Approximate mean absorption coefficients in the spectrum of water vapor between 10 and 22 microns at elevated temperatures

J. Quant. Spectrosc. Radiat. Transfer. Vol. 10, pp. 373 388. PergamonPress 1970. Printedin Great Britain

APPROXIMATE MEAN ABSORPTION COEFFICIENTS IN THE SPECTRUM OF WATER VAPOR BETWEEN 10 A N D 22 MICRONS AT ELEVATED TEMPERATURES* G. D. T. TEJWANI and P. VARANASI Department of Mechanics, State University of New York, Stony Brook, N.Y. 11790 (Received 29 September 1969)

Abstract--Mean absorption coefficients are calculated for the short wavelength wing (10-22 microns) of the pure rotation spectrum of water vapor at elevated temperatures (400-1200°K). Benedict's low temperature tabulations of spectral line intensities based on asymmetric rigid rotor approximation have been extended up to 1200°K. These calculations incorporate corrections due to first-order centrifugal stretching and the enhanced population of the first vibration level. Comparison with the available experimental measurements of absorption coefficients at 500°K by Varanasi and the spectral emissivity measurements of Ludwig et al. at 590°K, 850°K, 1040°K and 1640°K shows that our computations are in good agreement up to 1200°K and are, therefore, useful between 400°K and 1200°K. I. I N T R O D U C T I O N

THE PURE rotation spectrum of water vapor has been extensively studied, both experimentally and theoretically, at low temperatures tl 9) (below 320°K), since it plays an important role in atmospheric radiative transfer, The rotational spectrum contributes significantly to the total radiant emission from hot water vapor at moderately high temperatures (below 1200°K). However, due to lack of suitable window materials, convincing experimental measurements on the far-infrared spectra of hot water vapor in static systems, are not available at the present time. LUDWIGe t al. ~4' ~0) have used the supersonic burner technique in their measurements, which is not subject to window limitations. Identification of w~/ter vapor lines in terms of their rotational quantum states is practically complete. BENEDICTand KAPLANt 11) have computed line intensities of all the spectral lines arising due to transitions from states with rotational quantum number J less than 14 at low temperatures (180°K-320°K) using complete asymmetric top intensity formulae. But there are, at present, no reliable theoretical predictions of the rotational spectrum of water vapor at high t e m p e r a t u r e s . MALKMUSO3) has given an approximate theoretical analysis by combining results of symmetric top and most-asymmetric top models. Calculations based on this analysis do not show convincing agreement with experimental measurements, t ~o) Similar studies by THOMSON t14) w e r e also not too successful. A discussion of theoretical models for describing the pure rotation spectrum of heated water vapor and their short-comings may be found in a recent monograph by PENNER and OLFF,.t~s) At the present time, our understanding of the pure rotation spectrum of water vapor at elevated temperatures, from a quantitative point of view at least, is not very precise. * Supported by the National Science Foundation through Grant No. GK-5114. 373

374

G.D.T.

TEJWANI and P. VARANASI

We have computed a set of absorption coefficients in the pure rotation spectrum of water vapor between 10 and 22 microns, in the temperature range from 400°K to 1200°K, using the just-overlapping line model and the line intensity computations based on BENEDICT'S(ll) calculations at 300°K. These computations take into account the firstorder centrifugal stretching of an asymmetric rotor and the effect of the first excited vibrational state on the integrated line intensity of a rotational line. Contributions from the wings of strong lines in the v2--fundamental band and the pure rotation spectrum itself are found to be insignificant. II. T H E O R E T I C A L

CONSIDERATIONS

In this section, we develop the expression for the temperature dependence of rotational line intensities that we have used in our computations. The integrated intensity of an individual spectral line arising from transitions between degenerate energy levels at thermal equilibrium, is given by, tl~ 8TC3(DIu N!

~

2 k T 1]

(1)

where l and u denote the collection of quantum states for the lower and upper energy levels, respectively, ~ot. is the frequency of the line centre in cm- 1, T is absolute temperature, g~ is the statistical weight of initial state, k is Boltzmann's constant, h is Planck's constant, c represents velocity of light, N~ represents the number of molecules in lower energy level and b~tul2 is the absolute value of the square of the matrix element of the directional cosines of the dipole moment associated with transitions from lower level to upper level. In the temperature range of our present calculations (400°K-1200°K), the first vibrational level is strongly excited as the temperature increases and its contribution to the vibrational partition function becomes significant. If No and N1 are the number of molecules in the ground level and the first excited vibrational level, respectively, we have

where go and gl are the statistical weights of ground and first excited vibrational levels, respectively, with energies Eo and El. Eo and E1 are measured from the ground level. In first approximation, we may write Eo ~- hcF(d~)

(2)

E l ~- hc[co2 + F(J0]

(3)

and

where F(J~) is the rotational term value (in c m - ~) and co2 is the smallest vibrational frequency of the water molecule (in c m - 1). Fairly elaborate calculations are necessary to obtain a realistic representation of the rotational term value of an asymmetric top. One of the most widely used formulae in numerical calculations, due to Ray, as corrected by KING, HAINER and CROSS,O6} is F(J~) = ~(A + C)J(J + 1) +½(A - C)E~

(4)

Approximate mean absorption coefficients in the spectrum of water vapor

375

where the rotational constants A and C for water vapor in the ground state t17) are h A = ~ =

27.79cm-

81r'~clA

C = ~

h

= 9.96cm-

la and Ic being the smallest and largest principal moments of inertia respectively. The parameter defining rotational energy levels E~ depends in a complicated manner on the rotational constants, the quantum number for total angular momentum J, and the index of levels for asymmetric-top molecules z. The quantity E, assumes 2J + I different values, which are identified in the order of increasing values by the index z = - J , - J + 1. . . . . J - 1 , J . It has been computed by TURNER, HICKS and REITWIESNER (is) for all levels with J < 12, and for all values o f x from 0 to + 1 in steps of 0.01 using rigid-rotor approximation. This tabulation has been extended by ERLANDSSON (19) u p t o J --- 40. The actual asymmetric top molecules are not strictly rigid. Due to centrifugal distortion, the energy levels are perturbed depending on J and z as well as on the force constants in the molecule. If N is the total number of molecules, we have

where we have assumed that No+N~ ~- N for the range of temperature of our present interest. The statistical weights of both the ground and the first excited vibrational levels for water molecule are equal to unity, since they are non-degenerate vibrational levels. For the ground vibrational level, the term (Ndp) in equation (1) may be written as

Ni P

Noglexp(-- Eo/k T) pQ

(6)

while for the first excited vibrational level it is

Nt -

-

p

Nlgl exp(-E1/kT) =

PQ

(7)

In these expressions, Q is the total internal partition function, which is given by Q = 2 gint exp( - 8int/kT),

where elnt is internal energy and gint is the statistical weight of the internal state; el.t is the sum of the electronic, vibrational and rotational energies, and the statistical weight is the product of the three corresponding factors. However, practically for all polyatomic molecules, the Boltzmann factors of excited electronic states are entirely negligible compared to those of the ground state. For the pure rotation spectrum, interaction between vibration and rotation may be neglected, and the statistical sum may be written as the

376

G.D.T. TEJWANIand P. VARANASI

product of the vibrational and rotational partition functions,

Q =- Q,." Qr.

(7a)

Since only the first vibrational level is assumed to make any important contribution to the total vibration partition function in our calculations, Q,, may be calculated by using the harmonic-oscillator approximation ; it is given by, ~2°)

krj/

....

where dl, d2 . . . . are the degrees of degeneracy of the vibrations col, co2 . . . . . respectively. Up to 1200°K, only the v2--fundamental, which is non-degenerate, is of practical importance. Therefore,

1 Q'; - 1 - e x p ( - hcco2/kr)"

(8)

The rotational partition function for an asymmetric-top molecule is ~z°)

[x/(BC)hc]

7t

kT 3

] J ~ +/ For temperatures above IO0°K, this relation may be approximated by

Qr "~

A~

h~c

"

(9)

Modifying equation (1) to include transitions from the first vibrational level, we get the following equation for the integrated line intensity : St.-

3hcpcp t - Q e x p / - ~ ) + Q

-

--kT,]"

Using equations (5) and (7a), 8~3co,. I N \

2 [`

[

hcco,,,]]

exp( - Eo/kT) exp(-_Ej/kT) -I × -1 + e x p [ ( E o - E 1 ) / k T ] + 1 +exp[(E1 - E o ) / k T ] J "

(10)

In equation (10), the temperature dependence (for a fixed number of molecules, N) enters, through (a) the total pressure p, ( N / p = 1/kT), (b) the Boltzmann factor e x p ( - Eo/kT) exp( - E ffkT) 1 + exp[(Eo - E~)/kT] ~ 1 + exp[(E~ -

t

~o)/kTif'

(c) the induced emission factor [ 1 - exp(-hcco,./kT)], (d) the rotational partition function Q, = x/[(rc/ABC)(kT/hc)3], and (e) the vibrational partition function Q~ = [1 -- exp(-- hcco2/kT)]- 1

Approximate mean absorption coefficients in the spectrum of water vapor

377

The ratio of integrated line intensity at any temperature T1 with respect to line intensity at some reference temperature To is given by

~l-exp(-hcohu/kT1)]-exp(-hco92/kTo)]lTol

(S/u)r, -

~

exp( - Eo/k Tl) ×

+

~

J

~

5/z

1

exp( - E 1/k Tl)

(11)

1 +exp[(Eo-EO/kTt] 1 +exp[(Et-Eo)/kT1] exp( - Eo/k To) exp( - E 1/k To) + 1 +exp[(Eo-EO/kTo] 1+exp[(Ei-Eo)/kTo]

Ill. C O M P U T A T I O N S AND C O M P A R I S O N S WITH AVAILABLE EXPERIMENTAL RESULTS

We have employed equation (1 l) to compute spectral line intensities at higher temperaatures, using BENEDICT'S¢~1~ line intensities at 300°K. They constitute our reference data and are in very good agreement with the high-resolution experimental results published recently t12~ for temperatures below 400 ° K. We have considered 413 lines from 447.66 c m to 1001.09 cm 1 in our computations. These include all of the spectral lines arising as the result of transitions with J < 14. Some spectral lines with higher values of J, which make significant contribution to the spectrum, have also been considered in the higher wavenumber region. The value of Eo for each spectral line has been computed by using equations (2) and (4). Values of E¢ were computed by using the asymmetric rigid-rotor tables given in Ref. 17 for J < 12 and in Ref. 21 for J > 12. The value of E~ for each spectral line was obtained from Ref. 9. We have computed the local mean absorption coefficient (in c m - 1 a t m - 1) at spectral intervals of 5 cm-1 from 450 cm -1 to 1000 cm-1 at 400°K, 600°K, 800°K, 1000°K and 1200°K. In thecalculations, we have summed the spectral line intensities of all important lines which are within +2.5 c m - l of the wavenumber chosen and have averaged over the spectral interval. The interval was chosen to be 5 cm-~, because the spectral resolution of available experimental measurements is of this order. To get the mean absorption coefficient in c m - 1 a t m - t, the sum of line intensities must be divided by R~T, where Rg is gas constant per unit mass and T is the absolute temperature. The results are shown in Table 1. We have also plotted mean absorption coefficients at 400°K, 800°K and 1200°K in Figs. 1, 2 and 3, respectively. We first compare mean absorption coefficients computed at 500°K with self-broadening measurements of VARANASI(22) and PENNERand VARANASI(23~ at 500°K at pressures of 5, 10 and 20 atm, from 500 cm-1 to 1000 c m - i , in Fig. 4. The agreement at 5 atm is reasonable while, at higher pressures, hydrogen-bonding in the vapor phase probably produces experimental absorption coefficients that are higher than the c a l c u l a t e d values, t23'24~ We next compare our results with the experimental emissivity data obtained by LUDWIG et al. ~1°~ at these temperatures between 10~ to 22/~. All of these measurements were performed at a total pressure of 1 atm and a pathlength of 3.12 cm, with slit widths of 0.14# at 10/~ and 0.65p at 20/~. The optical depths were 0.62, 0.92 and 1.10 cm atm at 590°K, 850°K and 1040°K, respectively. Reference to Figs. 5, 6 and 7 shows that the agreement between theoretical and experimental results is good at these temperatures and pressures.

G. D. T. TEJWANI and P. VARANASI

378

TABLE I, MEAN ABSORPTION COEFFICIENTS OF WAFER VAPOR (PER ('M PER ATMJ TEMPERATIURE IN I)tqGREE KELVIN

Wavenumber (cm ~)

400.0

600.0

800.0

1000.0

450.00 455.00 460.00 465.00 470.00 475.00 480.00 485.00 490.00 495.00 500.00 505.00 510.00 515.0(I 520.00 525.00 530.00 535.00 540.00 545.00 550.00 555.00 560.00 565.00 570.00 575.00 580.00 585.00 590.00 595.00 600.00 605.00 610.00 615.0(I 620.00 625.00 630.00 635.00 640.00 645.00 650.00 655.00 660.00 665.00 670.00 675.00 680.00 685.00 690.00 695.00 700.00 705.00 710.00 715.00 720.00

.2111E-01 .1120E 00 .4136E 00 .1572E-02 .2256E 00 .1606E 00 .2048E-01 .9558E-01 .7800E-01 .1768E-01 .1535E 00 .4701E-01 .4375E-01 .1105E 00 .9086E-01 .9675E-01 .2346E-03 .1878E-01 .1096E-01 .1800E-01 .4597E-01 .6977E-02 .5262E-02 , 1398 E-01 .2612E-01 .4564E-01 .3015E-01 .2452E-01 .3979E-01 .4420E-02 ,2921E-01 .4792E-02 .1706E-03 .8576E-02 .1058E-01 ,1691E-01 .1073E-03 .2998E-01 .1584E-01 .4178E-02 .4013E-02 .3687E-04 .6762E-02 .1166E-02 .9399E-02 .2026E-03 .1389E-02 .4135E-02 .1259E-01 .7805E-02 .6549E-02 . 1250E-01 ,1722E-04 .3565E-02 .9651 E-03

.4687E-0l .1445E 00 .5339E 00 .1814E-02 .4739E 00 .3186E 00 .6842E-01 .2268E 00 .1984E 00 .7277E-01 .2573E 00 .5806E-01 .1785 E 00 .2876E 00 .9187E-01 .1533E 00 .2005E-03 .1027E 00 .8421E-01 ,2533E-01 . 1465E 00 .2214E-01 .5636E-01 .4362E-01 .3669E~01 .9084E-01 .4598E-01 .4130E-01 .4984E-01 .2478 E-02 ,6636E-01 .6656E-02 .1472E-03 ,4395E-02 ,2083E-01 .1813 E-01 .1833E-03 ,3629E-01 .3050E-01 .3267E-02 .7773E-02 .8941 E-04 .5388E-02 .7823E-03 .2855E-01 .821 I E-04 .1255E-02 .3435E-02 .1904E-01 .1699E-01 .8196E-02 .2326E-01 .3489E-04 .6585E-02 .4154E-02

.4622E-01 .1274E 00 .4735E 00 .1493 E-02 .4894E 00 .3527E 00 .8243E-01 .2631E 00 .2351E 00 .9706E-01 .2868 E 00 .5588 E-01 .2515E 00 .3963E 00 .7043E-01 .1981E 00 .1369E-03 .1779E 00 .1637E 00 .3993E-01 .2627 E 00 .3499E-01 . 1227E 00 .9490 E-01 .3312E-01 .1706E 00 .6437E-01 .3535 E-01 .7540E-01 .1231 E-02 .1204E 00 .5178 E-02 ,9060E-04 .2084E-02 .1931E-01 .1415 E-01 .1583 E-03 .2783E-0 l .2900E-01 .1918 E-02 .7150E-02 .9206E-04 .3223E-02 .4352E-03 .3636E-01 .3462E-04 .8378E-03 .2076E-02 .1550E-01 .! 981E-O 1 .6120E-02 ,2398 E-01 ,3292E-04 .6164E-02 .5835E-02

.3524E-01 .9722E-01 .3516E 00 .1064E-02 .3868E 00 .2947E 00 .7079E-01 .2233E 00 ,2032E 00 .8860E-01 .2496E (t(t .4881E-01 .2400E 00 ,3857E 00 .5029E-01 .1940E 00 ,8656E-04 .1914E 00 ,1800E 00 .4483E-01 .3084E 00 .3657E-01 .I 507E O0 . 1212E 00 .2464E-01 .22(/5 E 00 .7355E-01 .2477E-01 .9420E-01 .6237E-03 .1625E 00 ,3426E-02 .5216E-04 .1092E-02 .1421 E-01 .9790 E-02 .1116E-03 .1876E-01 .2l 82E-01 .1073E-02 .5237 E-02 .7213E-04 .1830E-02 .2388E-03 .3268E-01 .1588E-04 .5236E-03 .1182E-02 .1056E-01 .1749E-01 .3956E-02 . 1956E-01 .2450E-04 .4612E-02 .5545E-02

1200.0 .2461E-01 .7003E-01 .2442E 00 .7210E-0 .2772E 00 .2198E 0(t .5354E-0 I .1678E 00 . 1548E 01I .6976E-01 .1936E 0(1 .3150 E-0 I .1953E 0(1 .3206E 00 .3515E-0 I . 1634E 011 .5401 E-04 .1683E 0(1 .1727E 00 .4134E-01 .2924E 0(1 .3181 E-01 . 1448 E (10 . 12(11E 00 .1707f!-01 .2251E 00 .7108E-01 .1637E-01 .9660[:,-01 .3324E-03 .1754t" (~) .2179E-02 .3025E-04 .5388E-03 ,9702 F,-02 .6516E-02 .7404 E-04 .1225 E-0 I .1516E-OI .6109E-03 .3567E-02 .5138 E-04 .10521.!-02 .135(11:i-03 .2558E-01 .7911E-05 .3268E-03 .6798E-03 .6857E-02 .1371 F.-O I .2479E-02 . 1455 E-0 I . 1687 E-04 .3199E-02 .4506E-02

Approximate mean absorption coefficients in the spectrum of water vapor

379

TABLE 1 (Cont.) Wavenumber (cm ~) 725.00 730.00 735.00 740.00 745.00 750.00 755.00 760.00 765.00 770.00 775.00 780.00 785.00 790.00 795.00 800.00 805.00 810.00 815.00 820.00 825.00 830.00 835.00 840.00 845.00 850.00 855.00 860.00 865.00 870.00 875,00 880,00 885.00 890.00 895.00 900.00 905.00 910.00 915.00 920.00 925.00 930.00 935.00 940.00 945.00 950.00 955.00 960.00 965.00 970.00 975.00 980.00 985.00 990.00 995.00 1000.00

400,0

600.0

800.0

1000.0

1200,0

.5260E-03 .1466E-02 .2950E-02 .1313E-03 .6414E-02 .2301E-02 .2224E-02 ,1263E-03 .4869E-03 .6829E-03 .1609E-02 .3581E-03 .8661 E-03 .5216E-05 .2635E-02 .3321 E-02 .2258E-02 .7017E-03 .8675E-03 .4749E-05 .1673E-03 .2860E-03 .2357E-03 .5172E-03 .9831 E-06 .1075E-02 ,1473E-02 .2625E-03 .3157E-03 .1540E-03 .4888E-05 .1597E-03 .3855E-03 .2885E-03 .9715E-05 .2419E-04 .3523E-03 .5375E-03 .1124E-03 .5105 E-03 .7108E-04 .9005E-04 .1785E-04 .4007E-04 .8492E-04 .I 863E-03 .1456E-03 .1488E-03 .6577E-04 .1190E-03 .1836E-03 .9079E-05 .1552E-04 .1847E-04 .5827E-06 .1197E-03

.4052E-03 .4325E-02 .4443E-02 .1053E-02 .1317E-01 .7562E-02 .7231E-02 .4109E-03 .3891E-02 .1327E-02 .2199E-02 .4707E-03 .7563E-03 .3330E-05 .4414E-02 .7535E-02 .8510E-02 .3750E-02 .2461 E-02 .1018E-04 .5268E-03 .5930E-03 .1381 E-02 .1540E-02 .8415E-06 .2775E-02 .5716E-02 .2418E-02 ,6970E-03 .1367E-03 .7681E-05 .4366E-03 .1254E-02 .8493E-03 .3229E-04 .8908 E-04 ,2380E-02 ,I 958E-02 .5197E-03 .1534E-02 .7063E-04 .5395E-03 .1218E-03 ,1155E-03 .3951E-03 .2199E-03 .5547E-03 .1296E-02 .1600E-03 .5362E-03 .2932E-03 .1440E-04 .2328E-04 .1064E-03 .8758E-06 .3016E-03

.2364E-03 .6072E-02 .3622E-02 .1978E-02 .1253E-01 .9832E-02 .l 163E-01 .5383E-03 .7310E-02 .1270E-02 .1716E-02 .3594E-03 .4701E-03 .1829E-05 .4169E-02 .8187E-02 .1323E-01 .6063E-02 .3348E-02 .9929E-05 .6764E-03 .6043E-03 .2228E-02 .2100E-02 .5389E-06 .4163E-02 .7899E-02 .4891 E-02 .6928 E-03 .8604E-04 .6429E-05 .5913 E-03 .1954E-02 .9832E-03 .4025E-04 .1143 E-03 .4188E-02 .3666E-02 .7508E-03 .1961 E-02 .6573E-04 .9120E-03 .2131E-03 .1313E-03 .6301 E-03 .1602E-03 ,1058E-02 ,2667E-02 .1682E-03 .8481 E-03 .2548E-03 .1221 E-04 .1916E-04 .1729E-03 ,7216E-06 ,3651 E-03

.1318E-03 .5869E-02 .2470E-02 ,2228E-02 ,1033E-01 ,8955E-02 .1268E-01 .4954E-03 .8243E-02 .9575E-03 .1142E-02 ,2358E-03 .2727E-03 .9938 E-06 .3174E-02 .6735E-02 .1412E-01 .6274E-02 .3374E-02 .7556E-05 .6107E-03 .4765E-03 .2294E-02 .2054E-02 .3217E-06 .4555E-02 .7516E-02 .5776E-02 .5332E-03 .5037E-04 .4468E-05 .5698E-03 .2044E-02 .8313E-03 .3569E-04 . i 026E-03 .4567E-02 .4499E-02 .7249E-03 .1910E-02 .5533E-04 .9731 E-03 .2307E-03 .1099E-03 .6670E-03 .1025E-03 .1337E-02 .3208E-02 .1344E-03 ,8828E-03 .1826E-03 .8560E-05 .1320E-04 .1796E-03 .4978E-06 .1796E-03

.7482E-04 .4836E-02 .1604E-02 .2025E-02 .9393E-02 .7077E-02 .1147E-01 .3952E-03 .7504E-02 .6655E-03 .7310E-03 ,1492E-03 .1592E-03 .5570E-06 ,2235E-02 .4986E-02 ,1265E-01 .5399E-02 .2948E-02 .5291 E-05 .4785E-03 .3421 E-03 .1964E-02 .1731 E-02 .1921 E-06 .4214E-02 .6142E-02 .5429E-02 .3756E-03 .2960E-04 .2945E-05 .4718E-03 ,1786E-02 .6240E-03 ,2772E-04 .8026E-04 .4076E-02 .4439E-02 .5948 E-03 .1456 E-02 .4322E-04 .8551 E-03 .2045E-03 .8201 E-04 :5904E-03 ,6402E-04 .1355E-02 .3061 E-02 .9726E-04 .7675E-03 .I 232E-03 .5669E-05 .8653E-05 .1550E-03 .3267E-06 .2698E-03

380

G.D.T.

TEJWANI and P. VARANASI

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Approximate mean absorption coefficients in the spectrum of water vapor

381

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382

G.D.T.

TEJWANI and P. VARANASl

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Approximate mean absorption coefficients in the spectrum of water vapor

383

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We also show comparison of our emissivity computations with their experimental data at 1640°K in Fig. 8. We expect the computed values to be low because, for temperatures > 1200°K, contributiov~ rr,~m levels of high J-values (J > 14) become more and more important. IV. C O N C L U S I O N S

Comparison with the available experimental measurements (see Figs. 4-8) shows that our computations may be used to predict the pure rotation spectrum of water vapor, up to 1200°K, with accuracy comparable to that of the experimental measurements. Since our computations including the plots have been p r o g r a m m e d (IBM/360), results may be obtained on request from us for any intermediate temperatures. The contribution of the far wings of strong distant lines, in the temperature range of our interest, is found to be less than 2 per cent even in the region where the discrepancy between the experimental and theoretical results is large. Also, contributions due to strong

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lines in the v2--fundamental band (based on band-intensities given in Ref. 15) are found to be negligibly small. To extend our computations beyond 1200°K, Benedict's tables t15) need to be modified so as to include the spectral lines arising from transitions involving high J levels and corrections due to second-order centrifugal stretching. (12~ There is also a need to obtain more accurate and refined measurements of the pure rotation spectrum of water vapor at such high temperatures. REFERENCES 1. 2. 3. 4. 5. 6.

W. S. BENEDICT and L. D. KAPLAN, J Q S R T 4 , 453 (1964). H. M. RANDALL, D. M. DENNISON, N. GmSBURC; and L. R. WEBER, Phys. Rev. 52, 162 (1937). J. H. TAYLOR, W. S. BENEDICT and J. STRONG, J. Chem. Phys. 20, 528 (1952). C. B. LUDWIG, C. C. FERRISO and C. N. ABEVTA, J Q S R T S , 281 (1965). C. H. PAL~ER, J. Opt. Soc. Am. 47, 1024 (1957). N. G. YAROSLAVSKYand A. E. STANEVZCr~,Optics. Spectrosc. 7, 380 (1959).

388 7. 8. 9. 10. II. 12. 3. 4. 5. 16. 17. 18. 19. 20. 21. 22. 23. 24.

G.D.T.

TI~J,~,'Ayl and P. VARANASI

W. T. ROACH and R. M. G(×)DY, Quart..I.R. Met. Soc. 84, 319 11958L M. W. ELSASSER, Astrophys. J. 87, 497 (19381. W. S. BENEDICT, H. H. CLAASSENand J. H. SHaw, .I. Res. Natn. Bur. Stand. 49, 91 11952). C. B. LUDWIG, C. C. FERRmO, W. M ~,I.KMUSand F. P. BOYNTO~, J Q S R T S , 697 (1965). W. S. BENEDICT and L. D. KAPtAN, unpublished calculations, which are reproduced m part on p. 184 of Atmospheric Radiation 1. Theoretical Basis by R. M. GOODY. Oxford Univ. Press. London 11964). J. R. IZATL H. SAKAI and W. S. BENEDICT, J. Opt. ,S'oc. Am. 59, 19 (19691. W. MAI KMtJS, J Q S R T S , 621 (19651. S. S. PENNER, Quantitatire :lloh, cular ,S'pcctro.scopv amt (;a.s Etm~'sit'ilics. Addison-Wesley. Massachusetts (19591. S. S. PENNER and D. B. OI.FE, Radiation and Reentry, p. 83. Academic Press, New York (19681. P. C. CROSS, R. M. HAINER and G. W. KING. J. Ghent. Phys. 12, 210 119441. C. H. TOWNES and A. L. SCHAWLOW, Microwat'e Spectro~xcopy. p. 527. McGraw-Hill, New York (19551 T. E. Tt RNER, B. L. HI('KS and G. REITWIISNER, Reporl 878. Ballistics Research Laboratory, Aberdeen. Maryland (19531. E. ERLANDSSON,Ark. Fvs. 7, 65 (19551. G. HERZBERG, Moh'cular Spectra arid Moh,cular Structure I1, It!/~'ared at?d RclolUtt Spectra o/Poh'al