INDUSTRIAL AND ENGINEERING CHEMISTRY
May, 1942
- 6Ta/T2) - 9RT,(1128P,
or(T,P) =
Substitution of Equation 48 in 46 and integration gives 27 TZ - -- 2.303 log y =
(li8
2T
64
(49)
Acknowledgment
The writer is indebted to M. L. Schwartz for many of the calculations in this paper. Literature Cited (1) Appleby, M. P., Glass, J. V. S., and Horsely, G. F., J. SOC. Chem. Ind., 56, 279T (1937). (2) Aston, J . G., Chem. Rev., 27, 63 (1940). (3) Aston, J. G., Eidinoff, M. L., and Forster, W. S., J . Am. Chem. Soc., 61, 1539 (1939). (4) Aston, J. G., Kennedy, R. M., and Messerly, G. H., Ibid., 63, 2343 (1941). (5) Aston, J. G., Kennedy, R. M.,and Schumann, S. C., Ibid., 62, 2059 (1940). (6) Aston, J. G., and Messerly, G. H., Ibid., 58, 2354 (1936). (7) Ibid., 62, 1917 (1940). (8) Aston, J. G., and Schumann, S. C., unpublished calculations. ENG.CHEM.,29, 19 (1937). (9) Bliss, R. H., and Dodge, B. F., IND. (10) Egan, C. J., and Kemp, J. D., J . Am. Chem. floc., 59, 1264 (1937). (11) Fiock, E. F., Ginnings, D. C., and Holton, W. B., B u r . Standards J. Research, 6, 881 (1931). 12) Giauque. W. F., and Stout, J. W., J . Am. Chem. Soc., 58, 1144 (1936).
521
(13) Gordon, A. R., J . Chem. Phys., 2, 65, 649 (1934). (14) International Critical Tables, Vol. 111, p. 217, New York, McGraw-Hill Book Co., 1928. (16) Ibid., Vol. 111,p. 238. (16) Kassel, L. S., J . Chem. Phys., 4, 493 (1936). (17) Kelley, K. K., J . Am. Chem. Soc., 51, 779 (1929). (18) Kemp, J. D., and Egan, C. J., Ibid., 60, 1521 (1938). (19) Kemp, J. D., and Pitser, K. S., Ibid., 59, 276 (1937). (20) Kennedy, R. M., Sagenkahn, M. L., and Aston, J. G., Ibid., 63, 2267 (1941). (21) Moldavskil, B. L., and Nizovkina, T. V., Compt. rend. acad. S C i . u. R. 8. s., 23, 919 (1939). (22) Montgomery, C. W., McAteer, J. H., and Franke, N. W., J . Am. Chem. SOC.,59, 1768 (1937). (23) Osborne, D. W., Garner, C. S., and Yost, D. M., J . Chent. Phys., 8 , 131 (1940). (24) Parks, G. S., IND. ENQ.CHEM.,29, 845 (1937). (25) Pitzer, K. S., Chem. Rev., 27, 39 (1940). (26) Pitser, K. S., J . Chem. Phys., 5, 469 (1937). (27) Rossini, F. D., B u r . Standards J . Research, 6 , 1 (1931). (28) Ibid., 13, 189 (1934). (29) Rossini, F. D., J . Chem. Phys., 3, 438 (1935). (30) Rossini and Jessup, J . Research Natl. B u r . Standards, 21, 491 (1939); Rossini, Ibid., 22, 407 (1939). (31) Rossini, F. D., and Knowlton, J. W., Ibid., 19, 339 (1937). (32) Sage, B. H., Webster, W. N., and Lacey, W. N., IND.ENG. CHEM.,29, 1188 (1937): Sage, B. H., and Lacev. W. N.. Ibid., 30, 673 (1938). (33) Schumann, S. C., and Aston, J. G., J . Chem. Phys., 6,480 (1938). (34) Ibid., 6, 485 (1938). (35) Stanley, H. M., Youell, J. E., and Dymock, J. B., J . SOC.Chem Ind., 53, 205T (1934). (36) Wilson, E. B., Jr., Chem. Rev., 27, 17 (1940).
Prediction of Critical Constants J. /3,
M Massachusetts Institute of Technology, Cambridge, Mass.
Relations are proposed for predicting critical constants. They may be applied to any substance which is not highly associated and whose structural formula is known. As auxiliary data it is necessary to have either the normal boiling temperature of the substance in question or the liquid density end vapor pressure at some temperature. A method for testing the accuracy of the predicted constants is suggested.
that these methods either are inaccurate, require data which are difficult to obtain, or are not applicable to all types of substances. The more important of the methods will be mentioned below. CRITICAL VOLUME.Young (16) suggested that the critical volume could be computed from the relation,
D, = 3.7 M p J R T , This rule requires knowledge of p , and To. Natanson ( 1 1 ) suggested the relation
D, = 0.044 p o / T ,
T
HE critical temperature, pressure, and volume of a substance must be known before it is possible t o apply rela-
tions such as the reduced graphical equations of state for gases (I,,%') and saturated liquids (9). Methods for predicting the critical constants are of interest not only because experimental data are often lacking, but because decomposition a t lower temperatures may make the experimental determination of the critical point impossible. This paper presents new empirical relations for predicting the critical constants and suggests a method of testing the accuracy of these predicted constants. Previous Work
A considerable amount of work has been published on methods for predicting critical constants. Inspection shows
Sugden ( I S ) proposed an interesting relation,
cv, =
[PI
where C is a constant having a recommended value of 0.78, but which was found to vary from 0.93 for water to 0.635 for octadecane. CRITICALTEMPERATURE. Guldberg (3) pointed out that the ratio of the critical temperature to the normal boiling temperature is a constant. Experimental data show this constant to vary between 1.3 and 1.8, depending on the substance, with a mean value of about 1.5. A number of later investigators proposed modifications of this rule. Lautie (6), for example, related the critical temperature to the normal boiling temperature with a quadratic equation, which un-
522
COSSTANTS OF VARIOUS SUBSTANCES TABLE I. CRITICAL Acetone Acetic acid Acetylene Ammonia Isoamylformate
TB
[PI 161.7 141.2a 88.6 60.7 293.6Q
329 391 189.4 239.6 412
-Experimental-
To
216.5 171 112.5 72.5 407
7-Caloulated--To pe* 51.2 4976 561b 67.2 318b 61.5 406C 110.0 9820 30.0
210 182 116 83 407
pc
VC
594.8 309.2 405.6 576.2
47 57.2 62 111.5 34
508.2
Vc**
Ani 1in e Benzene n-Butane n-Butyric acid Carbon monoxide Carbon tetrachloride Chlorobensene Cyclohexane n-Decane Dodeoane
236.6'' 206.2 190.2Q 216.0 61.6
457 353 272.5 436.3 82.8
699.2 561.7 426.2 628.2 134.2
52.4 47.7 36.0 ? 35.0
? 267 258 292 90.1
709d 563d 409b 607b 134C
47.8 44.1 35 49.4 37.0
316 274 251 288 83.5
222.Oa 244.3a 240.0 424.20 502.0"
349.8 405 353.8 447 489.2
556.3 632.2 554.2 619.4 663.7
45 44.6 40.4 21.2 18.5
275 308 311.5 611 755
660. 6372 564 618b 661b
40.0 41.2 37.2 21.1 18.5
300 330 323 619 753.5
Ethyl alcohol Triethylamine Ethylbutyrate Ethylmercaptan Ethylpropylether
132.2a 351.0 2 9 7 . 8 362.5 293.9 394.3 307.7 162.9 249.2" 334.4
516.3 535.2 568.2 498.7 600.6
63.1 30.0 30.0 54.2 32.1
167 403 421 206 342
5196 532s 563s 500b 502b
66.6 27.5 29.7 51.1 31.9
170.0 411 403 211 335
Diethyl sulfide Ethylene Hydrogenchloride Methane 1-Methylisopropylbenaene
238.4a 99.5 67.8 73.2a
364.6 557.0 169.3 282.9 188 324.6 113 190.7
39.1 50.9 81.6 45.8
323 127.5 86.8 99.0
581: 284 318; 187
37.6 48.8 79.4 43.6
330 129 91 97.5
3800
448.5
28.6
?
665d
25.8
543
3-Methyl-1-butene Methyl alcohol Naphthalene Nitrous oxide Octadeoane
228.2" 93.2" 336.60 81.1 736"
464.5 293 337.5 513.2 491 749.7 183.3 309.7 694.4 763.7
460d 5056 759: 309 769b
32.2 92.6 33.5 65.9 13.5
305 121 479 105 1190
810d 81Od 708d 4486 316C 6126
38.8 28.0 51.1 58.6 60.8 35.5
543 610 296 170 116 363
651
33.9 7 78.7 11: 39.2 71.7 97.Q 1 3 . 8 1160
Stannic ohlori Sulfur trioxide Tetramethylbenzene Trifluorotriohloroethane Toluene Water m-Xylene
103.6
317.6
491.5
83.6
127
5146
85.0
134
380
469
675.5
28.6
?
682d
26.5
543
249.6 246.9 52.2 285.3
322 384 373 410
487.0 593.8 647.0 618.8
34.0 41.8 217.7 35.8
?
315 56.4
487' 6OOd 542b 6444
30.6 38.5 174 38.8
338 334 73 353
7
***Calculated from Equation GA or B. Calculated from Equation 3. a Computed from parachor values of Table 11. b 0
Caloulated from Eouation 5D. Calculated from Equation 5A.
d Calculated from Equation 5C. 6
Vol. 34, No. 5
INDUSTRIAL AND ENGINEERING CHEMISTRY
Calculated from Equation 5B.
ing the surface tension of a liquid to its temperature. For polar liquids k is found to deviate widely from the recommended value of 2.12. Unfortunately k has been found to vary seriously even for normal liquids (IS), depending upon molecular structure and size. For example, the value of k for tristearin is 5.95. For normal substances containing less than twelve carbon atoms, k usually lies between 2 and 2.5. Equation 1 should, therefore, be used with caution even for nonpolar liquids unless k has been evaluated from surface tension measurements. Of the various published methods for predicting the critical temperature, the Watson rule (16) seems most reliable. It is based upon Hildebrand's finding that the molal entropy change of vaporization of normal liquids is constant when in equilibrium with vapor at any fixed concentration. From this, Watson showed that In T. =
98 T
-'T B - 4.2
The critical temperature is then computed from the following relation: T./T, = 0.283
($)"""
The critical temperature can be computed with an accuracy of *2 per cent by means of this equation. Its limitations are that it applies only to nonpolar liquids, and that it requires a knowledge of the liquid density at the boiling point. CRITICALPRESSURE.Partington (fd) suggested a relation between the viscosity and the critical pressure. Mathias ( 8 ) pointed out regularities in the critical pressure of homologous series. Van der Waal (14) showed that, if the reduced vapor pressure curves coincide for all substances, then log P J P = a: ( T J T - 1 )
fortunately gives two alternative values for the critical temperature a t any given boiling temperature. Attempts have been made t o correlate the critical temperature with various other physical properties. Merkel (IO) suggested a relation between the critical temperature and the number of carbon atoms in homologous hydrocarbon series. Guye (4) related the critical temperature to the critical pressure and the molecular refraction. Lewis ('7) showed that the critical temperature is related to the parachor [PI as follows: T , = Kclog [PI - 0
For any given homologous series K , and /3 are constant, but vary from series t o series. For normal paraffin hydrocarbons with K , = 551 and /3 = 832, good agreement between the experimental and computed values of T,is obtained. Another equation relating the critical temperature to the parachor was presented by Lautie (6)
where k is the Ramsey-Shields constant having a value of 2.12 for normal liquids, and T is the temperature a t which the liquid density, D,is measured. This relation is derived from the well-known Eotvos-Ramsey and Shields equation relat-
where a! is about 3. By use of this equation the critical pressure can be computed from the critical temperature if a single point on the vapor pressure curve for the substance in question is known. Unfortunately the reduced vapor pressure curves do not coincide, but diverge so markedly that 01 varies between the limits of 1.5 and 4.0. Proposed Relations
The proposed relations are presented below. Their detailed development is not shown because of space limitations. The equations were developed from suitable graphs of the critical constants of over one hundred substances which differed as widely as possible in molecular weight, p.olarity, and structure; the critical constants themselves varied widely. Table I compares the experimental with the computed values for the critical constants of a representative group of these substances. This table is somewhat unbalanced, in that the compounds showing the greatest deviations were included in preference to those showing good agreement. However, with a few exceptions which will be discussed below, good agreement between the experimental and computed values is found. CRITICAL VOLUME. It was found that the critical volume can be accurately computed from the parachor by the following relation: V , = (0.377 [PI
+ 11.0)'~06
(3)
523
INDUSTRIAL AND ENGINEERING CHEMISTRY
May, 1942
good as it is for benzene. Only in the case of a few simple molecules such as CO, GOz, SO8, etc., is there disagreement, and these “anomalous” cases were discussed by Sugden (IS,page 170). The critical volume mav be comDuted from the Darachor with excellent accuracy by”Equation 3, as inspectionbf Table I makes evident. The only serious exceptions are water and biphenyl. Water is the one substance found which does not obey any of these relations and must be regarded as a special case. The 20 per cent error in the case of biphenyl, however, is surprising, particularly in view of the good subsequent check 207.1 obtained when calculating the critical pressure. I n the case of the simple compounds showing anomalous experimental parachor values, as mentioned above, the critical volume may be computed with good accuracy if the experimental parachor STRUCTURAL PARACHORS, FROM S U G D ~ N is used. (13) CRITICAL TEMPERATUR~. An analysis of the available data Triple bond 46.6 indicates that the critical temperature can be calculated from Double bond 23.2 3-membered ring 16.7 the normal boiling temperature for any polar or nonpolar 4-membered ring 11.6 5-membered ring 8.5 compound by using the following proposed relations, in which 6-membered ring 6.1 all temperatures are in degrees Kelvin: 02 in esters 60.0 For Compounds Boiling below K.:
This equation is perhaps easier to use than is at first evident, since the parachor is an additive property and can be evaluated if the structural formula for the compound in question is known. The parachor equivalents for the common elements and structures are presented in Table 11. From these values the parachor for benzene, for example, is computed 88 follows: Cs: 6 X 4.8 = 28.8 Hs: 6 X 17.1 102.6 3 double bonds: 3 X 23.2 = 69.6 &membered ring = 6.1
TABLB11. ATOMIC AND C
H
N P
0 8
81 si I
4.8 17.1 12.6 37.7 20 48.2 25.7 54 2
68.0 91.0
To = 1.70 T B - 2
This calculated value of 207.1 for the parachor agrees very well with the “experimental” value of 206.2 as computed by Sugden’s basic equation: ,
(4)
At ordinary temperatures, of course, d is negligible in comparison with D. Roughly speaking, then, the parachor may be regarded as a measure of the molal volume of the liquid in question a t a standard surface tension. Inspection of standard tables (6) contrasting the experimental and calculated values of the parachors for various substances shows agreement which is almost invariably as
(5-4)
Equation 5 apparently applies to all elements, regardless of boiling temperature. For Compounds Boiling above ,236’K.: 1. Compounds containing halogens or sulfur:
T, = 1.41 TB + 66
- 11 F
(5B)
where F is the number of fluorine atoms in the molecule. For example, in dichlorodifluoromethane, F is 2. 2. Aromatics and naphthenes (halogen- and sulfur-free) : Tc = 1.41 TB
+ 66 -
T
(0.383 TB - 93)
(5C)
where r is the ratio of the noncyclic carbon atoms to the total number of carbon atoms in the compound. For example,
Courtesy, E. B. Badger Condenser Room a t the Plant of the Calvert Distilling Company
& Sons Company
524
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 34, No. 5
since the errors in computing Vc and T, are retained in Equation 6. Associated Substances and the Elements
Pr Figure
I
isopropylbenzene contains 9 carbon atoms, of which 3 are noncylic; hence r is 0.33. When r is unity, as in the case of a straight-chain hydrocarbon, Equation 5C reduces to Equation 5D. 3. All compounds (halogen- and sulfur-free) other than aromatics and naphthenes: T , = 1.027 T B
+ 159
(5D)
With the exception of water, the maximum deviation between the computed and experimental critical temperatures of the compounds in Table I is about * 5 per cent. It should be noted that these equations have not been tested on substances boiling above 600' K. CRITICAL PRESSURE. The proposed relation for calculating the critical pressure is as follows:
Table I presents values of p., calculated from Equation 6A, using as values of T,and V , those computed from Equations 3 and 5 for the substances in question. With the exception of water, phenol, acetic acid, naphthalene, and methyl alcohol, maximum errors of 11 per cent are evident. It is to be expected that the critical pressure calculation using Equation 6 will be less accurate than the calculation of the critical volume and critical temperature by Equations 3 and 5 , respectively,
Regardless of whether the substance in question i s associated or not, the critical temperature can be computed from Equation 5 without serious error. This is probably to be expected, since no terms involving molecular weight enter into these equations. The value of To for acetic acid, for example, c o m p u t e s t o 561" K. by Equation 5D, which is within 6 per cent of the experimental value of 595' K. The critical temperature for other associated liquids, such as methyl and ethyl alcohols, can be computed with even better accuracy by this equation, as Table I shows. Water is the only s u b s t a n c e f o u n d whose critical temperature as calculated does not agree closely with the experimental value. Obviously, it is necessary t o know the correct structural formula of a substance in order to compute its parachor. However, Equation 3 apparently makes it possible to compute the critical density of an associated compound by assuming the substance to be unassociated. The reported critical density for acetic acid, for example, is 0.351 gram per cc. Assuming the formula for acetic acid to be C€I&O,H, the critical volume computes to 171 cc. per gram mole by Equation 3, which is equivalent to a critical density of 0.351 gram per cc. The same good results are obtained with other associated substances, such as methyl and ethyl alcohols. Water again does not obey Equation 3. The critical pressure for highly associated liquids cannot be accurately obtained from Equation 6. This is to be expected, since Equation 6 is merely a refinement of the wellknown expression (pc&l/DCETc)= po. I n the case of associated substances, pc deviates considerably from its average value of 0.27 when the molecular weight, ill, is assumed to be that of the unassociated molecule. Table I1 shows errors of 20 per cent in t,he predicted critical pressures of acetic acid and methyl alcohol. These errors are probably due to the use of too small a value for M in Equation 68. Water again is the only substance showing really large errors. Data on the critical constants are available for only a few of the elements. It was found that the critical temperature of the elements could be predicted with fair accuracy by Equation 5A, regardless of the boiling point. The critical volume and pressure of the elements were predicted successfully by the relations proposed above except for hydrogen and helium, for which errors as great as 20 per cent were encountered.
May, 1942
INDUSTRIAL A N D ENGINEERING CHEMISTRY Recommended Procedure
The critical constants for either a polar or a nonpolar substance can therefore be computed if its boiling point and structural formula are known. The critical volume and critical temperature should be computed first, and the critical pressure derived from them. The following procedures are recommended: I n predicting the critical volume for either polar or nonpolar substances, the parachor must first be estimated from the structural values presented in Table I. The critical volume is then computed from the parachor by Equation 3. The critical temperature of nonpolar substances probably can be computed most accurately by the Watson Equations 2A and B, which require knowledge of the normal boiling temperature and liquid density a t the boiling point. For both polar and nonpolar substances, the proposed relations 5A to 5D can be used with an accuracy very nearly that of the Watson relation. Their use requires a knowledge of only the normal boiling temperature. The critical pressure can be computed for either a polar or a nonpolar substance from proposed Equation 6, using for V , and Tothe values predicted by the above mentioned methods. Caution must be used in applying this relation to associated substances. Substances will be found for which boiling temperatures are not obtainable. Their critical constants can be computed by Equations 1, 3, and 6 if the liquid density is known at some temperature. However, as pointed out above, Equation l may be used with confidence only if the value of the Ramsey-Shields constant k for the substance in question is known. The critical constants for a substance of unknown boiling point may also be computed by solving Equations 7 and 6 simultaneously, with the help of the relation between pL, p,, and T, shown in Figure 1. This method requires an experimental vapor pressure and liquid density at some known temperature, and is described in some detail below. It may be applied only to such substances which obey the relation shown in Figure 1. Test Procedure
Since the proposed methods of computing the critical constants are largely empirical, it is desirable to test the accuracy of the predicted constants. This can be done if experimental data are available on the vapor pressure and liquid density a t some temperature for the substance in question. The test procedure involves comparison of the experimental density with the density computed from the equation of state for saturated liquids (9):
puted by Equation 1 is 426" K., while Equations 2A and B predict 429" K.) Using these constants, the reduced temperature at the boiling point is 0.618, while the reduced vapor pressure is 0.0235. From Figure 1 pL is found to be 0.0040. Then substituting in Equation 7,
D = - -PM PL
= vapor pressure PI, = a universal function o€p , and T , as shown in Figure 1
(expanded to cover a somewhat larger range than the original PI,chart, 9)
The predicted values of the critical pressure and temperature are used in obtaining the value of pL from Figure 1. If the density computed from Equation 7 checks the experimental density, then the predicted critical constants are probably not far in error. As an example of this method, consider the case of butadiene, whose normal boiling point is -4.5' C., a t which temperature its liquid density is about 0.65 gram per cc. Using the values reported in Table 11, the parachor computes to 168.2, giving a value for V, of 220 cc. per gram mole by Equation 3. From Equations 5 D and 6, the critical temperature and pressure are found to be 435" K. and 42.6 atmospheres, respectively. (The critical temperature com-
RT
- 0.004 x
1 x 54 82.07 x 268.5
0*615grm'cc*
This compares well with the experimental value for the density at -4.5' C. of 0.65 gram per cc. and is evidence that the predicted constants are not far in error. As another example of this method, the critical constants of crotonaldehyde (M = 70) were computed. The normal boiling point is reported as 102.3' C., the vapor pressure a t 20' C. is 30.0 mm. of mercury, and the liquid density is 0.8531 gram per cc. The parachor is found to be 165, the critical volume is 214 cc. per gram mole, the critical temperature is 544" K., and the critical pressure is 55 atmospheres. Using these predicted constants, the liquid density computes to 0.90 gram per cc. a t 20" C. from Equation 7, which is within 10 per cent of the experimental density. (It is interesting to note that T,equals 444' K. by Equation 1, when taking k as 2.12. By Equation 6 pa is then found to be 44 atmospheres. The liquid density predicted from these values is incorrect, indicating that crotonaldehyde probably does not have a normal value of k.) Examination of this test procedure shows that an error of, say, 10 per cent in either pa or T. causes about a 10 per cent error in pL and, therefore, in the predicted density. However, a successful check by this method is not conclusive, for an error in pa may be offset by a compensating error in Towhen finding pL from Figure 1. A successful check should therefore be regarded as evidence that the predicted criticals are not far in error but are not final proof. It is obvious that this test method cannot be applied to highly polar or associated substances, to which Equation 7 does not apply without modification. This test method represents another procedure of finding the critical pressure and temperature for a substance of unknown boiling point, whose vapor pressure and liquid density are known a t some temperature. This would be done as follows: Using the vapor pressure and density data mentioned, pL can be computed from Equation 7. By trial and error the critical temperature and pressure can then be found by solving simultaneously Equation 6 and the relation between pL, p,, and T, presented in Figure 1. As an example, consider the case of ethyl alcohol. At 20" C. the vapor pressure is 43.9 mm. and the density is 0.7893 gram per cc. From Equation 7 pL is found to be 0.00014 €or these conditions. The parachor of ethyl alcohol is 132.2; hence its critical volume computes to 170 cc. Substituting in Equation 6,
(7)
where p
525
p , = 0.128
T,
Upon guessing a critical temperature, this relation fixes the corresponding critical pressure. From these tentative critical constants, the values of pr and T,corresponding to 43.9 mm. vapor pressure and 20" C. are calculated. If the value of pL found on Figure 1 from pr and T,checks the computed value of pL, then the desired values of p , and Tohave been found. From the ethyl alcohol data mentioned, the values of T, and pa are found to be 533" K. and 68 atmospheres, which is fair agreement with the experimental values of 513" C. and 63 atmospheres. Nomenclature d
D
DB
= vapor density, grams/cc. = liquid density, rams/cc. = density of l i q d at normal boiling temperature, grams/
cc.
= 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)
