Gray, L. D.

Penner, S. S.

English

Caltech Authors - Main

Reprinted from JOURNAL OF THE OPTICAL SOCIETY OF AMERICA, Vol. 51, No.4, 460-462, April, 1961 
Printed in U. S. A. 
Approximate Infrared Emissivity Calculations for 
HCl at Elevated Temperatures* 
S. S. PENNER AND L. D. GRAY 
Daniel and Florence Guggenheim Jet Propulsion Center, California Institute 
of Technology. Pasadena. California 
(Received November 15. 1960) 
I N a recent paper,' Plass has presented the results of infrared emissivity calculations for HCI at elevated temperatures. The 
data were obtained through the use of machine computations by 
employing the best available spectroscopic data. We wish to point 
out that numerical values in good accord with Plass' spectral 
absorption coefficients may be obtained, when the weak-line 
approximation holds, by utilizing the results of a relatively simple 
analytical treatment to the harmonic oscillator approximation. 
Our method is similar to an analysis first used by Kivel, Mayer, 
and Bethe2 in emissivity calculations on the ultraviolet bands of 
NO. This method of treatment for electronic band systems has 
been refined recently by Keck, Camm, Kivel, and Wentink3 in 
order to account for the influence of vibrational structure on 
emissivities. 
O. 
0.7 
X=IO em-atm 
0.6 
€w I 
0.5 I I 
I 
I 
0.4 I I 
I I 
I I I I I 1 0.3 I I I 
I I 
0.2 X=I 
em-aIm 
0.1 
0 2200 3400 
w (em-I) 
FIG. 1. Comparison of spectral emissivities in the weak-line approxi-
mation obtained by numerical calculations l (dotted curves) and spectral 
emissivities obtained by using EQ. (3a) (solid curves) for various optical 
depths at 600° K. 
1.0 
0.9 
O. 
I 
f 
I 
X= 100 em-aIm I 
O. f 
I 
Ew I 
O. 
0.4 
0.3 
X=IOem-alm 
0.2 
X=lem-atm 
0.1 
1400 1800 3800 
FIG. 2. Comparison of spectral emissivities in the weak-line approxi-
mation obtained by numerical calculations l (dotted curves) and spectral 
emissivities obtained by using Eq. (3a) (solid curves) for various optical 
depths at 2400°K. 
Consider' transitions between the ground state with vibrational 
quantum number n" and rotational quantum number j" and the 
upper state n', j'. Then 
hCWc::>'.(En·- En,,)±2j'hcBe, 
where w is the observed wave number for the transition (n",j") -> 
(n',j') if En" and En' are the corresponding vibrational energies. If 
the dominant contributions are made by the higher rotational 
energy levels, then 
Ei"c::>'.(j')2hcBe= [w/2Be- (En'- En ,,)/2hcBe]hcBe 
and· 
{ [hcw-(En'-En,,)]2} Xexp - 4hcBekT . (1) 
2 LETTERS TO THE EDITOR Vol. 51 
In Eq. (1), jn',n" is the total jnumber for the n' --> nil transition, 
k is the Boltzmann constant, and T denotes the temperature. 
It is convenient to write 
(Z) 
where jn'-n", O is the j number for the transition from the ground 
state that falls in the same spectral region as the transition 
n' --> nil. Since6 
7re2 N n" djn' n" Pw, ,,~- ----'- (l_e-hcw/kT) 
n,n me2p dw ' 
where PWn', n" is the spectral absorption coefficient at the wave 
number w produced by the vibrational transition n' --> nil (e and m 
denote, respectively, the electronic charge and mass; Nn,,/p 
represents the total number of molecules in the ground vibrational 
state per unit volume per unit pressure), we find for the spectral 
absorption coefficient P w the following explicit relation: 
1 7re2 NT he ~ PWn' n"~--- jn' n" 0- --[l-exp(-hew/kT)] 
n" ' Z me2 - 'pQv ZBekT 
X ~ x2n',n,,[exp(-<n,,/kT)]lw-wn',n,,1 
n" 
[ -he(w-wn' n,,)2J (3) Xexp 4BekT' , 
where NT/p is the total number of molecules of the emitting 
species per unit pressure per unit volume. Equation (3) takes a 
particularly simple form to the harmonic oscillator approximation 
since only the transitions n' --> n'-l are allowed in emission, 
wn',n"~wO=constant, <n"= (n"+!)hewo,X2n', n,,=n' = nil +1. Thus 
17re2 NT . 
Pw,h.o.=-Z 2iJ.0-Q [l-exp( -hew/kT)][exp( -hewo/ZkT)] 
me p v 
he1w-wol [ X[l-exp( -hewo/kT)]-2 exp 
2BekT 
he(w-wo)2] 
4BekT .' 
(3a) 
where the subscript h.o. identifies results that are applicable to 
the harmonic oscillator approximation and w~wo for harmonic 
transitions. 7 
Spectral emissivities 
<w~l-exp( -pw,h.o.pl), 
computed by the use of Eq. (3a), are compared with Plass' data 
in Figs. 1 and z.s Reference to Figs. 1 and 2 shows that Eq. (3a) 
represents an acceptable approximation for the spectral absorption 
coefficient. 
* Supported by the Office of Naval Research under contract. 
IV. R. Stull and G. N. Plass, "Spectral emissivity of hydrogen chloride 
from 1000-3400 em-I," Pub!. No. U-461. (Aeronutronic, Newport Beach, 
California, May 15, 1959). 
2 B. Kivel, H. Mayer, and H. A. Bethe, Ann. phys. (N. V.) 2, 57 (1957). 
3 J. C. Keck, J. C. Camm, B. Kivel. and T. Wentink, Ann. phys. (N. V) 
7, 1 (1959). 
4 A more complete account of the theoretical treatment will be published 
elsewhere by S. S. Penner. K. G. P. Sulzmann, and C. B. Ludwig. 
6 S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities 
(Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 
1959), p. 398. 
6 Reference 4, Eqs. (14-7) and (2-19). 
7 Equation (3a) is identical with Eq. (11-141) of reference 4. 
8 In our calculations we used a numerical estimate for /t,o equivalent to 
that employed by Plass for the sum of the isotopic species Reps and HC137. 
This value is probably somewhat low (d. reference 4), 


Approximate infrared emissivity calculations for HCl at elevated temperatures

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Approximate infrared emissivity calculations for HCl at elevated temperatures

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