Empirical Molecular Heat Equations from Spectroscopic Data W. M. D. BRYANT, E. I. du Pont de Nemours & Company, Wilmington, Del.
T
HE study of molecular
w i t h i n the m o l e c u l e . This The calculation of molecular heats of gaseous structure and quanta has vibratory motion of the atoms substances f r o m spectroscopic cibration frehad a marked influence may be r e s o l v e d into one or quencies is briejly discussed. The methods outupon the theoretical treatment of more component vibrational delined are applied to eighteen gases and vapors. gaseous molecular heats a t high grees of freedom, each of which The theoretical molecular heats are found to be t e m p e r a t u r e s . Today it is has R as its limiting value a t high possible to calculate the molecutemperatures. Diatomic s u b in close agreement with the more reliable of the lar heats of a number of gases stances have but a single vibraexperimental results. with a precision approaching that tional degree of freedom. The A group of sixteen empirical second-power of the most exact experimental number of vibrational degrees of equations is derived to jit the theoretical molecular work. The c a l c u l a t e d heat freedom associated with polyheat curves f r o m 300" to 2000" K. The maxicapacities are obtained from the at.omic molecules is 3n - 6, n bespectroscopic v a 1e n c e and defmum deviation of the empirical f r o m the theoing the number of component ormation f r e q u e n c i e s of the atoms. For the special case of retical seldom exceeds 3 per cent. It is believed molecules with the aid of a few linear polyatomic molecules the that these equations as a group may be more s i m p l e r u l e s based on the bevibrational degrees of freedom reliable than previous empirical expressions havior of molecular models. are 3n - 5 . The rotational heat based on experimental spec@ heats and covering The contradictory nature of is almost always completely despecific heat data obtained by veloped a t room temperatures a similar temperature range. s e v e r a 1 different experimental and may be considered constant. methods has been the subject of The limiting molecular heat a t considerable comment (5,11,21)and does much to encourage very high temperatures can then be expressed by the two the use of the theoretical values calculated from spectra, following equations (14) : especially since most of the recent experimental work is in Diatomic and linear polyatomic molecules: agreement with the latter (5, 7, 15, 35). Several sets of empirical equations representing the experimental specific c,, = I2R R (3n - 5) (1) heat data are now available (6, 11, 28, 32, 33). These, in 8 xonlinear PolYatomic molecules: CP = Z R f R (3n - 6) ( 2 ) general, represent the experimental work satisfactorily enough, but almost necessarily involve a degree of partiality in the selection of the data. This disadvantage is less pro- Except for a smaIl positive correction arising from the stretchnounced in the equations of the Planck-Einstein type derived ing of the valence bonds and the accompanying increase in from spectroscopic vibration frequencies (12, 16, 19, 2.2). potential energy of the molecule ( I S ) ,no other types of motion However, these expressions are more cumbersome than simple should affect the specific heat in the range of terrestrially power series and require tables for their convenient solution. attainable temperatures, The increase in vibrational heat with rising temperature A set of empirical equations of the quadratic type has therefore been fitted to the theoretical curves of a group of may be expressed with fair approximation by means of the eighteen gases and vapors with only slight loss of accuracy, Planck-Einstein equation for the heat absorbed by a linear the maximum deviation from the theoretical seldom exceeding harmonic oscillator: 3 per cent between the limits 300" to 2000" K. Better hv agreement could, no doubt, have been gained by the use of higher powers, but this was considered undesirable from the standpoint of simplicity. The heat capacity of a gas a t zero pressure is made up of a where h = Planck's constant number of separate energy parcels arising from the motion of k = Boltzmann's constant the molecule as a whole, together with the vibration and shift R = gas constant per gram mol., calories T = absolute temperature in the equilibrium positions of its component atoms. All Y = vibrational frequency of molecule as deduced from gaseous molecules have a translational heat of 3/2 R to which spectra may be added an additional R , representing the difference between the molecular heat a t constant volume and constant ( h v / k ) , or 0, in practice reduces itself to 1,4327~',where is pressure. (The Value Of R employed in this paper is 1.9869 the characteristic wave number in cm.-~. 8 has the dimen15"-calories per molecule.) NO other heat than the 5/2 sions of temperature and is sometimes known as the characabove is absorbed by stable monatomic molecules (17). In teristic temperature of a vibration. the case of di- and polyatoniic gases, additional heat iS abThe complete molecular heat equation for a diatomic gas sorbed with the production of other types of motion. At is then: room temperature, rotation of the molecule may account for another R calories per molecule in diatomic and linear polyC,, = 6.95 P (4) atomic molecules, or 3/2 R in the case of other polyatomic molecules. Xost of the increase in specific heat above room for righthand member of Equation 3. temperature is due to the vibration of atoms and atom groups
+
+
(;)
820
(g)
In the case of polyatomic molecules where several vibrational frequencies are involved, a separate Planck-Einstein term may be used for each different frequency. (It is assumed that vibration is independent of rotation, an approximation which becomes less exact the lower the vibration frequencies.) The equations for two types of triatomic molecule-the linear or carbon dioxide type and the bent or water type-are: linear: C ,,
bent: C,,
821
I N D U S T R I ,4L A N D E N G I N E E R I N G C H E hI I S T R Y
July, 1933
=
6.96
I
I
85
?2 Bo
LE
5
+
= 7.95 f
-?
P(F)+F'(!$)+F'($)
(6)
$.
R
h
7c
Equation 1 is the limiting case of Equations 4 and 5 ; Equation 2 bears a similar relation to I I I 1 Equation 6. 800 1000 /zoo 1400 1600 iaoo 400 LOO The normal vibration frequencies of a number Emperature , i k p v e s & / v i a of diatomic and simple polyatomic molecules have FIGURE 1. h'fOLECUL.4R HEATS OF DI.4TOhlIC GASES been assembled bv Eucken ( I $ . 16). Mecke (29. Broken lines, theoretical curves; full lines, curves from empirical equations. SO), and others 22). In the present paper: preference has been given wherever possible to frequencies obtained directly from absorption and Raman constructed from Equations 4-6 without correction for mospectra without any adjustment to fit experimental specific lecular stretching or other minor effects. In the single case of hydrogen the small positive correcting term, 0.00008T, was heats. These normal frequencies, V I , YZ, v3 ..... (or e,, e3. . . . . to which they are proportional) are the only data appended to Equation 4 ( I S ) . Although the molecular heats calculated by means of the characteristic of the individual chemical substance that are required in the above method of calculating molecular heats Planck-Einstein equations are strictly valid only for zero pressure, little error is introduced in assuming their validity a t high temperatures. It is customary to divide normal vibrations into two classes: a t one atmosphere. Somewhat larger discrepancies may be valence vibrations acting along the chemical bonds, and expected below 400" K. in the cases of water vapor, ammonia, deformation vibrations acting a t right angles to the chemical hydrogen cyanide, and carbon disulfide, but these may be bonds. The latter cttn be present only in polyatomic mole- corrected, if necessary, by a suitable equation of state. cules, and are always of lower frequency than the correspond- AGREEMEXTBETWEEX THEORETIC.4L A S D EXPERIMENTAL ing valence vibrations. This distinction does not alter the RESULTS treatment as long as the correct number of degrees of freedom It would seem worth while before proceeding further to are assigned. It has consequently not been emphasized in ascertain what order of agreement exists between the theothe present article but is fully discussed by Mecke (SO). Among diatomic gases, satisfactory data for the calculation retical molecular heats and the more reliable of the experiof molecular heats have been found for hydrogen, nitrogen, mental results. Recent experimental work of improved precision by Henry oxygen, carbon monoxide, nitric oxide, hydrogen chloride, hydrogen bromide, and hydrogen iodide; among polyatomic ( d l ) , Eucken and Lude ( I j ) , and Chopin (7) a t moderate molecules, for water vapor, hydrogen sulfide, sulfur dioxide, temperatures, and by Wohl and Magat (35) a t very high hydrogen cyanide, carbon dioxide, carbon oxysulfide, carbon temperatures is indicative of a close agreement with theory disulfide, ammonia, acetylene, and methane. The numerical throughout the entire range covered by these investigators, values of the vibrational frequencies of these substances are the maximum deviation being about 3 per cent. A similar given in Tables I and I1 where they are expressed as (hvlk) order of agreement is observed between the experimental specific heat of water vapor by Knoblauch and co-workers (25) or 8. and the data obtained from the corresponding Planck-EinTABLE I. VIBRATIOS FREQUENCIES OF DIATOhlIC GASES stein curve (19). The theoretical equation for ammonia (9) Substance HC1 IIBr 0 2 CO NO HI H2 N2 is in harmony with the Bureau of Standards measurements R 6100 3360 2242 3090 2710 4135 3665 3199 ( S I ) upon this gas. Its values are, however, a few per cent Refmence (18) (29) (89) (15, 20) (89) (1) (1) (1) greater than those of Haber and Tamaru (18) a t higher Molecular heats a t zero pressure, C,,, obtained from the temperatures. values of 8 tabulated above, appear as broken lines in Figures For the better known gases a t room temperature, experi1 and 2. All of these curves except that of hydrogen were mental molecular heats (26) are reproduced by theory within
GI
e,,
TABLE 81
H?O 2292
Degrees of freedom
H2S 1690
1
1
Substance
11. VIBRATION FREQUESCIES so2 HCNo C025 752
1017
1
2
e2
5300
3750
1647
2994
ea
5370
3850 1
1920 1
4714 1
(do)
.. (SO,
Degrees of freedom Degrees of freedom
er
1
1
Degrees of freedom
1
.. ..
9 5
Degrees of freedom Reference a Linear molecules.
(19; 'SO) b
'Theoretical frequencies.
1
..
1
.. .. ..
(SO)
974 2 1838 1 3367 1
..
..
OF ~OLYATOMICGASES
COP 755 2 1332 1
2979 1
..
..
,.
(30)
(2)
CSP 569 2 93s 1
2185 1 .
I
(SO)
NHab 1346 1 2330 2 4810 2 6370 1
.. ..
(0)
C*H?a 860 2 1044 2
2830 1 4695 1
4828 1 (90)
CHI IS70 3 2180 2 4180 1 4310 3
..
(4','8)'
822
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 25, No. 7
about one per cent or better.' The spectroscopic equation Lewis and Randall without too great a sacrifice of accuracy. for methane agrees well with the experimental work a t room These empirical quadratic expressions were obtained by temperature. At higher temperatures this equation yields tabulation of the theoretical specific heats for even hundreds values of C,m lower than those of Dixon (10) and of Eucken of degrees between 300" and 2000" K. and subsequent treatand Lude (16). Since the interpretation of the methane ment by the method of least squares. The final empirical spectrum does not appear to be in error to this extent, the equations and the maximum deviation from the theoretical spectroscopic results have been chosen provisionally as the curves are as follows: more reliable. The e x p e r i m e n t a l w o r k of EQU.4TION GAS CD Max. DEVIATION FROM TEEORETICAL % Partington and Shilling (as, 3% 7 H2 6.88 4-0.0000662' + 0.0000002792'2
