INDUSTRIAL AND ENGINEERING CHEMISTRY
532
Subscripts
g = gas phase
IC = any component from 1 t o n 1 = liquid phase = any section from 1 to n at an entrance or exit in a steadyflow process
T
Literature Cited (1) Beattie and Bridgeman, Proc. Am. Acad. Arts Sci., 63, 229 (1928). (2) Benediot, Webb, and Rubin, J . Chem. Phys., 8, 334 (1940). (3) Brown, “Deviation of Natural Gases from Ideal Gas Laws”, Olean. N. Y.. Clark Bros. (4) Budenholzer, Sage, and Laoey, to be published. (5) Cope, Lewis, and Weber, IND. ENQ.CHEM.,23, 887 (1931). (6) Cragoe, U. S. Bur. Mines, Misc. Pub. 97 (1929). (7) Dodge and Newton, IND. ENQ.CREM.,29, 719 (1937). (8) Edmister, I h i d . , 30, 352 (1938). (9) Epstein, “Textbook of Thermodynamics”, p. 82, New York, John‘Wiley & Sons, 1937. (10) Gibbs, Collected Works”, Vol. I, p. 63, New York, Longmans, Green and Co., 1931.
Vol. 34, No. 5
(11) Ibid., Vol. I, p. 118. (12) Gibbs, Trans. Connecticut Acad., 111, 108 (Oot., 1875). (13) Goranson, “Thermodynamic Relations in LMdticomponent Systems”, Carnegie Inst. Washington, 1930. (14) Kay, IND.ENG.CHEM.,28, 1014 (1936). (15) Konz and Brown, Ibid., 33, 617 (1941). (16) Lewis, G. N., J . Am. Chem. Soc., 30, 668 (1908). (17) Lewis, G. N., Pmc. Am. Acad. Arts Sci., 37, 49 (1901). (18) Ibid., 43, 273 (1907). (19) Nellis, master’s thesis, Calif. Inst. Tech., 1938. (20) Sage, Baakus, and Laoey, IND. ENQ.CHEM.,27,686 (1935). (21) Sage, Budenholm-, and Lamy, I b X , 32, 1262 (1940). (22) Sage and Lacey, Ibid.. 30, 673 (1938). (23) Ibid., 31, 1497 (1939). (24) Ibid., 34, to be published. (25) Sage, Olds, and Lacey, to be published. ENQ.CHEM.,26, 1218 (1934), (26) Sage, Schaafsma, and Lacey, IND. (27) Sage, Webster, and Lacey, Ibdd., 29, 658 (1937). (28) Ibid., 29, 1188 (1937). (29) Scatchard, Chem. Rev.,8, 321 (1931). (30) Seltz, J . Am. Chem. Soc., 56, 307 (1907). (31) Watson and Nelson, IND.ENO.CHEM.,25, 779 (1933). ,
I
Heat Capacity of Certain Halomethanes The State University of Iowa, Iowa City, Iowa
A
semiempirical method of calculating heat capacities based on the additive contributions of the bonds and angles which make up the molecule is presented. It is shown that this method reproduces statistically calculated heat capacities of twentythree halomethanes and three temperatures with an average deviation of 0.6 pet cent. H e a t capacities of seven compounds are calculated statistically to the rigid rotatorharmonic oscillator approximation in the light of most recent developments.
AN INTERVAL
of time has always existed between theoretical developments in any given field of science and their use by industry. I n recent years this period has become increasingly shorter, and a t present industry is making use of the latest developments in theory and in some cases is itself actually engaged in the making of theoretical progress. The powerful tool of thermodynamics finds almost universal application in the solution of many industrial problems ; hence it is important to have reliable values for the thermodynamical properties of all substances. When complete spectroscopic data for a substance are available, it is possible t o calculate its thermal properties (for the gaseous state) more accurately than they can be measured. However such data are known for only a few molecules, and so it is necessary to turn t o less accurate approximations. The amount of research being carried out in Raman and infrared spectroscopy makes possible many calculations based on the rigid rotator-harmonic oscillator approximation. This paper is concerned with heat capacity calculations for some halomethanes for which new spectroscopic and theo-
retical data have recently become available, and the presentation of a semiempirical method by which such calculations can be extended to molecules whose fundamental vibrations are still uncertain or even unknown. Theoretical Considerations
The energy of a polyatomic molecule of this type may be divided between translation, rotation, vibration, and a small term involving the coupling of rotation and vibration. Neglect of this last term and the assumption of simple harmonic vibrations and constant moments of inertia result in the well known Einstein equation,
where x = hcv/kT, in cm.-l, and the summation is over the vibrational degrees of freedom. The constant, 4R, contains the classical contributions of translation and rotation plus R for C, - C., That little error is made in using the classical rotational contribution t o heat capacity has been demonstrated by many investigators, including MacDougall, Gordon, and Barnes (7). This equation, as commonly used, is accurate to 3 per cent and requires only the fundamental vibrations (and their degeneracies). If these data are uncertain or unknown, it is still possible t o obtain reliable heat capacity values by means of a semiempirical equation first derived elsewhere (6). Although the vibration of a polyatomic molecule is a complicated process involving, in general, the motion of all atoms in each fundamental vibration, nevertheless certain modes of vibration exist whose frequency is largely determined by only a few atoms. Thus in the halomethane C X Y Z T there is a frequency associated with the stretching of each valence
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
May, 1942
bond which remains nearly independent of the molecule in which the bond is located. The fundamentals for the remaining five degrees of vibrational freedom result primarily from the deformation of the six angles and are more or less characteristic of them. Equation 1 may be partitioned to give:
where Cii and Ci, are Einstein functions for a stretching and a deformation vibration, respectively. By choosing values of vi most nearly consistent with a large number of molecules, it is possible to approximate the second term with considerable accuracy. A summary of the characteristic C-X vibrations in some halomethanes has been published (6) ; Table I11 gives the values of vi for each bond as well as its contribution to heat capacity a t several temperatures. The deformation vibrations are, in general, more nearly functions of all the angles rather than any one; hence a different procedure must be adopted for them. The third term in Equation 2 may be satisfactorily approximated by expanding it into a summation over the six angles, weighted according to the total number of angles. Thus Equation 2 becomes: (3)
where niis the number of j-angles, and the summation is over all different angles. Cisis now the Einstein function for the hypothetical deformation frequency, 6i,to be associated with a j-angle and must be evaluated. a By Equation 3 the heat capacity of any halomethane is solely determined by the bonds and angles of the molecule and is made up of additive contributions from them. The success of this treatment is evident from Table IV. Bennewitz and Rossner (1) successfully applied a similar treatment to molecules containing C, H, and 0, in which deformation frequencies were assigned to bonds rather than angles. Such a procedure assumes that the forces resisting deformation arise primarily from the directed nature of the bond but is also valid whenever the bond appears only in angles of the same kind, as in their case. I n view of the fact that the forces between the halogen atoms in the halomethanes play an important part in deformation, it is not surprising that the modification outlined above is more successful in this case. Calculations Recent calculations (based on a normal coordinate treatment) performed in this laboratory have indicated that the fundamentals used in the previous heat capacity calculations on the methylene halides-CHzFz (6),CHzClz (9),CHzBr2 @)-were in error. (The CHzBrzvalues havedready been corrected, 2). The fundamentals given for them in Table I are more satisfactory than any so far proposed (II), and are used in evaluating the heat capacities in Table 11.
TABLE11. C; T OK.
CHzFa
250 273.1 298.1 300 350 373.1 400 460 473.1 500 550 600
9.44 9.81 10.24 10.28 10.22 11.67 12.19 13.14 13.56 14.04 14.93 15.66
CHzClz
c
11.23 11.74 12.28 12.32 13.38 13.85 14.37 15.26 15.64 16.06 16.78 17.42
CHzI2
FOR
533
HALOMETHANES
CHzFCl
Small calories ver 12.76 10:45 13.27 10.88 13.81 11.37 13.85 11.41 14.86 12.42 15.29 '12.88 13.41 15.75 16.54 14.33 14.73 16.87 17.23 15.18 17.83 15.95 18.37 16.66
CClaBr CCltBrz CBraCl K . ver mole 19.36 19.99 20.60 20.64 21.62 21.99 22.38 22.96 23.18 23.42 23.77 24.06
19.72 20.33 20.92 20.96 21.90 22.28 22.62 23.17 23.39 23.60 23.94 24.20
20.11 20.69 21.24 21.28 22.18 22.51 22.85 23.37 23.56 23.76 24.OS 24.33
I n addition, spectroscopic data are now available for CHzFCl (3), CC12Bi-Z (IO), CClaBr (IO), and CBraCl (IO). The fundamentals in CHzFCl were chosen to harmonize with those of CHZFZand CHzClz to which they bear a close resemblance; and as pointed out by Leader (@,the data for CHzFC1 are better interpreted by the inclusion of the shift Av = 1237 cm.-'. All these data appear in Table I, while their heat capacities calculated statistically by Equation 1 are found in Table 11. The standard state is the usual one of unit fugacity and zero pressure. All physical constants used were from the International Critical Tables. Before Equation 3 can be applied, it is necessary to evaluate the hypothetical deformation frequencies to be associated with each angle. As indicated previously (6) these constants were evaluated from statistically calculated heat capacities (4) and appear in Table 111. ~
TABLE111. CONTRIBUTION OB ANGLESAND BONDSTO HEAT CAPACITY Angle or
Bond HCH HCCl ClCCl HCF FCF HCBr BrCBr FCCl FCBr ClCBr C-H C-F
c-c1 C-Br
Y
or 8 ,
Cm.-1 1508 1040 223 1219 467 973 54 377 359 166 3000 1050 730 610
C; or Cz, Small Cal./O K./Mole 298.1' K. 373.1' K. 473.1" K. 0.075 0.205 0,441 0.340 0.607 0.922 1.869 1.806 1.913 0.710 0.411 0.196 1.686 1.530 1.328 1.010 0.412 0.694 1.983 1.980 1.976 1.784 1.522 1.673 1.802 1.701 1.569 1.945 1.920 1.884 0.0188 0.0026 0.0002 0.910 0.595 0.330 0.778 1.344 1.073 1.017 1.507 1.284
The new values for the methylene halides have a marked influence upon these frequencies, with the result that they differ considerably from those previously proposed (6). Not only are the individual and average deviations greatly reduced, but it is now possible to treat molecules of the type CX4with considerable accuracy by this method. Probably the most useful application of Equation 3 is to molecules for which no data are available. Thus the heat capacity of CHBrFz at 373.1' K. may be calculated to be
TABLEI. FUNDAMENTAL FREQUENCIES OF HALOMETHANES cg = 7.948 f (0.0026 f 1.284 -!- 2 x 0.595) f '/a (0.694 f (IN CM.-') 1.530 + 2 x 0.411 2 X 1.701) = 15.80 cal./O K./mole
+
CHzFa
CHzClz
CHzIz
CHiFCl
CClzBrz
CClaBr
CBraCl
1390 141 187' 243" 266" 164 766' 674" 230 210 289 250 326 418 318 710 734 370 670 720 750 a These frequencies have a Statistical weight of 2,all others have a weight of 1. 532 1054 1079 1262 1294 1488 1509 2963 3030
284 700 736 896 1149 1266 1425 2985 3046
119 487 573 713 1028 1129 1345 2970 3091
368 743 1004 1046 1238 1352 1468 2993 3048
A oomparison is made in Table IV between values calculated by Equation 3, in which constant stretching and deformation frequencies were used, and those calculated by the more rigorous statistical method using experimentally determined frequencies. The largest deviation is 2.8 per cent while the average is only 0.6 per cent. This is to be compared with 5.1 and 1.6 per cent for these same molecules using the earlier values (6),which were based on the older methylene
INDUSTRIAL AND ENGINEERING CHEMISTRY
534
TABLEIv.
COMPARISON BETWEEN
K.---
,---298.l0
ComDound
~i
calcd.
c; stat.
SEMIEMPIRICAL AND STATISTICALHEATCAPACITIES" ,-----373.1°
Deviation
c; calcd.
ci
K.-
stat.
Small cal./
Smal2 c a L / K./mole
K-.
c---473.1e
Deviation
ci
oalcd.
c;
Deviation
stat.
Small cal./ K./mole 12.76 12.77 15.71 15.64 19.09 18.99 13.14 13.17 16.39 16.34 19.97 19.87 11.79 11.80 13.94 13.56 16.69 16.68 14.81 14.73 17.46 17.52 18.26 18.11 22.13 21.95 18.82 18.71 19.38 19.28 18.54 18.40 19.67 19.54 21.85 21.85 23.38 23.39 23.63 23.56 23.13 23.18 22.89 22.84 23.89 23.86
% K./mole % % 11.05 11.05 9.76 9.74 +0.2 -0.1 CHaCl 12.28 13.85 13.85 12.21 -0.6 +0.4 CHiClr 17.36 15.73 -0.5 17.36 15.65 4-0.5 CHCl: 11.48 10.18 11.48 10.18 0 -0.2 CHaBr 13.14 14.66 13.07 14.66 -0.5 0 +0.3 CHrBrr 18.48 18.48 17.08 +0.5 -0.6 16.97 CHBr: 9.82 9.82 -0.1 8.95 4-0.1 CHrF 8.96 10.24 11.96 11.67 42:5 +2.8 10.43 +1.9 CHaF: 14.58 12.68 14.58 12.75 4-0.6 +O.l CHF: 12.88 12.88 11.28 11.37 -0.8 CHtFCl +0.5 15.59 15.47 -1.1 -0.8 3.3.64 13.79 -0 3 CHClF, 14.64 16.39 16.34 4-0.3 14.61 -0.2 $0.8 CHClaF 18.70 20.62 20.41 +1.0 18.93 +l.2 +0.8 CFCla 17.10 17.10 15.41 ... +0.6 CHBriF 17.73 17.73 16.08 10.5 CHBrClt 16.74 16.70 15:09 40:2 -0.6 +0.8 CHBrClF 15.00 18.10 18.07 4-0.2 +0.7 16.52 CHBrtCl 20.30 20.05 +1.2 16145 18.59 46:s f1.3 CFaBri 22.26 22.27 20.92 +0.2 0.0 20.97 0.0 CClrBrr 22.51 22.62 21.24 $0.5 21.43 +0.9 f0.3 CClBrr 21.92 21.99 -0.3 20.52 -0.4 20.60 -0.2 CBrCls 21.52 4-0.3 21.58 fO.2 19.96 +0.7 20.09 CClr 22.98 22.94 10.2 21.90 +0.3 21.84 10.1 CBrd 5 Besides those in Table 11, the statistically calculated heat capacities are from the following citations: CHaCl CHCL, CClr ( 8 ) ' CHsBr, CBraH, CBrr (8);CHsF, CHzBrz ( 8 ) ; CHFI. CHFCh, CHFiCl ( 4 ) ; all othe;s are from Leade; ( 6 ) .
... ... ... ...
... ... ...
...
...
...
halide heat capacities. A dotted line in the per cent de&tion columnsindicates that this value was used in determining the deformation frequencies in Table 111. Vold (9) showed that heat capacities calculated to the rigid rotator-harmonic oscillator approximation, using liquid Raman frequencies, are accurateto 5 per centwhen corrected t o finite pressures. Table IV indicates that the use of Equation 3 and Table 111 will yield values accurate to 8
Vol. 34, No. 5
per cent in the temperature range T = 250" to 560" K. This is more reliable than most of the. meager thermal data available at present. Acknowledgment
We gratefully acknowledge financial assistance received from the Committee on Grants of the American Association for the Advancement of Science, which enabled us to construct a wave-length comparator used in this research. Literature Cited
(1) Bennewitz, K., and Rossner, W., 2. physik. Chcm., B39, 126 (1938). (2) Edgell, W. F., and Glockler, G., J . Chcm. Phys., 9, 484 (1941). (3) Glockler, G., and Bachmann, J., Phys. Rev., 55, 1273 (1939). (4) Glockler, G.. and Edgell. W. F., J . Chem. Phys., 9, 224 (1941). (5) Ibid.. 9. 527 (1941). (6) Leader, G. R., Ph.D. thesh, Univ. of Minnesota, 1940. (7) MacDougalL D. p., Phus. Be%, 38, 2074. 2296 (1931); Gordon, A. R., and Barnes, C.,J . Phys. Chem., 36, 2601 (1932). (8) Stevenson, D. P., and Beach, J. Y . , J . Chem. Phys., 6 , 25 (1938). (9) Void, R. D., J . Am. Chem. SOC.,57, 1192 (1935). (10) WU, Ta-You, "Vibrational Spectra and Structure of Polyatomic Molecules", pp. 243, 250, Shanghai, China Science Corp., 1939. (11) Ibid., pp. 246, 248.
Covrter,,
Cracking U n i t in an
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