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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
F1. Cloudy. Reddish brown. Deposit on walls and sediment, both light and not compact nor flaky. Few oily particles with metallic sheen. F2. Cloudy. Reddish brown. Adherent uniform deposit on walls. Light sediment. Several metallic particles. F3. Cloudy. Yellowish brown. Adherent flaky deposit on walls. Adherent sediment definitely granular like El. Numerous metallic particles. F4. Numerous metallic particles. Dark amber cloudy deoosit on walls, heavy but mobile and formless. Granular and amorphous sediment: G1. Slightly cloudy. Yellowish brown. Reddish-brown, adherent deposit on walls. Granular sediment. Numerous metallic particles. G3. Clear reddish brown. Adherent deposit on walls. Flaky and granular sediment. Numerous metallic particles. Y1. Cloudy. Reddish brown. Adherent amorphous deposit on walls. Numerous metallic particles. Heavy granular and amorphous sediment. Y2. Cloudy. Reddish brown. Numerous metallic particles. Li ht flaky deposit on walls. Granular and flaky sediment. %3. Cloudy. Yellowish brown. Adherent granular deposit on walls. Granular sediment. Numerous metallic particles. Y4. Slightly cloudy. Yellowish brown. Numerous metallic particles. Heavy brown flaky deposit on walls. Heavy granular sediment. Z1. Slightly cloudy. Reddish brown. No deposit on walls. Kumerous metallic particles. Mobile flaky sediment. 22. Clear. Reddish brown. Konadherent flaky deposit on walls. Flaky and granular sediment. Numerous metallic particles.
The final results are summarized in Table XI.
Acknowledgment The authors gratefully acknowledge the assistance of J. D. Ponting and L. K. Nobusada in analyzing the samples.
Vol. 33, No. 7
Nontechnical clerical assistance in typing of this paper was furnished by the personnel of the WPA Official Project 8876-B5, District 8, Northern California, The photographs used with this article were provided through the courtesy of the Wine Institute of California.
Literature Cited
__
Assoc. Official Am. - Chem.. Methods of Analvsis. 4th ed... DD. 163-9 (1935). Boutaric, A., Ferrit, L., and Roy, &I., Ann. fals., 30, 196-209 (1937). Dolgov, E. N., Kotin, M. M., and Lel’chuls, S. D., Org. Chem. Ind. (U. S. S. R.), 1, 70-5 (1936). Drews, B., and Illies, R., 2. Spiritusind., 59, 362 (1936). Evelyn, K. A., J . Biol. Chem., 115, 63-75 (1936). Joslyn, M.A., IND. ENG.CHEM., 30, 568-77 (1938). Joslyn, M.A., Wine Reb., 6,No. 4, 16-17 (1938). Joslyn, M. A., and Comar, C. L., IND.EXG.CHEM., Anal. Ed., 10, 364-6 (1938). Kayser, E., and Demolon, A., Compt. rend., 146, 783 (1908); Rev. vit., 31, 6 1 4 (1909). Laborde, J., Ann. inst. Pasteur, 31, 215 (1917). Laborde, J., Rev. vit., 30, 169-73, 200-3, 533-7, 564-9 (1908); 31, 47-51, 178-84, 204-7 (1909); 32, 453-7, 480-4, 561-7, 706-11, 733-41 (1909); 33, 206-11, 238-42 (1910). Ibid., 48, 225-30, 241-4, 385-90 (1918). Mathieu, L., Bull. ussoc, chim. sucr. dist., 22, 891-5, 1283-93 (1904-5). Peynaud, E . , and Maurie, A , , Bull. intern. d u vin, No. 118, 33-8 (1938). RibBreau-Gayon, J., “Contribution 6. l’ktude des oxydations et ritductions dans les vins”, 2nd ed., pp. 1-213, Bordeaux, Delrnas-kditeur. 1933. Roques, X., Rev. wit., 12, 95-9 (1809). I b i d . , 19, 260-1 (1903). Trillat, A., Ann. inst. Pasteur, 22, 704-19, 753-62, 876-95 (1908). Winkler, A. J., and Amerine, A., Food Research,3, 429-47 (1938).
Calculation of Partial Pressures inary Mixtures J
ROBERT RI. LEVY1 Armour Institute of Technology, Chicago, Ill.
N THE past a great variety of equations, mostly of a n
I
empirical nature, have been introduced for the calculation of liquid-vapor equilibrium of binary mixtures. I n all of the methods a small amount of experimental data is necessary. This is not a limitation of these methods, but the manner in which the final result is obtained seriously limits them in their application, since cumbersome graphical or analytical computations are involved in most cases. The author has found a semiempirical expression which closely approximates the experimental results for a variety of mixtures.
The activity coefficients y1and yz of the components of a binary liquid solution are defined as follows:
1
Present address, Ecusta Paper Corporation, Pisgah
(1)
Forest, N.
- 5)
(2)
wherefl, fi
= fugacities of components whose mole fractions are x and (1 - x) f:, fg = fugacities in the standard state
Then, if we may assume that the vapors behave as perfect gases, which is usually the case a t ordinary temperatures and pressures, pl = fl and p : = f!, so that (3)
(4) where p l , pa = partial pressures of components in mixture p : , p ; = vapor pressures of pure components
Improved Method of Calculation
Y1 = fi/SPX
Yn = f&(l
C.
The Duhem equation, which gives an exact relation between the partial pressures of a binary mixture, may be written in the following form, d l d In Y Z
- -0
-2) X
(5)
-
1
A pair of solution functions for this differential equation have been proposed by hfargules (IS). These functions are as follows:
in
y1 =
a2
(1
- s)* + ?(I
-
5)s
+....
(6)
(7) where a, B
=
constants
Some systems given later are accounted for rather well by neglecting all coefficients beyond the first, with the following symmetrical expressions resulting: In
y1 = a(1
In y2
=
- 2)'
(8)
az2
(9)
Equations 8 and 9 are limited t o mixtures whose deviation curves are symmetrical or nearly so. Figure 1illustrates a pair of such curves for the system ethyl iodideethyl acetate. Also Figure 1 is an example of the type of curves (the system acetone-water, 6) which cannot be predicted by Equations 8 and 9. Starting with Equations 8 and 9, we make the assumption that I n y is approximately equal to (y 1); i. e., the solutions do not deviate to a great extent from the ideal. By substituting the values of y from Equations 3 and 4,
-
[gs] -
[-$ (l"t41 -
By substituting the value of a in Equations 10 and 11, and solving for p l and pz,
To obtain the vapor composition, y, we use the following relations:
+ pn - PYX -[pfzp: (1(1 -- =+ p: 2)
(1
2)2
- z)z2]a
-
(17)
Pl = yrr
(12)
pn
=
(1
- l/)n
(18)
I n the case where the solutions deviate to a considerable extent from the ideal solution, we may use an alternate method for calculation of liquid-vapor equilibria. I n this case we use Equations 6 and 7 in the following form:
ETHYL ACETATE -ETHYL IODIDE
a-0- A C E T O N E
Solving for a ,
=
Clearing the fractions and adding,
*
Two simplified methods of calculating the relation between partial vapor pressure and liquid composition of binary liquid mixtures are given. The equations developed depend upon the total pressure or boiling point of the mixture. A number of examples are given illustrating the use and validity of the derived equations, and comparing the results with those obtained by other methods of calculation. The case of two-phase liquid mixtures is discussed briefly.
- z)2
1 = a(l
p1
929
INDUSTRIAL AND ENGINEERING CHEMISTRY
July, 1941
WATER
' 5I-
Iny1 =
(1
-
+ ?(I - z)*
(19)
6 ;
(20)
5:
I-
42 4
3k" 2
These equations were shown to apply to a wide range of mixtures ( W ) , and their application is based on the assumption that the two-term Margules expansion is an adequate solution for the Duhem equation. Margules ( I S ) gives the relation between constants a and p as follows: @a =
I
FIGURE
1. ACTIVITYCOEFFICIENTSFOR THE SYSTEMS ACETONEWATER AND ETHYLACETATE-ETHYL IODIDE
+
Since T,the total pressure, is defined as p1 pp, and the total pressure calculated for a perfect solution [p:z pg(1 - z)], we may write Equation 12 as follows: T
-m
AT = [p:% (1
- z)* f- p $
(1
- Z)zz]a
a2
+ c r ~ ; Pa
5
-aa
I n order t o evaluate the constants, the experimentally determined partial pressure curve of one of the components may be used. Margules (13) devised another method which involves the determination of the tangents to the total pressure curve a t each end. His relations are as follows:
7rs,
+
(13)
where
TO=
[g]
z=o
TI
- [E]
S=l
INDUSTRIAL AND ENGINEERING CHEMISTRY
930
Equations 21 and 22 thus afford a method of determining the values of 01 and @ from the slopes of the total pressure curves a t each end. Zawidzki (21) measured these slopes and based his calculations on the values of the constants so obtained. This method of measuring slopes is subject to error, but the following method can be used t o overcome this difficulty. At infinite dilution, we may w i t e these equations for solvent and solute: p1 = k1x; x = 0
- 2)
p2 = p ; ( l 7r
kin
+ pi(l
p , = pyx;
p , = k2(1 7r
= pyx
- Z)
= 1
2
- z)
+ kn(1 -
2)
Inspection of partial-pressure curves in the literature (14, 17, 21) shows these equations to be true for a small range in composition. Therefore me may solve for TOand T I using Equations 23 through 28:
Vol. 33, No. 7
by Zawidaki (21). The agreement is excellent. The equation of Beatty and Calingaert ( 2 ) , however, does not agree so well with the less ideal system (Table 11). Their equations may be compared with the author’s in the form,
PYX Pi p , = p;(l
-+
(I - X ) A r
- x) + X A T
TABLEI. EQUILIBRIUN DATAFOR MIXTURES OF ETHYL ACETATE AND ETHYL IODIDE(21) AT 49.99’ C.
--
LIole 4 CzHd in Liiuid
Observed
0 5.90 11 48 13.76 19.46 22 I88 30.57 37.46 45.88 54.86 83.40 73.88 82.53 90.98 100.00
0 9.8 17.7 20 9 27.9 31.7 39.0 48.3 52.2 59.7 66.1 74.0 81.5 89.3 100.0
Mole 70 CzHd in VaporCalod. by Calod. by Beatty author and Calingaert
0 10.2 18.0 21.0 28.0 31.7 39.2 43.3 52.4 59.5 66.1 74.3 81.6 89.3 100.0
0 10.2 18.0 20.8 27.8 31.9 39.2 45.1 51.8 58.8 65.3 74.2 81.4 89.5 100.0
TABLE11. EQUILIBRIUM D A T A FOR MIXTURESOF CARBON DISULFIDE AND ACETOXE (81) AT 35.17’ C. Mole
7
Mole % ’ CSs in Liquid 0
Solving Equations 25 and 28 for kl and kz, 7r - p g ( 1 - 2)
ki
X
7.11 18.60 27.60 35.00 40.60 49.30 57.30 66.70 82.80 91.90 100.00
To CS2 in Vapor
Observed
Calcd. by author
0 27.2 46.5 54.0 57.6 59.9 62.4 64.4 66.7 72.1 79.9 100.0
28.0 45.3 52.5 55.0 57.0 59.5 62.6 65.0 71.7 78.6 100.0
Calod. by Lewis and Murphree
Calcd. by Beatty and Calingaert
0 28.7 47.7 54.6 58.4 60.5 63.1 64.1
0 27.7 43.4 48.6 51.4 53.6 56.3 58.8 62.5 70.0 78.0
0
7i:l 78.8 100.0
LOO.0
we find by substitution into Equations 30 and 31,
By measuring two values of ?r a t very small values of 2 and (1 - x), values of oc and p may be calculated from Equations 21 and 22 and the known relation between a! and 0. From these values we may solve for y1 and y2 in Equations 19 and 20 a t all values of x. With this information, values of the partial pressures may then be determined by means of Equations 3 and 4. The accuracy of this method depends only on the precision of measurement in determining the values of R . I n using the above equations for the calculation of isobaric data, one must be careful that the temperature spread is not too great, since in this case values of a and p will not be constant. Equations 8, 9, 19, and 20 can be applied t o the soluble portions of partially miscible systems, provided they follow the assumptions set forth in the derivations of the equations. Hildebrand ( 7 ) illustrates such systems graphically. Other methods of calculating liquid-vapor equilibrium have been reported by various authors, (1-5, 7-18, 1 4 , 15, 16, 19,ZO).
Application of Equations to Calculation of Liquid-Vapor Equilibrium Table I compares the calculated results obtained by the use of Equations 15 and 16 with those experimentally determined
Lewis and Murphree (12) apply an integrated form of the Duhem equation to the calculation of liquid-vapor equilibrium (Table 11). The agreement is good but their method of calculation is tedious. The values in column 3 were obtained by means of Equations 15 and 16. I n this system the deviation curves for each component are not symmetrical; hence the difference between calculated and experimental values is large. Table I11 illustrates the application of Equations 15 and 16 to the calculation of partial pressures. The values obtained by means of Rosanoff’s equation (16) do not agree so well with the experimental data as do those given by the author.
TABLE111.
VAPOR PRESSURE DATA FOR MIXTURES OF ZENE AND CARBON DISULFIDE (18) .4T 25’
c.
BEN-
Partial Pressure CSn, Partial Pressure CeH6, Mm. Hg Mole To Mm. Hg CeHe in Calod. by Calcd. by Calod. by Calcd. by Liquid Observed author Rosanoff Observed author Rosanoff
0 2.11 4.68 11.53 18.24 18.86 30.02 43.88 49.86 57.38 63.02 76.63 87.14 94.19 100 .oo
0 3.5 7.1 16.0 24.0 24.2 35.1 48.2 53.7 59.8 63.5 75.3 83.8 89.5 94.9
0
2.2 6.8 15.9 23.1 24.1 35.1 47.5 52.3 59.5 63.4 74.2 83.3 89.5 94.9
0 2.1 5.7 14.1 21.5 22.4 34.2 47.9 52.9 59.5 64.5 75.0 83.6 89.6 94.9
361.1 352.7 344.1 321.4 299.1 297.9 263.6 218.7 197.4 173.8 156.6 102.5 60.2 27.9 0
361.1 353.8 344.4 321.5 300.0
298.0 263.6 219.4 198.8 174.1 156.7 103.6 60.7 27.9 0
361.1 353.8 344.3 320.6 298.6 296.5 262.3 220.5 201.4 180.1 104.1 113.1 69.7 32 .9 0
July, 1941
INDUSTRIAL AND ENGINEERING CHEMISTRY
931
Table IV compares the results obtained using Equations 15 * and 16 with the two-term solution functions used by Zawidzki (21). It may also be noted that this system shows a wide deviation from the ideal solution. DATAFOR MIXTURESOF CHLOROTABLE IV. VAPORPRESSURE FORM AND ACETONE(81) AT 35.17' C. Mole yo CHCla in Liquid n 5.95 18.35 26.30 36.13 42.40 50.83 57.12 66.22 80.22 91.77 100.00
Partial Pressure CHCla, Mm. H g Calcd. by Calcd by Observed author Zawidaki n 0 0 8.9 9.3 8.8 31.2 32.0 31 . 8 48.5 48.8 50.4 72.6 73.3 73.5 91.2 92.2 89 4 117.7 119.5 115.3 142.7 144.8 139.9 172.2 175.4 169.9 225.1 227.6 224.3 266.8 267.2 266.3 293.1 293.1 293.1
Partial Pressure CaHeO, Mm. Hg Calcd. by Calcd. by Observed author Zawidzkx 344.5 344.5 344.5 322.4 322.9 323.3 275.3 275.8 276.0 242.2 240.6 243.5 198.9 200.3 201.9 170.9 173.7 175.4 137.6 140.3 133.1 104.0 108.5 111.4 74.2 79.1 81.8 34.5 37.9 38.8 12.6 13.5 12.9 0 0 0
FIGURE 2.
Table V gives the results obtained using Equations 19 and 20. The values obtained by the author were found as outlined previously. The values of the constants obtained were The values given by Zawidzki are cyz = 4.78, ag = -3.19. az = 4.82, a8 = -3.12. Thus, the author's method gives values which are in slightly better agreement with the experimental data. Table VI compares the calculated values of Bancroft and Davis (1) with the results calculated by Equations 15 and 16. I n Figure 2 the points represent values obtained by the use of Equations 19 and 20, the constants being determined as described previously. The curve represents the experimental values (6).
EQUILIBRIUM RELATIONS FOR SYSTEMACETONE-WATER
THE
Curve represents experimental d a t a iven by Hausbrand (6);the points were oalculated by Jfquations 19 and 20.
Summary
Two sets of equations for the calculation of liquid-vapor equilibrium have been given. The first set (Equations 15 and 16) applies to a wide range of mixtures which do not deviate too far from the ideal solution and whose deviations are symmetrical. Equations 15 and 16 seem to have a much wider range of applicability than Equations 8 and 9 from which they are derived. The second set of equations (19 and 20) apply to systems whose deviations from the ideal solution are greater and need not be symmetrical. These equations can be used if two special accurately determined points on the total DATAFOR MIXTURESOF CARBON pressure or boiling point curve are known. The case of twoTABLF, V. VAPOR PRESSURE DISULFIDE AND ACETONE(81) AT 35.17' C. phase liquid mixtures is mentioned. Mole yo CSz in Liquid 0 6.24 6.90 12.71 18.57 20.38 28.15 35.26 41.00 45.02 49.53 57.16 61.44 437.13 72.08 82.80 92.16 93.78 95.84 96.92 100.00
Partial Pressure CSs, Mm. Hg Calcd. b y Calcyt. b y Observed author Zawidski 0 0 0 112.9 109.0 110.7 123.4 119.4 121.4 204.7 196.3 199.1 268.5 259.1 258.4 285.0 275.6 277.9 340.7 332.2 326.0 374.1 366.6 359.8 392.8 380.8 386.5 402.3 397.1 392.3 410.3 403.6 406.6 416.7 420.1 419,s 424.6 427.6 422.0 430.1 428.8 437.6 435.7 434.4 447.2 453.2 452.8 464.9 479.0 479.0 490.3 484.9 484.9 492.0 493.2 493.2 498.5 497.7 497.7 502.0 512.3 512.3 512.3
Partial Pressure CaHaO. Mm. Hg Calcd. by Calcd. b y Observed author Zawidski 343.8 343,s 343 8 323.4 323.4 331.0 321.6 328.2 321 5 305.0 310.9 304.8 289.3 295.4 290.0 285.3 287.0 286 1 269.1 274.7 270.5 263.0 257.0 259 .O 253.7 248.9 251.5 243.9 248.9 246.9 242.4 239.0 242 5 231.7 232.4 236.1 226.1 227.8 232.7 217.3 221.7 227.2 214.5 207.4 220.5 184.9 180.2 191.2 120.6 121.8 125.4 103.4 106.4 106.5 79.6 77.5 78.1 62.0 60.1 60.7 0 0 0
DATAFOR MIXTURESOF METHATABLE VI. VAPORPRESSURE NOL AND WATER(1) AT 39.90" C. Mole % MeOH in Liquid 0 5 10 20 30 40 50 60
70 80 90 100
Partial Pressure MeOH, Mrn. Hg Calcd. Calcd. by by Banoroft Observed author and Davis 0 0 0 21.3 20.2 20.6 40.7 38.7 39.8 69.2 74.4 74.3 96.4 101.1 102.4 121.5 124.8 122.8 145.1 143.7 145.9 167.9 165.3 167.7 190.1 189.7 188.0 212.1 210.4 212.0 234.7 235.7 235.0 260.5 260.5 260.5
Partial Pressure Hs0, Mm. Hg Calcd. Calcd. by b Bancroft Observed autgor and Davis 55.0 55.0 56.0 52.4 52.8 52.7 49.8 50.7 50.7 45.2 45.1 46.5 40.6 41.9 42.2 35.7 37.7 40.3 30.5 32.8 32.8 25.1 27.5 27.4 19.3 21.0 21.7 13.2 14.8 12.1 7.8 7.1 6.7 0 0 0
Acknowledgment The author wishes to express his thanks to Harry McCormack and to Bruce Longtin of Armour Institute of Technology for suggestions concerning this work.
Literature Cited (1) Bancroft, W. D., and Davis, H. L., J. P h y s . Chem., 33, 361 (1924). (2) Beatty and Calingaert, IND. ENO.CHEM.,26, 504, 904 (1934). (3) Bose, E., P h y s i k . Z.,8, 353-8 (1908). (4) Brown, J. W., J. SOC.Chem. I n d . , 46, 482T (1927). (5) Dolezalek, F., 2. P h y s . Chem., 64, 727 (1908). (6) Hausbrand, E., "Principles and Practice of Industrial Distillation'', p. 215, New York, John Wiley & Sons, 1926. (7) Hildebrand, J. H., "Solubility of Non-Electrolytes", A. C. S. Monograph, 2nd ed., New York, Reinhold Publishing Corp., 1936. (8) Keyes, D. B., and Hildebrand, J. H., J . Am. Chem. Soc., 39, 2126 (1917). (9) Kritschwesky and Kasamowsky, Z . anorg. allgem. Chem., 218, 44-59 (1934). (10) Laar, J. J. van, 2.physilc. Chem., 72, 723 (1910); 83, 599 (1913). (11) Leslie and Carr, IND. ENO.CHEM.,17, 810 (1925). (12) Lewis, W. K., and Murphree, E. V., J . Am. C h m . Soc., 46, 1 (1924). (13) Margules, Sitzber. Alcad., Wiss. Wzen, 104, 1243 (1895). (14) Marshall, A., J. Chem. SOC.,89, 1350 (1906). (15) Porter, C. W., Trans. Faraday SOC.,16, 336 (1921). (16) Rosanoff, Bacon, and Schultz, J. Am. Chem. Soc., 36, 1993 (1914). (17) Rosanoff and Easley, Ibid., 31, 953-87 (1909). (18) Sameshima, J., Zbid., 40, 1483 (1918). (19) Story, Z. physik. Chem., 71, 129 (1910). (20) Tinker, F., Phil. Mag., 32, 295 (1916). (21) Zawidzki, J., Z. physik. Chem., 35, 129 (1900).
