‘lo4
constants by their arithmetic or geometric mean. This technique has been appliedo2) to the calculation of the absorption coefficient of H20, using the prolate spheroidal expression. On the other hand, we have found that the energy levels of the “most asymmetric” molecule may be represented by a simple approximate formula. The values of S*/d for a given class of transitions J-+7’ are not greatly different for the two cases, and we assume that a reasonable value is given by the weighted mean. Symmetric top model. The expression for the mean absorption symmetric top is as followso3) &(J)
=
1 hc $p,, To ABC T (1~)~‘~kT ( A -(BC)1’2
)“2
coefficient of a nearly
(exp[ -f$BC)l/2~(~+
1)] (7)
-
exp
- g(BC)‘!‘(J+
l)(J+2)]]
x (J+ 1)2G[ (A -(BC)‘,‘)
l/2 hc (=)
112
(JS l)].
A similar, but slightly different, expression has previously been given,(12) and a discussion of the differences between that expression and the one used here is given in Ref. 13. The function G(z) in Eq. (7) is defined as G(z) = (w)lj2( 1 - -$) for which the power series representation
erf(z) + i exp( -z2),
(8)
is found to be
( - l)n z2n+l n= IJ n!(2n + 1)(2n f 3) *
G(z) = 4 i
(9)
This series is absolutely convergent, and, hence, valid for all z; however, for convenience in computation, the following expressions, namely, G(z) FZfz-
4 2 -23 + -25 15 35
- &z7
(for z < 1.45)
(10)
and G(Z) z (,)112( 1 - --&)
(for z 2 l-45)
(11)
are found to approximate G(z) to within 3 per cent for all z. The closest representation of the energy levels of the Hz0 molecule by the symmetric top model is obtained by choosing the A axis as the top axis and selecting an effective value for the remaining constant, such as (BC)l12, as has been done previously.(12) A difficulty arises in the case of the rotational spectrum in that strict application of the selection rules (AJ = 1, AK = 0) yields the expression w = 2(BC)1’2(Jf
1)
(12)
for the frequency as a function of quantum number. The permanent dipole moment, however, lies along the B axis, which must be considered the top axis in applying the selection rule AK = 0. On the basis of this argument, the effective rotational constant (BC)1’2 in Eq. (12) would be better replaced by (AC)‘12 or 4 (A + C).
High-temperature spectra of the pure rotational band of H20
705
For an asymmetric top, a reasonable value for the (average) frequency is given by the difference of the average term values for each value of J. This assumption gives for the average frequency w = 2(A+;+C)(J+l).
(13)
This relationship is consistent with the observed frequencies of identified lines. We note that for Ha0 the values of + (A + B + C), 4 (A + C), and (AC)lj2 do not differ greatly. We will use Eq. (7) for evaluating F&(J), but will use Eq. (13) as the relationship between o and J. In effect, this procedure retains the characteristic structure of the symmetric top energy levels but releases the restriction on the frequency which results from the symmetric top selection rule. Most asymmetric top model. The development for the most asymmetric top model proceeds as follows: The energy levels of a rigid asymmetric rotor are given by’14) E(J,T,A,B,C)
A+C = hc ---J(J+ 2
1) +-
A-C
2
&(J,T,K)
1 ,
(14)
where E, is a complicated function of J, r, and K, and for a given Jassumes 2J+ 1 different values. For the most asymmetric rotor (K = 0), it is found that the energy levels calculated by CROSS, HAINER and KING ( w h’rch are odd functions of 7) can be reasonably well approximated by E,(J,T,O)
z JT.
(15)
The approximation is especially good at high values of 1~1,as one can see from Fig. 9, where E, is plotted against T for J = 12. It is assumed that the relationship of Eq. (15) also applies to higher J’s. We assume a simplified pattern of transitions, namely, that only those transitions for which IAT] = 0 or 2 are considered and that the total intensity is divided equally among these three groups. This assumption permits explicit evaluation of the summations in Eq. (3). In particular, we .&id
=
exp[(hc/kT)&(A-C)J(J+l)]-exp[-(hclkT)&(A-C).72] exp[(hc/k
kT 2 z -__hc A_C
T)3( A - C)Jj - 1
(16)
1 J+4{ ex p[(hc/kT)B(A-C)(J+a)2l-exp[-(hc/kT)b(A-C)(J+~)2i).
If the rotational partition function is calculated using Eq. (16), and replacing the sum over J by an integral, and using J(J+ l)=(J+ Q2, we obtain QRA = (“)1’2(~)3’2~AB~c~~,2~
(17)
C. B. LUDWIQ, C.
706
C. FERRISO,W.
I, II -12 -10 -6
I
-6
-4
I -2
MALKMUSand F. P. BOYNTON
II 0 T
2
I 4
I
6
I 6
I IO
I 12
14
FIG. 9. Plot of &( K = 0) versus 7 for J = 12.
where B* = $[B+(LIC)~‘~].
(18)
The development was carried through in powers of the parameter e = (A -C)/2B,
(19)
which for the most asymmetric rotor may have values from 0 to 1. The limiting case of e = 0 is that of the spherical top. Because of the approximations involved in setting up the model, the development was carried out only to the second order in e. To this approximation, we note that the rotational partition function becomes Q,,
=
(TT)~~~(~)~‘~--&-[~ +ge2].
(20)
The expression for S$+l becomes 2B(J+ 1)[*(2.7+ 1)(25+3)]
x[
1 -exp(
-%)I(
1 ++?(l
exp
-~B.J2)+$2[~B(J++)2]2).
-FTBJ(J+
1
1)
(21)
We note that the second term in the brace is nearly independent of J. Since the population is a maximum when (k/H’) BJ2 ~1, the second term has a value of approximately (23/36) e2= #e2, which will cancel the second order term in QRA[Eq. (20)].
High-temperature spectra of the pure rotational band of Ha0
707
The third term in the brace remains as a significant function of J. When Eqs. (20) and (21) are combined, we obtain ,;+I
= [ffp$;]-$(;)3’z2B(J+
x exp
+2J+
I)][ 1 -exp(
--$BJ(J+
1)(2J+3)
-$)I
(22) or, to the same (second-order) s;+l
accuracy in e,
= [ffp~~]-$~(~)3’z2B(~+
x exp
I$(~J+
I)]( 1 -exp[
-sBJ(J+
sinh[(hclkT)B(A
1)(2~+3)
-$I)
-C)(J+8)21
(23)
’
‘-(hc/kT)+(A-C)(J+Q2
Since for the most asymmetric top B = Q (A + B-I- C), we have ,;+l
= 2B(J+
1)
(24)
and d = 2B.
We find, then, that the mean absorption p(J)
(25)
coefficient is given by
= S;+l/d
l)3exp[ --$BJ(J+
= [ gP$]$($r’2$J+
-$2B(J+
I)]
1) I)
x
sinh[(hc/k T)#(A - C)(J-t
4 )2]
(he/k T)+(A - C)(J++)2
’
(26)
In applying the results of the most asymmetric top model, we define an “equivalent most asymmetric top” by the conditions A’B’C’
and
A’+B’+C’ A’+C’-2B’
= ABC = A+B+C = 0.
(27)
708
C. B. LUDWIG, C. C. FERRISO,W. MALKMUS and F. P. BOYNTON
From the valueso5) of the equilibrium rotational constants for HaO, A = 27.33 cm-l B = 14.58 cm-l
C = we obtain for the “equivalent
9.50cm-l,
most asymmetric top” A’ = 25.68 cm-l B’ = 17.14cm-l
C’ =
8.60cm-I.
When the two expressions, Eq. (7) and Eq. (26), are evaluated using the values of A’, B’, and C’ for HaO, it is found that the resulting values of pj& and PAS are not greatly different for a given value of J, when P* is large. In the wing of the band, they differ by
FIG. 10.
Plot of j$,
(dashed line) and FX, (solid line) versus J forj12OO”K.
an increasingly large factor, as can be seen in Fig. 10. For high values of J, P&(J) from Eq. (7) is proportional to J2 exp
and P&(J) from Eq. (27) is proportional Jexp[
- ( BQ~~“&J~]
to -G$J2],
High- temperature spectra of the pure rotational band of Ha0
709
Since C < (BC)lj2, it is evident that P& > P;r, for sufficiently high J. In other words, the lowest levels of the asymmetric rotor with rotational constants, A, B, and C of water are lower than the lowest levels of the symmetric top having rotational constants A and (BC)l12. Therefore, the asymmetric top has a greater total population of the energy levels of a given J, when J is large. We use for P* of water the value given by the weighted mean of P& and P&, p&o = (1 +‘%o)& where
~~~~
=
-0.436.
The local mean absorption P
Ha0
=
coefficient is then given by
P&o(l-ew[ --g-J)3
which is plotted in Fig. 11 as a function of temperature
WAVENUMBER
FIG. 11.
and wavenumber.
(CM-‘)
Plot of the local mean absorption coefficient &9 between 300”and 3000°K.
COMPARISON
(28)
-‘Qrgo%
OF CALCULATION
versus wavenumber
AND EXPERIMENT
In Figs. 12 through 15, the calculated and measured emissivities are compared for temperatures of 540”, 1040”, 1640”, and 2200°K. The dots represent the actual observed maxima and minima of the spectral structure. The solid lines were calculated using the relation r = 1 -exp( -&I), (30) where pl = optical depth and P is the local mean absorption coefficient in the weightedaverage asymmetric top expression, Eq. (29). Since the spectral emissivities calculated according to Eq. (30) are appropriate only to highly pressure-broadened gases, and since it is obvious that our experimental spectra are not highly pressure-broadened, we should expect Eq. (30) to provide an overestimate
C. B. LUDWIG, C. C. FERRISO,W. MALKMUS and F. P. BOYNTON
710
of the emissivity. In order to account for the spectral structure, statistical band model with exponential intensity distribution
the equation for the
E = 1 - exp[ - PpZ/( 1 + Ppr/4Z)l9
(31)
was used for the emissivity. The parameter
a = ~(yzpT[c(~)1’2+eg+(l +&]
(32)
is the ratio of collision-broadened line half-width to average line spacing, y0 is the line width at one atm and 300°K averaged over the band, c is the mole fraction of water vapor in the sample, CT* is the ratio of the optical cross-section for “inert” gases to that for water vapor, and CJthe ratio for non-resonant to resonant dipoles. Effective band-averaged values of 6 have been determined at various temperatures for the 6+3- and 2.7-p fundamental bands of water vapor. 06) These calculations are uncertain within a factor of 5 to 7. We have therefore calculated the emissivities by Eq. (31) in the upper and lower limits of d. The results are shown in Figs. 12 through 15 as dashed lines. The agreement with the measured values becomes better, especially at the longer wavelengths. I.0
I
,
,
SPECTRAL
,
,
,
EMISSIVITIES
I
,
OFTHE
,
,
Hz0
WAVENUMBER
,
,
I
ROTATIONAL
(CM-’
,
1
BAND AT
,
,
,
I
540°K
)
FIG. 12. Comparison of theory with present experiment, where the dots represent the maxima and minima of the experimental spectra; the solid line represents “thin gas” calculation, the dashed lines represent “thick gas*’ calculation with z = 0.085 (upper) and 0.0106 (lower). T = 540”K, pT = 1 atm, pl = 0.66 cm.atm.
The rigid rotor approach does not consider the actual spread of the lines in a group defined by a given J (which would tend to augment the emission at higher wavenumbers) nor the effect of centrifugal stretching (which would tend to decrease it). Thus any close agreement with experiment in the far wing of the band must be considered to some extent fortuitous. However, for the purpose of extrapolating to lower frequencies to determine total emissivities, the use of the calculated values is considered quite reasonable.
High-temperature
1.0
I
,
I
SPECTRAL
,
spectra of the pure rotational
I.,
1
EMISSIVITIES
,
I
OFTHE
(
H2b
I
,
I
ROTATIONAL
WAVENUMBER
(CM-’
711
band of Ha0
,
,
BAND
,
AT
I
,
’
,
1040°K
)
FIG. 13. Comparison of theory with present experiment, where the dots represent the maxima and minima of the experimental spectra; the solid line represents “thin gas” calculation, the dashed lines represent “thick gas” calculation with ii = 0.10 (upper) and 0.023 (lower). T = 1040°K,pT = 1 atm,pl = 1.1 cmeatm.
I.0
I
,
,
SPECTRAL
02
,
I
,
EMISSIVITIES
I
,
OF THE
I
,
Hz0
I
,
ROTATIONAL
I
,
BAND
I
,
AT
I
,
,
1640°K
. .
300
400
500
600
700
600
WAVENUMBER
900
1000
1100
l
-
L
*
1200
1300
(CM-‘)
FIG. 14. Comparison of theory with present experiment, where the dots represent the maxima and minhna of the experimental spectra; the solid line represents “thin gas” calculation, the dashed lines represent “thick gas” calculation with rT = 0.26 (upper) and 0.065 (lower). T = 164O”K, pT = 1 atm, pl = l-68 cmsatm.
712
C. I.0
,
B. LUDWIG,C.C.FERRISO,W.MALKMUS and F.P.BOYNTON
,
,
,
SPECTRAL
I
,
I
EMISSIVITIES
,
,
OF THE
,
I
Hz0
,
I
,
ROTATIONAL
1
BAND
,
AT
,
2200
,
,
OK
0.8-
300
500
400
700
600
WAVENUMBER
1100
1000
900
800
1200
13oc
(CM-‘)
FIG.15. Comparison of theorywith present experiment, where the dots represent the
maxima and minima of the experimental spectra; the solid line represents “thin gas” calculation, the dashed lines represent “thick gas” calculation with ii = 0.62 (upper) and 0.177 (lower). T = 2200”K,pT = 1 atm,pZ = 2.34 cmeatm.
As a further comparison, we show in Figs. 16 through 18 a similar comparison of our calculations with the data of NELSON(lo) at 555”, 832”, and 1111°K for p = 1 atm. The agreement appears to be quite good, particularly at the longer wavelengths. At the shorter wavelengths, there is a deviation of the calculated values from the measured data which we cannot explain, even on the basis of deviation from rigidity. Similar agreement is found with Nelson’s data for 2 atm. I -
E
’
I
SPECTRAL
’
I
’
EMISSIVITIES
I
’
I
OF THE
’ Hz0
I
’
I
ROTATIONAL
’
I
I
I’
BAND AT 555*K
-
-
0.5 -
0
’ 400
*
” 500
E WAVENUMBER
(CM-‘1
’s(lo) experiment, where the dashed-pointFIG.16. Comparison of theory with NELSON curve represents the experimental data, the solid line reprcscnts “thin gas” calculatbn, the dashed lines represent “thick gas” calculation, with Z = 0.281 (upper) and 0.0351 (lower). T = 555”K,p~ = 1 atm,pl = 38.6 cmsatm.
High-temperature spectra of the pure rotational band of Ha0 I
‘I(
I
_ SPECTRAL
‘II
EMISSIVITIES
1’1,
OF THE
Hz0
WAVENUMBER
I ‘I’ll ROTATIONAL BAND
AT 1lfl.K
713
-
(CM-‘)
Fro. 17. Comparison of theory with NELSON’S@)experiment, where the dashed-pointcurve represents the experimental data, the solid line represents “thin gas” calculation, the dashed lines represent “thick gas” calculation with ii = O-241 (upper) and 0.0482 (lower). T = 832”K,pT = 1 atm,pZ = 38.6 cm-atm. I
-
’
SPECTRAL
1’1’1
II
0
EMISSIVITIES
OF THE
I Hz0
n
1’1’1’
ROTATIONAL
I 1000 WAVENUMBER
BAND
I
I 1100
AT B32’K
I
I 1200
-
I
(CM-‘)
FIG. 18. Comparison of theory with NELWN’S(~~)experiment, where the dashed-pointcurve represents the experimental data, the solid line represents “thin gas” calculation, the dashed lines represent “thick gas” calculation with Z = O-295 (upper) and 0.059 (lower). T= llll”K,p, = 1 atm,pl= 38.6cmsatm. TOTAL
ROTATIONAL
BAND
INTENSITY
The total intensity of the pure rotational band system of a symmetric top is given by’ls’ a(T)
=
16= 3P,,B 2 3hckT ’
(33)
714
C. B. LUDWIG, C. C. FERRISO,W. MALTS
and F. P. BOYNT~N
In particular, Eq. (33) holds for a spherical top. If the expression for S,J+l, Eq. (23) is summed over J, retaining terms to the second power in e, we obtain the most asymmetric top 16+; a( T) w-B
3hckT
From the quoted values for the rotational
l+zez. [
1
(34)
constants, we find that e2 = 0.248 and
a(273”K) = 1350 [l-362] = 1840cmb2
-atmel.
Approximately the same value is obtained by a numerical summation of Eq. (7), the modified symmetric top model; an analogous series expansion has been developed for this case, but it is not as rapidly convergent. The effect of centrifugal stretching has not been considered here. An approximate treatment indicates that the overall effect of centrifugal stretching on the band intensity is quite small.(13) The experimental emissivity results (corrected to 273°K) appear consistent with this band intensity value, although the data are not sufficiently extensive to determine an experimental value. authors wish to acknowledge their indebtedness to assistancein obtaining and reducing the experimental data.
Acknowledgement-The
Mr. C. N. ABEYTA for
REFERENCES 1. C. B. LUD‘WIG, C. C. FERRISOand C. N. ABEYTA,JQSRT5,281(1965).
2. H. M. RANDALL,D. M. DENNISON,N. GINSBURGand L. R. WEBER,Phys. Reu. 52,160 (1937). 3. J. H. TAYLOR,W. S. BENEDICT and J. STRONG,J. Chem. Phys. 20,528 and 1884 (1952). 4. W. S. BENEDICT,H. H. CLAASSEN and J. H. SHAW, J. Res. Nat. Bur. Stand., Wash. 49,91 (1952). 5. C. H. PALMER,J. Opt. Sot. Amer. 47,1024 and 1029 (1957). 6. W. S. BENEDICT.M&n. Sot. Rov. Sci. LiPae. Suecial Vol. 2. 18 (1957). 7. N. G. YAROSLA~SKYand A. E. STANEVIC,, bp;. Spectrosc. 5,380 (1959). 8. S. S. PENNER, Quantitative Molecular Spectroscopy and Gas Emissivities, p. 322. Addison-Wesley, Massachusetts (1959). 9. C. C. FERRISO,C. B. LUDWIG and F. P. BOYNTON,submitted to Znt. J. Heat Mass Transfer. 10. K. E. NELSON, Experimental determination of the baud absorptivities of water vapor at elevated pressures and temperatures, M.S. Thesis, University of California (1959). 11. P. C. CROSS,R. M. HAINFZR and G. W. KING, J. Chem. Phys. 12,210 (1944). 12. S. S. PENNER,ibid. pp. 327-329. 13. W. MALKMUS,Intensities of pure rotational band systems, JQSRTS, 621-631(1965). 14. P. C. CROSS,R. M. HAINERand G. W. KING, J. Chem. Phys. l&27 (1943). 15. G. HERZBERG,Infraredand Raman Spectra, p. 488, Van Nostmnd, New York (1962). 16. C. C. FERRISO,C. B. LUDWIG and A. THOMSON,Empirical infrared absorption coefficients of Hz0 from 300” to 3000”K, to be published.