1568
Vol. 45, No. 7
INDUSTRIAL AND ENGINEERING CHEMISTRY LITERATURE ClTED
Bridgman, P. W., Proc. Am. Acad. Arts. Sci., 70, 1 (1935). (2) Dodge, B. F., “Chemical Engineering Thermodynamics,” p. 160, New Pork, McGraw-Hill Book Co., 1944. International Critical Tables, vel. 111, P. 1-4, New York, ArcGraw-Hill Book Co., 1929. Jacyna, V., Z. P h y s i k , 95,246-51 (1935). Leiden Univ. Comm. 2276 (1933); American Inst. Physics, “Temperature, Its Measurement and Control in Science and Industry,” Xew York, Reinhold Publishing Corp., 1941. Michels, 8., and Goudeket, &I., Physica, 8, 347 (1941). (1)
I
(7)
Michels, A , , Nijhoff, G. P., and Gerver, -4. J. J., Ann. P h y s i k , 12,
582 (1932). ( 8 ) Morgen, R. A , and Childs, J. H., IND.ENG.CHEM.,37, 671 (1945).
(9) Newton, R.H., Ibid., 27, 302 (1936). (10) Wiebe, R., Gaddy, V,L., and Heins, C., Jr., J . A m . Chcm. Soc., 53. 1721 11931). (11) Woolley, H.’W., Scott, R. R., and Brickwedde, F. G., J . Rcseawh S a t l . Bur. Standards, 41, 379 (1948). (12) Zlunitsyn, S. A., and Rudenko, N. S., J . Esptl. Theowt. Pltys. ( U . S. S. R.), 16,776 (1946). RECEIVED for review September 30, 1950.
ACCEPTED April 13, 1953.
Binary Vapor-Liquid Equilibria Trial and Error Prediction and Correlation ARTHUR ROSE, E. T. WILLIAMS, W. W. SANDERS, R. L. REIKY, AND J. F. RYAN The Pennsylvania State College, State College, Pa.
I
N ORDER to predict distillation phenomcna and to design distillation equipment, i t is necessary to have information on vapor-liquid equilibrium for the system under consideration. -4great variety of apparatus for obtaining vapor-liquid equilibrium data by direct experiment has been described. It is often difficult t o obtain equilibrium samples from such an apparatus, and t o analyze these samples with the required precision and accuracy. It has become standard practice to test the consistency of vapor-liquid equilibrium data obtained 1% ith such apparatus, by procedures based on thermodynamic equations derived from the Gibbs-Duhem equation. For this purpose, additional information, such as the vapor pressure-temperature relationships of the pure compounds and the boiling point-composition diagram of the mixture, is required., It has long been recognized t h a t the ability to predict the equilibrium data from such additional information only would be of great value. Various methods of prediction for binary systems (1-3, 5 , 6 , 11-13, 16, 16, 19, $3-26) have been proposed, but thesc are usually applicable only when the mixture under consideration behaves in accord with some particular simplified thermodynamic equation. The method proposed in this paper is niorc general. It involves a straightforward trial and error procedure, and makes use of such information as boiling point-composition data, vapor pressure-temperature data on the pure components, and a thermodynamic equation applicable t o the system. For the systems tested, it has been possible to obtain industrially useful results with moderate effort, and excellent results with more effort but with no increase in mathematical complexity. The method is most useful for systems that behave in accord with the van Laar or Margules equations, or equally simple equations. When more complex thermodynamic equations are required, the method is applicable in principle, but excessive computation is required. Fortunately, a great many common systems follow the van Laar, Margules, or similar equations. The following calculation outlines and illustrates the method in ts simplest form. PREDICTION O F ISOBARIC VAPOR-LIQUID EQUILIBRIA OF ETHYL ALCOHOL-WATER AT ATMOSPHERIC PRESSURE
1. Boiling Point-Composition Data from several sources for ethyl alcohol-water at 760 mm. of mercury were plotted, and found to be in close agreement (see Figure 1). The visually
smoothed data of Carey and Lewis ( 4 ) were used for the ralcu1~tion. Vapor pressure-temperature data for ethyl alcohol (8) and water (IO)were also plotted and visually smoothed. If extreme precision had been desired, the smoothing n o d d have been done by the method of least mean squares. 2. Base Point. The boiling temperature ( t = 79.85’ C.) of a 0.50 mole fraction ethyl alcohol solution and the vapor pressures (Pea = 800 mni. of mercury, Po*= 353 mm. of ~ncrrury) of the purc coniponents a t the boiling point were read fiom the graphs.
’“Ob 95
P
90 0’ I
+
85
80
75
O X, = Mol % EtOH in Liquid
Figure 1.
Ethyl Alcohol-Water at 760 Mm. of Mercury Q.
-.
Experimental ( 7 , 1 4 ) Carey a n d Lewis (4)
This particulai liquid composition is referred to below as the “base point.” The purpose of the ensuing procedure (steps 3 to 9 ) is t o find, by trial and error repetition o€ these steps, tjhe “correct” value of the equilibrium vapor composition a t thc base point liquid composition. “Correctness” here implies consistency with the experimental vapor pressure and boiling point data and with the thermodynamic equation that has been chosen as applicable. When the correct equilibrium vapor composition for this base point is established, the other equilibrium vapor compositions are obtained by step 10 of the procedure.
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 1953
3. Method of Calculation. The method of calculation will be illustrated in detail for a first trial value of the vapor composition (mole fraction of ethyl alcohol) a t the base point ( 2 , = 0.50). The choice of this first trial value of the equilibrium vapor composition a t the base point was made by assuming that the activity coefficients are equal a t 2, = 0.5. This makes the relative volatility equal to the ratio of the vapor pressures. Since, in general,
6. Values of the Activity Coefficients for each component a t each check point (designated yo', etc.) were calculated as illustrated here for 2,' = 0.10. log ya' =
c1
A
E]'
+
Yb'
0.50, ya =
yb,
_-
0.4803
(0.4803)(0.10)]* = 0'3892 (7)
[l + (0.4803)(0.90)
z]'
B
=
-
[l =
-
y a t = 2.450
log
it follows that when xa = xb
1569
then
0.4803
1
(0.4803)(0.9OP = 0'00480(8) [l 4-(0.4803)(0.10)
Yb' =
1.010
+.
7. The Total Pressure a t each check point was then calculated, as shown here for xo' = 0.10,
0
4. Activity Coefficients for each component were then calculated as follows:
(3)
(4) log
yo =
log
yb
0.12008
= 0.12008
The equality of yo and Y b holds for only the first trial calculations. On the second and following trials y a f yb. Equations 3 and 4 are applicable when the system is a t modera t e pressure, and Dalton's law may be expected to apply in the vapor phase. 5. The van Laar Equation should satisfactorily express the equilibrium relationships for ethyl alcohol-water a t atmospheric pressure. This choice was made on the basis of the criteria suggested by Carlson and Colburn (6). The values of the constants, A and B , were calculated for the Carlson and Colburn modifications of the van Laar equations, using the trial activity coefficients obtained in the previous step.
Poaxa'Ya'
2I-: 1-
xa log yb[l
f
= 0.12008[1
P,$ = 458 mm. of .mercury. Therefore, Equation 9 gives: (1037)(0.10)(2.450)
+ (458)(0.90)(1.010) = 670.4
8. The Calculated Pressures were compared with the experimental pressures a t each check point by calculating AT = vexpTcalcd. The values obtained are listed in Table I. The sign and magnitude of AT indicate the direction and extent of error in the choice of the first trial ya.
TABLJG I. A T
AND
Arrrn, VALUESFOR ETHYLALCOHOL-WATER AT
0.694
0 670
760 MM. OF MERCCRY 0.665
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Trial y a 0.660
0.665
0.650
0.640
7.0 11.2 10.7 7.9
21.8 17.8 12.6 8.4
54.3
a -w
0'" ..I
-89.6
...
-34.7
-17.4
...
...
1.7
...
...01 . 9
-2.7
1.0
-3.6
...
. .0.
...
-21.9 -3.7 4.8 6.3 0
-1.4 -3.0 -2.4 -4.0 8.06
-8.0 3.8 7.8 7.1 0 -1.6 -4.1 -3.2 -5.1 5.21
0
-1.5 -4.5 -4.0 -5.5 6.8
0
-2.1 -4.3 -5.0 -6.7 11.3
...
16.7 ,
..
0
...
-6.1
-8:l
+ (0.50)(0.12008)1 (5)
= 0.12008[1
*
+ (0.50)(0.12008)1 (0.50)(0.12008)
Ya
B = 0.4803 x
(9)
(0.50)(0.12008)
+
A = 0.4803 B = log
= Tcaicd
= 0.10, t = 86.5" C., Po, = 1037 mm. of mercury and
At '.2
ATTIW
A = log y.[l
+ PobXb'Yb'
(6)
The arithmetic of this step is very simple because 2. = 2 ) and y, = 71,. This simplicity does not continue for repetitions of the calculation, as they involve 7, # i.6. The above equations are written for constant pressure and constant temperature. If, in the prediction of isobaric equilibria, the boiling point range and/or the nature of the components make it probable that the activity coefficients will change appreciably with temperature, the original van Laar equations, which contain a temperature term, may be used. The procedure remains unchanged. The next portion of the first trial computation is concerned with the calculation of the total pressure a t several widely spaced liquid compositions, again employing the van Laar equations, and using the values of constants A and B obtained above. The calculated total pressure a t these several liquid compositions (called ((checkpoints" and designated by the symbol xa') will be 760 mm. if and only if the trial value of vapor composition is actually the equilibrium value corresponding to the base point liquid composition. The check points chosen for calculations in this paper were x.' = 0.10, 0.30, 0.70, and 0.90.
The next phase of the calculation procedure involves the repetition of steps 4 through 8, using a new trial value of vapor composition a t the base point liquid composition xo = 0.5. The new trial value may be obtained from the A r values by an orderly procedure as described in step 9. This gives a rapid approach to the desired correct value 0: y,. These calculations are repeated until successive trial values of ya are within the range of precision warranted by the experimental data, or within the range required for the application of the vapor liquid equilibrium values to distillation design or other purposes. Step 9 deals with the exact manner of choosing the second and following trial values of yo; 10 covers the calculation of final values of ya a t other than the base point, to obtain points on t h e desired vapor-liquid equilibrium composition curve. A 9. Choice of Second and Following Trial Values of y,. quick but approximate method is used as long as it gives a clear indication as to whether the new trial value of y a is to be larger or smaller than the previous value. When this short method is inconclusive, it is necessary to calculate values of ArrmS and t o use these as a guide to successive choices of ya. The discussion of the ethyl alcohol-water system calculation is continued to illustrate these methods. SHORTAPPROXIMATE METHOD.The first trial values of T as given in step 8 and Table I are low a t the low check points and high a t the high check points.
1570
INDUSTRIAL AND ENGINEERING CHEMISTRY
The value of r c a l o d . depends on the quantities in Equation 9. However, a t low check points the value of Koaiod. is dependent chiefly on the value of ya', because a t low values of 5,' the value of yb' is always near unity, regardless of other circumstances. This is a general characteristic of y - zrelations. Thus, Toalod. can be made larger by choosing the new trial ye so that y,' is larger a t low check points. I n order to achieve a larger -ya' a t low check points, it is necessary to choose a new trial value of ya t h a t gives a smaller value of */e' at za = 0.5. This conclusion follows from another general property of y - z relations t h a t causes a lower value of ya' a t 2, = 0.5 to be always associated with higher values of -tal a t low check points. Finally, the desired lower value of a t za = 0.50 may be shown to require a lower value of ya, because a t za = xb = 0.50
Thus, the second trial value for the ethyl alcohol-water calculations was made with ya = 0.67 compared with the first trial value of ya = 0.694. The preceding discussion ignored the high values of Tcsicd. a t high check points. Similar reasoning may be used to show that these also call for the second trial value of y a to be lower than the first. The preceding explanation of the reasoning underlying this short approximate method is lengthy and applies only to cases where and are greater than unity. The actual application of this method is easy and rapid, and may be summarized for all cases by the following rules on which way to go in the choice of new trial values of ya. When -ya and Y b are greater than unity, if values of A T are negative a t low check points and positive a t high check points, the new trial value of ya should be less than the previous trial value, and vice yersa. When y a and yb are less than 1, the preceding requirements are reversed. When both positive and negative values of AT occur a t both low and high check points, and there is no clear-cut trend in these values, or when the A T values are small, the short approximate method must be abandoned in favor of the A r r m s procedure.
Vol. 45, No. 7
values of Table I. This gives the desired vapor-liquid equilibrium composition curve corresponding t o the experimental vapor pressure and boiling point data, and the chosen thermodynamic equation. No use whatever was made of directly determined experimental vapor-liquid composition data. COMPARISON OF PREDICTED AND EXPERIMENTAL RESULTS
For Ethyl Alcohol-Water System. The predicted values for ethyl alcohol-water were compared with the visually smoothed experimental values in Table 11. The root-mean-square difference between the predicted and experimental results was Ayrms = 0.84 mole %. The visually smoothed experimental vapor-liquid equilibrium values listed in Table I1 are those of Carey and Lewis (4). These agree closely n-ith the experimental data of several other investigators ( 7 , 14, 17). For Other Binary Mixtures. I n Table I11 the predicted data for methanol-water ( 1 8 ) , chloroform-ethyl alcohol ( 2 6 ) , and
TABLE11. COXPARISOKOF PREDICTED WITH VISUALLY SMOOTHED EXPERIMENTAL EQCILIBRIUX CTRVE FOR ETHYL ALCOHOL-WATER AT 760 h l x OF MERCURY Visually Smoothed Experimental ( 4 ) XO
0.10 0.20 0.30 0.40 0 . 5 0 (base point) 0.60 0.70 0.80 0.90
Predicted
tJa
71
ya
7
0.441 0.527 0.573 0.614 0.653
760 760 760 760 760
0.439 0.540 0.689 0.627 0.660
752.0 763.8 767.8 767.1 760.0
0.696 0.751 0.817 0.899
760 760 760 760
0.702 0.753 0.817 0.897 0.0084
758. 4 755,Q
Ayrrns
756,5
754.9
TABLE111. COMP.4RISON O F PREDICTED WITH VISUALLY SXOOTHED EXPERIMEXTAL EQUILIBRICM CURVE Visually Smoothed Experimentala Ua
%a
0.10 0.20 0.30 0.40
In the example under discussion, the second trial value of A n = -34.7 (see Table I ) at za = 0.1 was considered as indicative of a trend, and the third trial ya was again made smaller (0.66). -4s the resulting A n values gave no clear trend, the A r r m s procedure was used for choice of succeeding trial values of y a . This depends merely on calculation of ArrrnsPROCEDURE.
0.50
0.60
0.70 0.80 0.90
Methanol-Water a t 0.421 0.586 0.665 0.727 (base point) 0.778 0.825 0.870 0.914 0.956
i7
Predicted Ua
760 Mm. of Mercury 760 0.432 760 0.584 760 0.669 0.730 760 760 760 760 760 760 Agrms
0.781 0.829 0.872 0.916 0. 958 0.00456
7r
773.6 763.0 761.8 760.0 757.6 750.7 746.7 748.3 745.3
Chloroform-Ethyl Alcohol a t 35' C.
0 . 4 0 (base
0.360 0.576 0.686 0.746
144.0 189.9 228.3 258.0
0.364 0.568 0.684 0.750
145.2 189.3 228.5 258.0
0.50 0.60 0.70 0.80 0.90
0.787 0.814 0.836 0.862 0.890
277.1 289.7 299.8 305.0 306.0
0.793 0.816 0.829 0.857 0.890 0.00486
278.0 289. 9 297.2 301.4 303.5
0.10
0.20 0.30
for check points a t xa = 0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 0.8, and 0.9, and choice of successive trial values of ya so as t o minimize
.point)
Anrrna.
The procedure outlined above is repeated for new trial values of the vapor composition chosen as directed. The trial value
AtJrrna
resulting in the lowest A n r m nis thus the value in best thermodynamic agreement with the Po-t and 2-t data. As shown in Table I, the smallest root mean square A T for the example under discussion occurred a t a trial value of 0.660. x Curve. The values of 10. Calculation of Comple,te y partial pressure, P , = Poox,ya, and the calculated total pressure, Kcalcd., are used t o obtain values for ya a t each check point in the best nine-point check by means of the equation
Acetone-Chloroform a t 35.17' C. 0.10
0.20 0.30 0.40 (base point) 0.50
-
0.60
0.70 0.80 0.90
0.058 0.141 0.266 0.410
277.4 262.3 251.6 248.0
0.084 0.143 0.266 0.410
275.1 259.7 250.2 248.0
0.556 0.889 0.796 0.883 0.949
254.7 266.9 283.6 304.1 324.1
0.558 0.693 0.805 0.891 0.955 0.00495
253.1 264.7 281.3 301.2 322.7
Ayrrna
Visually smoothed experimental data for methanol-water (181, for chloroform-ethyl alcohol (sa), and for acetone-chloroform (86). b Predicted data for methanol-water a n d acetone-chloroform from van Laar, for chloroform-ethyl alcohol from Margules. a
For the example under discussion the best nine-point check was with yo = 0.660, as can be seen by comparison of the Airrrns
July 1953
INDUSTRIAL AND ENGINEERING CHEMISTRY
acetone-chloroform ( 2 6 ) are compared with the corresponding visually smoothed experimental results. The calculations were performed as described for ethyl alcohol-water, integral mole per cents only being used for trial values of vapor composition a t the base point liquid composition, with the exception that xa = 0.40 was used for the base point. These calculations were completed before the advantages of using 0.50 for the base point were noted.
1571
TABLEIV. SUMMARY OF NINE-POINTA T CHECKSFOR SYNTHETIC VAN LAARSYSTEM Trial vG at x a 0.50 0.1 0.2 0.3 0.638 7.2 2.5 1.2 0.640 0.2-0.1 0.5 0.642 -6.9-2.3-0.1 I
AT at xa = 0.4 0.5 0.6 1.5 0.1 0.3 0.3 0.0 0.4 3.5 0.0 0.5
0.7 0 8 0.9 0.2-1.6-0.5 0.2 0.6-0.4 0.6 1.3 0.6
A.?rrmB
2.682 0.35 2.74
BASIS FOR COMPARISON
.
The evaluation of the predicted results requires the choice of a basis for comparison in order to obtain values of Ay and Ayrmaas i n Tables I1 and 111. The choice of values for comparison is not simple because of the general lack of close agreement between the experimental data of various investigators, and the variety of methods for smoothing their data, For the present purpose comparisons were made only with visually smoothed experimental data that were in close agreement with the corresponding ‘%hermodynamically smoothed” data. The constant pressure methanol-water data were thermodynamically smoothed in this investigation using the van Laar equation as modified by Carlson and Colburn ( 5 , 9 ) . The isothermal chloroform-ethyl alcohol data had been previously smoothed with the Margules equation ( 5 ) , and the isothermal acetonechloroform data with the van Laar equation ( 5 ) . I n each case the root-mean-square deviation between the thermodynamically smoothed and visually smoothed experimental data ( AyrmB) was less than 0.5 mole %. These visually smoothed data were used for calculating Ay and Ayrmavalues in the tables. ARITHMETIC ACCURACY OF METHOD
Some of the previously proposed prediction methods lead to erroneous results because of the nature of the mathematical operations involved. For example, a loss in significant figures by subtraction may spoil a theoretically sound procedure. It was established that this does not occur for the method proposed herein by the following test. A synthetic set of x-y-t-P values were prepared in such a way as t o be completely consistent with the van Laar equation to three significant figures. The t-x and P-t data were then used to predict the x-y data by the procedure under discussion. Table IV shows the results of the nine-point checks. The final values of yo (see Table V) agree exactly to three significant figures with the yo values of the synthetic van Laar system. COMPARISON WITH OTHER PREDICTION METHODS
Tables VI, VII, and VI11 compare experimental equilibrium curves with the curves predicted by various methods (11). The calculation of Ayrmaindicates the superiority of the present method. CHOICE OF APPROPRIATE THERMODYNAMIC EQUATIONS
The method described herein is not limited to one thermodynamic equation. Carlson and Colburn ( 5 )have indicated how the choice between the Margules and van Laar equations may be made on the basis of molar volumes. In case of doubt, the Scatchard-Hamer (21) or some other less restricted equation should be used. However, the less restricted equations generally require additional experimental data and may greatly complicate the trial and error procedure. USE OF AUTOMATIC COMPUTERS
The method proposed is suitable for calculation on existing automatic computers. With these computers the total pressurecomposition curves may be compared at a great many points, and the calculations may be continued until the experimental total pressure-composition curve has been fitted as closely as the experimental data justify.
COMPARISON OF PREDICTED WITH EXPERIMENTAL EQUJLIBRIUM CURVEOF SYNTHETIC VAN LAARSYSTEM
TABLEV.
[Trial values of yo (at z a = 0.50) carried to 0.2 mole % I Synthetic Predicted
0.10 0.20 0.30 0.40 0.50 (base point) 0.60
0.70 0.80 0.90
Ya
li
Ya
T
0.5337 0.5845 0.6001 0.6162 0.6397
760 760 760 760 760
0.5338 0.5847 0.6000 0.6162 0.6400
760.2 759.9 760.5 760.3 760.0
0.6755 0.7258 0.7924 0.8808
760 760 760 760
0.6753 0.7259 0.7923 0.8809
760.4 760.2 760.6 759.6
OF VARIOUS PREDICTED EQUILIBRIUM VI. COMPARISON CURVESFOR ACETONE-CHLOROFORM AT 35.17” C.
ABLE
v
Vis&y Smoothed Experixa mental 0.1 0.058 0.2 0.140 0.3 0.265 0.4 (base 0.410 point) 0.5 0.555 0.6 0.690 0.7 0.790 0.8 0.883 0.9 0.950 Avrms
Authors 0,0542 0.1432 0.2558 0.4100
y o Predicted Levy Zawidzki 0.059 0.059 0.132 0.151 0,239 0.275 0.380 0.416
0.5583 0.6929 0,8045 0.8911 0.9548 0.00473
0.540 0.680 0.796 0.883 0.949 0.01473
Rosanoff 0.062 0.144 0.260 0.408
0.557 0.690 0.799
0.565 0.704 0.811
0.885
0.888
0.949 0.00541
0.945 0.0157
CORRELATION O F EXPERIMENTAL VAPOR-LIQUID EQUILIBRIUM DATA
I n addition to prediction of x-y data from P-t-x data in cases where no experimental z-y values are available, the prediction method may serve to check the consistency of experimental x-y data. In order to correlate experimental x-y-l values, by any of these previously described processes, it is necessary to derive from the experimental x-y data and auxiliary P-t-x data an average value of the constants A and B of the thermodynamic equation that is t o be used to represent the system. One method in common use (EO) involves plotting certain linear (or nearly linear) functions of the activity coefficients against the ratios of the mole fractions of the two components, and determining the constants for the thermodynamic equation from the values of the intercepts and slopes. This method admittedly (EO) may throw too much weight on the activity coefficients of the component present in least amount a t each end of the composition range. Such values are more likely to be erroneous than middle range values. Therefore, if reasonable agreement between the thermodynamic and experimental equilibrium vapor-liquid curves is not obtained, arbitrary changes are usually made in the weighting of the points. This gives new straight lines with different intercepts, and therefore, different values to the constants of the thermodynamic equation. This;of course, amounts to a rather unsatisfactory type of trial and error procedure, in that the end result depends upon a subjective judgment rather than a definite objective procedure. Another correlation procedure involves the calculation of A and B values for each experimental point, averaging these values, and finally calculating an equilibrium curve using the average A and B and comparing this curve with the experimental one.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1572
TABLE VII.
CO31PdRISON O F \'ARIOUS P R E D I C T E D ~ Q U I L I B R I U , ? f CURVESFOR CARBOSDISULFIDE--ACETONE AT 35.17" C. ua Predicted
Ya.
Visually Smoothed ExperiZa mental .4uthors 0.1 0.342 0,3481 0.2 0,479 0.4871 0.3 0.554 0.5609 0.4 0.596 0.6034 0 . 5 (base 0,625 0,6300 point) 0.651 0.6484 0.6 0.675 0.6668 0.7 0,707 0.6967 0.8 0.7678 0.9 0.776 4vrms 0.00730
Lewis
Levy 0.3386 0.4884 0.5636 0.6994 0.634
Zawideki 0,3277 0.4960 0.5661 0.6065 0.628
0.6471 0.6664 0.6987 0.7691 0.00740
0.6422 0.6608 0.6911 0.7691 0,0122
Beatty and Calingaert 0.336 0.444
and Murphree 0.357 0.491 0.560 0.601 0.632
0.496
0.634 0.565
0.645 0.675 ' 0.716 0.771 0.00828
0.599 0.639 0.686 0.756 0.0433
Vol. 45, No. 7
method, however, he knows that he obtains a therniodynamic curve as consistent as possible with the t-2, T-z, and P-t data, and that the degree of agreement between this curve and the experimental one is a direct measure of the degree of consistency of the experimental data. 4CKNOWLEDGRIEYT
The method proposed in this papei was developed by the graduate students in chemical engineering as a result of a seiic:: of seminars and the extraclass discussion which they inspired. The authors acknowledge their indebtedness in partirular t o John Cusack, Robert Manning, and Stanley Speaker for thcir P H It in the many long discussions 1% hich laid the foundation for this papc'r. NOXI ENC LATURE
TABLE
VIII. COAfPhRISOX O F VARIOUS P R E D I C T E D I
