= density at critical point, grams/cc.
De IC
K, M
P Pa
=
Ramsey-Shields constant
=
molecular weight
=
critical pressure, atmospheres
= a constant
= pressure or vapor pressure, atmospheres = reduced pressure (pressure/critical pressure)
parachor as defined by Equation 4 gas law constant, (cc.) (atmospheres)/’ K. temperature, OK. normal boiling temperature, ’ K. = critical temperature, K. = temperature at which ,1 gram mole of saturated vapor occupies 22.4 liters, K. = reduced temperature (temperature/critical temperature) = critical molecular volume, cc./gram mole = constants = surface tension, dynes/cm. = PCV~M I R T c = Mp/DRT = = = =
O
Y Pc
PL
Vol. 34, No. 5
INDUSTRIAL AND ENGINEERING CHEMISTRY
526
Literature Cited (1) Brown, Souders, and Smith, IND.ENQ.C H ~ M 24, . , 513 (1932). (2) Cope, Lewis, and Weber, Ibid., 23, 887 (1931). (3) Guldberg, 2. physik. Chem., 5, 134 (1880). (4) Guye, J. phys. 9,312 (1890). (5) Landolt-Bornstein, Physikalisch-Chemische Tabellen, 6th ed., Vol. IIa. p. 175, Berlin, Julius Springer, 1923. (6) Lautie, Bull. SOC. chim., [51 2, 155, 2234 (1935). (7) L e w k Nature, 145,551 (1940); J. Chem. SOC., 1066 (1938). (8) Mathias, “Le point critique”, Paris, 1904. (9) Meissner and Paddison, IND.E m . CHEM.,33, 1189 (1941). (10) Merkel, Proc. Acad. Sci. Amsterdam, 40, 183 (1937). (11) Natanson, J. phys., 4,219 (1895). (12) Partington, Trans. Faraday SOC.,17,734 (1922). (13) Sugden, “The Parachor and Valency”, London, Routledge & Sons, 1930. (14) Waal, van der, “Kontinuitkt des Gas formigen und flussinen Zustandes”, Leipzig, 1899. (15) Watson, IND.ENG.CHEM.,23, 361 (1931). (16) Young, Phil. Mag., 33, 153 (1892).
Applications of Therm 18. A. SqZe ta4d R. A. O& California Institute of Technology, Pasadena, Calif.
Some of the simpler applications of thermodynamics to multicomponent systems have been indicated. The utility of the chemical potential or the fugacity of a component in the estimation of the phase behavior of a complex system is discussed. Experimental values of the partial enthalpy of methane, ethane, and propane in liquid-phase systems containing a hydrocarbon liquid of high molecular weight are recorded for a range of compositions and temperatures. The application of the general energy equation for steady flow to the passage of heterogeneous mixtures through branch conduits is illustrated.
TofHERMODYNAMICSof
has been applied to the solution many problems engineering interest, but its extension to systems containing more than one component has been hampered by the paucity of experimental facts relating to the behavior of such systems. Furthermore, the engineer has not had occasion to be greatly interested in the thermodynamic relations of such systems. Recent industrial developments in the processing of fluids have caused the subject t o assume new interest, particularly in the petroleum and other chemical industries. It is believed that a suficient background of factual information is available at present to make it desirable to present the elements of the application of multicomponent thermodynamics to problems of industrial interest.
Gibbs ( l a ) was perhaps the first t o appreciate the broad application of thermodynamic reasoning to multicomponent systems, especially in connection with the understanding of heterogeneous equilibrium. His ideas have been extended and applied to many special problems with success by other workers. Goranson (13) presented an excellent tabulation of pertinent thermodynamic relations and a brief review of the more fundamental parts of the science from a somewhat mathematical viewpoint. Until recently the application of multicomponent thermodynamics in its general sense has been limited for the most part to isobaric conditions. This has resulted from the general interest of chemists in reactions which are carried out a t constant pressure. However, Scatchard (99) and Dodge and Newton ( 7 ) have carried the development forward along more general lines. These workers, together with others, have prepared a satisfactory theoretical and usable experimental background upon which the engineer may build many procedures of practical value. The objective of this discussion is to indicate briefly the thermodynamic approach to the problems which may be treated with such information as is currently available. Heterogeneous Systems
The application of thermodynamics to heterogeneous multicomponent equilibrium appears to offer promise in the solution of certain problems. I n any homogeneous rnulticomponent phase of variable weight the following equation obtains (IO): d E = TCES
- Pdk’ +
k=n pkdmk
k=l
(1)
INDUSTRIAL AND ENGINEERING CHEMISTRY
May, 1942
527
From the first and second laws of thermodynamics it follows for equilibrium in a heterogeneous system consisting of j phases that
In Equation 14 subscripts A and B refer to the system at com-
P-j
D B ) =0
(2)
p=l
(3)
If the auxiliary restraints of constancy of volume and weigh4 are imposed,
positions which correspond to those of the dew and bubble points, respectively, at pressure P . The residual partial specific volumes of methane in the m e t h a n m b u t a n e system for two such compositions a t 100" F. are depicted as a function of pressure in Figure 1. The shaded area between these two curves represents the value of the integral in Equation 13 and illustrates the geometric relation between the volumetric and phase behavior of a component. Equation 14 is general in its application and applies to both binary and multicomponent systems.
0
A combination of Equation 1 applied to a virtual change in state (9) with Equation 2 results in: n= i
d
5
1
-0.01
5
V
p=l
J
p=l
PI
In order that Equations 3 to 6 be simultaneously applicable, it follows from the principles of the variation calculus that it is necessary and sufficient for the following conditions to obtain a t equilibrium : TU) = T ( * ) = T ( i ) P(') = p w = pci)
J$
(7) (8) (9)
=
The foregoing equalities are restricted to cases in which gravitational and surface forces may be neglected. Lewis (17) defined the fugacity as follows: bkT In f x =
+
WL~
I¶
The quantity p in Equation 10 is solely a function of temperature and therefore it follows from Equations 9 and 10 that
The fugacity of a component in any system at equilibrium, whether homogeneous or heterogeneous, is related to the volumetric behavior of that system by the following expression (23):
This equation is based upon rigorous thermodynamic expressions and the application of kinetic probability considerations with regard to the change in entropy with composition a t infinite attenuation. The only physical hypothesis involved is that relating to the existence of the material as an infinitely attenuated gas a t pressures in the neighborhood of zero, From Equations 12 and 13 it follows that the gas-liquid equilibrium constant is related to the partial ( I S ) volumetric behavior of the component by the following expression:
-
0.1 5
250 PRESSURE
500
750
IO00
LB. PER SQ. IN.
Figure I. Residual Partial SpeciFic Volume of M e t h ane i n the Methane-n-Butane System at 100" F.
Figure 2 shows the influence of pressure upon the composition of bubble-point liquid and dew-point gas of the methane-n-butane system a t 100" E'. The compositions of the liquid and gas phases which are a t equilibrium a t 1000 pounds per square inch are indicated by vertical dotted lines. These correspond to the compositions shown in Figure 1. The ratio of the mole fraction of methane in the gas phase to that in the liquid phase a t this pressure is called the "gas-liquid equilibrium constant" which may be evaluated by use of Equation 14 with the information available in Figure l. This relationship is of primary utility in connection with the study of the thermodynamic consistency of data determined in several ways and in illustrating the geometric significance of the relation between volumetric and phase equilibrium data. The equation, in itself, does not permit the direct evaluation of the equilibrium constant from volumetric data without information for all of the components involved, and even in that case an indirect approach must be employed. Several methods of estimating the composition of coexisting phases at equilibrium have been proposed (11, 89, SO). However, in order to illustrate, it appears desirable to plot the fugacity of the component in each phase as a function of its concentration a t a given temperature and pressure. The compositions a t which the fugacities of each component in all of the several phases are equal correspond to the compositions of the coexisting phases. If this information is es-
I
1500
s (L
& w
4
1000
I
I
U w I
yi
~
YI
a w
I
500
1
/ .
ID
-
,
DEW*/
01
Fi 01
00
-
I *I El 01 ,I
r-
02
03
04
05
OB
for both the liquid and gas phases have been extrapolated into the unstable regions where they are indicated by dotted lines. The coexisting phases as determined experimentally are connected by straight lines which are sometimes called "combining lines". It is apparent that the gas phase approaches the behavior ascribed to an ideal solution a t the lower pressures. -4similar diagram for pentane is given in Figure 4; again it is apparent that a t lower pressures the behavior approaches that attributed to an ideal solution. It is of interest to note that the fugacity of pentane in the heterogeneous region is relatively independent of the pressure a t a given tempera-
-
which accounts for the-rectilinear nature of the isobars in Figure 3. Dotted lines have been employed to designate the coexisting phases which would be expected to obtain in the methane-pentane system if it followed the relationship shown in this diagram. The necessary and sufficient conditions for equilibrium are stated by Equations 7 , 8, and 9 and the following relationships : k=n
*CYk = 1
(15)
= 1
(16)
k=l
k=n
Ex. k=l
Geometrically these facts are illustrated by the rectangle of dotted lines. The compositions corresponding to the dew-point and bubble-point lines represent the compositions of the coexisting gas and liquid phases, respectively. To illustrate the behavior of the components of an actual system, the fugacity of methane in the liquid and gas phases of the methane-n-pentane system is presented in Figure 4 as a function of the mole fraction of methane for 280" F. The curvature of the isobaric-isothermal lines shows that the deviations from ideal solutions a t the higher pressures are significant, especially at the lower concentrations of methane. The curves
MOLE FRACTION METHANE
Figure 3.
Fugacity of Methane and n-Pentane in an Ideal Solution dt 280"
F.
May, 1942
I N ~ ~ J S T R I AALN D E N G I N E E R I N G C H E M I S T R Y
529
three components. However, the fugacity of a component is a function of the state of the phase involved; therefore, in the case of a multicomponent system, it is necessary to take into account t h e nature and amount of all of the other components present. The foregoing discussion illustrates i n a general fashion the way in which fugacities may be used in predicting the composition of coexisting phases. However, the prediction of the fugacity of the components of a given system as a function of state requires detailed experimental information upon which may be based a satisfactory tabular, graphical, or analytical generalization. It is to be expected, however, that the Courtesy, Texas Gulf Sulphur Company prediction of the fugacity Jet-Type Heaters for Heating Treated Water Used in Sulfur M i n i n g b y Steam of a component in a phase Condensation will be somewhat simpler than the estimation of an equilibrium constant for a heterogeneous system. The former necessary. I n the latter instance for situations involving a binary system with a nonvolatile component, the partial quantity is dependent only upon the state of a single phase, while the value of the equilibrium constant is a function of enthalpy of the volatile component in the liquid phase may be the properties of both the coexisting phases. calculated by application of the following rigorous expression: Either the fugacity or the chemical potential of a component in a phase is, strictly speaking, a function of the state. Many simplifications of behavior have been proposed, and among them the concept of the ideal solution and the simple Experimental calorimetric measurements were made over theory of nonpolar mixtures (209) are worthy of note. Fura range of temperatures and pressures to determine the enthermore, the Beattie-Bridgeman equation of state (1) has thalpy changes for solution of ethane and propane in a heavy, been applied with reasonable success t o mixtures. The equawater-white oil (4). Although these data are not generally tion of Benedict (9) has been noteworthy in connection with applicable, it is believed that the result may be useful in conthe prediction of the fugacity of pure hydrocarbons. nection with the prediction of the partial enthalpy of these components in the liquid phase of other multicomponent hydrocarbon liquids. The experimental results yielded values Homogeneous Systems of the changes in partial enthalpy of these two paraffin hyI n homogeneous systems the thermodynamic properties drocarbons in going from the gas to the liquid phase. may be completely ascertained from experimental measureThese data were combined with the existing information ments relating to the volumetric behavior of the system and concerning the enthalpy of gaseous ethane (97) and propane to the heat capacity of the components a t infinite attenuation (19, 96) t o yield the partial enthalpy of these components throughout the range of temperatures of interest. It suffices in the liquid phase. All of the enthalpy values obtained in for most engineering purposes to possess a knowledge of the this fashion are based upon a datum value of zero for the gas enthalpy, volume, and entropy of the fluid as a function of phase a t 60" F. and infinite volume. The results of these state. I n the case of gaseous multicomponent systems, reacalculations are recorded in Table I. It is believed that their sonable success in the prediction of these quantities has been use will not often involve uncertainties larger than 3 B. t. u. obtained (8, 8, 14, 26). Apparently these data are adeper pound, when applied to situations comparable to those obqiirtte to permit the thermodynamic properties of gaseous taining in the experimental work upon which the values were hydrocarbon mixtures to be estimated with an accuracy sufbased. ficient for many engineering needs. No pertinent calorimetric data appear to be available conFor the present it appears that the concept of the ideal solucerning the change in enthalpy resulting from the solution of tion may be applied in many cases to the estimation of the methane in a hydrocarbon liquid. However, this quantity enthalpy of hydrocarbon liquids from the enthalpy of the may be estimated from existing volumetric and phase becomponents. This generalization is more nearly true below havior data for mixtures of methane and a nonvolatile oil the critical temperature of the component in question. I n (go), by application of Equation 17. A combination of these such cases isothermal extrapolations to pressures below vapor results for the change in the partial enthalpy of methane as a pressure at the temperature in question may be made with result of solution in a liquid phase with data for the enthalpy good accuracy. I n the case of components in the liquid phase of gaseous methane in the pure state (21) yields the values of at temperatures above their critical temperature, recourse to partial enthalpy recorded in Table I. The results for methane calorimetric measurements or to complete volumetric data is may involve uncertainties of as much as 5 B. t. u. per pound,
I N D U S T R I A L A N D E N G I N E E R I N GC H E M I S T R Y
530
because of difficulty in establishing the change in the bubble-point pressure with temperature with the necessary precision. The results in Table I should be used with caution for systems other than those from which the experimental data were obtained. Their use for other systems entails the assumption of ideal solution, and serious discrepancy from the values given may exist a t states in proximity to the critical state of the mixture in question. I n calculations involving heterogeneous systems, it is imperative that the enthalpy datum for each component be the same for the gas and liquid phases. I n general, the enthalpy of a phase may be calculated from thepartialenthalpyof the components in the followingfashion:
3000
2500
t 54
g
2000
0
p! >
C
I500
y
LL
1000
500
k=n
H
0.0
= x n k H ~
Vol. 34, No. 5
01
0.2
0.3
1
I
I
0.4
0.5
0.6
0.7
0.8
0.9
1.0
(18)
k=l 300
As has been indicated, the enthalpy of some liquid phases may be computed from the enthalpy of the pure components in the same type of phase, and a t the same pressure and temperature upon the assumption of ideal solution. At present data are available relating to the enthalpies of propane (19, d6), butane (28), isobutane (ZW),and pentane (15,2 4 ) , while the enthalpies of the hydrocarbons of higher molecular weight may be estimated from existing generalizations (6,6,31).Although the available experimental work leaves much to be desired, it is sufficient to permit the enthalpies of many liquid and gaseous multicomponent hydrocarbon phases to be predicted with reasonable accuracy.
250
z d cc
2
200
f >
loo/
50
I
1 -
I ,I l I l I I i M
I 0
ai
I
I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.Q
MOLE FRACTION METHANE
Figure
Applications
\
4.
Fugacity of Methane (above) and of Pentane and Gas Phases of Methane-n-Pentane System at
The primary utility of enthalpy i n engineering c a l c u l a t i o n s arises &om its use in connection with the over-all energy balance equation for steady flow. The application of this relation to the steady flow of fluid through a system involving a multiplicity of entrances and exits results in the following expression, when the energy associated with the interfacial area between phases is neglected: r=m r 1
(below) in Liquid 280' F.
If the entrance and exit velocities are low, the kinetic energy terms may be neglected. Since there is no mechanical energy transfer by devices such as agitators or pumps and since the changes in elevation are small, Equation 20 simplifies to the form,
(19) r=l
L
J
To illustrate the application of Equation 19, the partial condenser schematically illustrated in Figure 5 may be employed. If it is assumed that the material is a quaternary system, Equation 19 becomes:
The evaluation of the thermal energy transferred from the surroundings per unit time, which is directly ascertainable from Equation 21, may be made without knowledge of the path of the process or the degree t o which equilibrium between the phases is attained. However, it is necessary that the conditions obtaining within the homogeneous phases at entrance and exit be sufficiently uniform across each section to permit single values of the temperature, pressure, and
I N D U S T R I ,AL A N D E N G I N E E R I N G C H E M I S T R Y
May, 1942
of one-component systems, and data for such systems are widely applied in engineering practice. However, the application of the concepts of multicomponent thermodynamics, especially in connection with the estimation of phase behavior and in the evaluation of work and heat associated with changes in state under both transient and steady-flow conditions, does not appear to be utilized to the same extent. Existing data permit the estimation of the fugacity or the chemical potential of many of the common paraffin hydrocarbons in both gas and liquid phases throughout a wide range of conditions, When additional information of this nature accumulates, it will be possible to estimate the composition of coexisting phases from the equality of the fugacities of each of the components in the gas and liquid phases. Likewise, the enthalpy of both gaseous and liquid hydrocarbon mixtures may be estimated from experimental measurements upon binary mixtures and from application of the behavior of the ideal solution, which apparently applies with reasonable accuracy in this case. The information regarding the enthalpy is of special value in estimating directly the transfers of thermal and mechanical energy associated with changes in state of both homogeneous and heterogeneous systems under steady-flow conditions.
T E R IN
Figure 5.
531
Diagram of a Partial C o n d e n s e r
composition t o be ascribed to it. If desired, similar methods may be employed in distillation column calculation by the step-by-step method without the need of assuming constant latent heat of vaporization and constant heat capacities for the components. It is necessary in the use of Equation 19 that the enthalpy for each component be based upon a single datum.
Acknowledgment
This work constitutes a part of the activities of Research Project No. 37 of the American Petroleum Institute whose financial support has permitted the accumulation of experimental information pertinent to the prediction bf the behavior of hydrocarbon fluids encountered in practice. L. Fay Prescott assisted with many of the calculations reported here.
Summary
The thermodynamic behavior of multicomponent homogeneous systems of constant composition is analogous to that
I
TABLE I. PARTIAL ENTEALPIES OF METHANE, ETHANE, AND PROPANE IN THE LIQUID PHASE 7-100” Pressure Lb./Sq. Ih. Weight Aba. fraction 0
200 400 600 800 1000 1250 1600 1750 2000 2250 2500 2750 3000 0
14.7 20 40 60 80 100 200 300
400
500 600 700
0 14.7 26 50 100 150 200 250 300 860 400
0
b c
0
0.0035 0.0079 0.0106 0.0142 0.0179 0.0227 0.0278 0.0332 0.0392 0.0454 0.0522
.... .... .... .... .... .... .... .... .... .... .... .... .... .... ....
0 0.0190 0.0340 0.0710 0.1640 0.3160
.... .... .... .... ....
F-. Partial
-160’
F
.
7 c-220’
Partial enthalp B.t. u A .
B”pt”’%.
Weight fraotion
-25.6 -29.5 -34.9 -37.8 -42.0 -46.4 -52.0 -58.0 -64.2 -71.0 -78.2 -86.0
Methane4 0 0.0033 0.0073 0,0098 0.0131 0,0166 0.0209 0,0254 0.0301 0.0351 0,0403 0.0459 0.0520 0.0580
-1.0 -3.6 -7.0 -9.8 -12.0 -15.6 -19.0 -23.6 -28.9 -33.0 -38.8 -45.0 -61.0
Ethaneb 0 0.0012 0.0022 0.0046 0.0070 0.0090 0.0120 0.0256 0.0404 0.0580 0.0804 0.1120 0.1516
-126.9 -126.9 -126.9 -126.9 -126.9 -126.8 -126.7 -126.6 -127.1 -127.7 -128.1 -128.4
..... .....
..... ..... ..... .....
..... ..... ..... ..... .....
. . I . .
..*.. ..... ..... -140.6 -140.6 -140.6 -140.8 -141.6 -142.2
..... ..... ..... ..... .....
Propane0 0 0.0100 0.0160 0.0340 0.0740 0.1200 0.1740 0.2410 0.3320 0.4660
....
.....
.....
-106.0 -106.0 -106.0 -105.7 -105.4 -106.2 -106.0 -104.9 -104.8 -104.8
.....
F
.
7
Partial Weight enthalp fraotion B.t. u./iyd. 0
0.0030 0.0067 0.0091 0.0123 0.0158 0.0194 0.0235 0.0277 0.0321 0.0366 0.0413 0.0464 0.0516
... ... ... .... .... .... .... .... .... .... .... .... .... 0 0,0045 0.00s0 0.0170
0.0365 0.0580 0.0810 0.1070 0.1360 0.1770 0.2100
Experimental values based on studies of methane-oryatal oil emtern. Experimental valuar based on studies of ethane-orystal oil system. Experimental values baaed on studies of propane-arystal oil Eystem.
26.0 24.0 21.8 20.0 18.0
16.0 12.4 9.2 6.6 2.5 -3.0 -8.2 -14.2 -21.0
.... .... ....
.... .... ..I.
.... .... .... .... .... .... .... -69.7 -69.7 -69.7 -69.7 -69.7 -69.8 -69.8 -69.9 -69.9 -70.0 -70.0
Nomenclature = specific gas constant ( b = R / M ) E_ = internal energy of a system, B. t. u. f = fugacity, lb./sq. in. = acceleration due to @;ravity, ft./sec.P h = elevation above datum, ft. b
S_ T
B. t. u./lb. partial enthalpy, B. t. u./lb. gas-liquid equilibrium constant ( K = Y/X) = weight, lb. = weight rate of flow (positive for material entering, negative for material leaving), lb./sec. = molecular weight = weight fraction = mole fraction = pressure, lb./sq. in. abs. = rate of thermal transfer to fluid during a steadyflow process = universal as constant = entropy oPa system, B. t. u . / o ~ . = thermodynamic temperature, O R. (” F.
u
= velocity, ft./sec.
€€ = enthalpy,
B K
m
& M TI
n P
4
R
V
= =
+
459.69)
=
specificpolume, cu. ft./lb.
V = volume of a system, cu. ft.
v
= partial specific volume, cu. ft./lb. = residual partial specific volume
P
- T), CU. ft./lb.
(y =
bT/
0
W
rate of mechanical energy transfer from fluid between sections during a steady-flow process X = mole fraction of component in liquid phase Y = mole fraction of component in gas phase 6 = s virtual infinitesimal change of an independent variable p = chemical potential, B. t. u./lb. =
Superscripts Lp) = any phase from 1 t o j infinite attenuation 5
INDUSTRIAL AND ENGINEERING CHEMISTRY
532
Subscripts
Vol. 34, No. 5
(11) Ibid., Vol. I, p. 118.
g = gas phase
IC = any component from 1 t o n 1 = liquid phase T = any section from 1 to n at an entrance or exit in a steadyflow process
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.
(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. ENQ.CHEM.,27,686 (1935). (20) Sage, Baakus, and Laoey, IND. (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. Heat 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
