An Algebraic Representation of Vapor-Liquid- Equilibria WALTER H. PRAHL Durez; Plastics and Chemicals, Inc., North Tonawanda, N. Y . I n his scientific and technical work with distillation problems, the author felt the need for a simple algebraic relation between y, the mole fraction in the vapor phase, and x, that in the liquid phase, of binary systems. An empirical equation was developed that expresses this relation by means of three constants, which can be derived easily from a straight-line plot. Because this equation connects y and x directly by a simple relation, not through some derived function such as a, y , etc., it permits simple algebraic treatment of many calculations that otherwise require a very complicated calculation, or have to be handled graphically. It can be used to smooth out the experimental data in preparation for a thermodynamical treatment. It can replace the usual graphical procedure in the McCabe-Thiele method of determining the number of theoretical plates, by a faster and more accurate arithmetical procedure. It permits accurate intra- and extrapolation of experimental x - y curves, and calculation of azeotropic composition and many other data connected with distillation behavior of binary mixtures.
molar ratios, unfamiliar to most technical men and trailing off into infinity at both ends, instead of mole fractions; and that it does not give a direct and simple relation between x and y. Another empirical relation: a =
( A - Bx)/(x
+ C)(1 - 2c + C X )
was suggested and explained by Kretschmer and Wiebe (6, 6). In this method, various values of C are tried until a straight line results. It has the disadvantages common to trial and error methods. Owing to the drawbacks of these methods, most industrial men prefer the old method of drawing a smooth curve through the z-y points and performing the desired operations graphically on this curve. The obvious disadvantage of this method, familiar to anyone who has ever tried to draw curves through a system of points, indicates the need for a method which is almost as easily handled as the purely graphical method but offers the obvious advantages of an algebraic method. The author has used such a method for many years and feels that a description of his method might fill the gap obviously existing between theoretical methods and practical needs. ALGEBRAIC METHOD
F
OR the industrial chemist looking for a quick and simple application of vapor pressure data for nonideal mixtures to practical problems, such as minimum reflux, plate efficiencies, fractionation economies, etc., the methods suggested so far are sometimes disappointing. Most of them emphasize the theoretical, normally thermodynamic, background, require data sometimes not easily available, tend to simplify the conditions by operating at constant temperature instead of constant pressure normally found in practice, and try to facilitate the mathematical treatment by resorting to derived functions such as a,y, y I / y z , etc., instead of a simple, easily handled, algebraic relation directly connecting x and I/. I t is obvious, on theoretical grounds, that no simple relation between z and y can be applicable over the whole range of every system under all conditions, nor can it aspire to the heights of scientific standards normally applied to theoretically derived methods. It is also obvious that for scientific purposes, especially where the experimental results are to be checked against thermodynamic principles, other methods designed for that purpose are preferable (2). For practical, technical purposes, however, a fast, easily handled, simple equation relating directly x and y, applicable to many systems over the whole range, to all over certain ranges, would be useful in spite of these drawbacks. Methods designed for a similar purpose have been suggested before. Clark (3) suggested one which represents the x-y relationship by t w o hyperbolas: y/(l
- Y)
=
~ x / ( l- X)
(1 - y)/g = ~ ' ( 1- X)/X
+b
The algebraic method uses the empirical relation:
The constants are derived by a simple graphical procedure which gives, in many systems, one straight line covering the whole range, while practically any system can be covered, within the limits of experimental accuracy, by two, or rarely three, lines. The practical procedure is illustrated here by means of the carbon tetrachloride-methanol system a t 35' C., measured by Scatchard (8). TABLEI. CARBONTETRA SZXLORIDE-METHANOL SYSTEM AT 35" c. 2 ~
X
0.0169 0.0189 0.1349 0.3560 0.4776 0.4939 0.6557 0.7912 0,9120
Y
a
x-xi -0,4770 -0.4750 -0.3690 -0.1379 -0,0163
......
$0.1618 +0.2973 -I-0.4181
a - a 1
a
- 2, - ai
27.553 25.373 4.483 0.700 0.060
-0.0173 -0.0187 -0,0918 -0.1970 -0.2717
- 0.4535 - 0.6838 - 0.8194
-0,3568 -0.4348 -0.6103
... .. . .
......
In Table I, columns 1 and 2 give the x and y data, found by Scatchard (8). Column 3 gives the relative volatility for each x-y pair, calculated by the following equation:
+ b'
A great advantage of this method is that it permits the derivation of the constants by means of straight lines in a simple plot. Disadvantages are that it needs two lines for a system; that it uses
The next step is to select one pair of x-y data, through which the x-y curve is to pass. This selection can be based on such considerations as experimental reliability, or location near that
1767
INDUSTRIAL AND ENGINEERING CHEMISTRY
1768
part of the curve for which maximum accuracy is desired; two or more experimental points may lie close to each other, for which an average point of greater reliability than that of a single point with the can be substituted, etc. The z selected is designated zl, corresponding 011. In the example, the z value of 0.4939 was selected as xl, with the corresponding value 1.047 as cyI. z - 2 1 are talThe values z - x l , a - cyL, and their quotient, cy - 011 culated for each experimental point. These values are given in x - XI columns 4, 5, and 6, respectively, of Table I. The values 01 - a1 are then plotted versus z,as shown in Figure 1. If the 01 versus x versus z would curve were a perfect hyperbola, the plot of cy - 01, be a straight lihe. I n the present example that i s the case within the limits of experimental errors. Consequently, the 01 versus x curve can be described by an equation of the form: ~
here can describe them as simply as nonassociating as long as their cy versus x curve can be represented, within the limits of experimental error, by a hyperbola. The acetic acid-benaene system measured a t 760 mm. by Rosanoff ( 7 ) can serve (ts an example. Table I1 gives in columns 1, 2, and 3, the experimental x, y, and a values, respectively. Taking the values of the first line, because of their considerable distance from 0 and from the next x, as x1 and cyll columns 4, 5 . and 6, give the x - xl,
~
cy
=
c (-) A -
x (3)
B + x 0
-
-
-0.1 -
-0.2
I;I
-Os3 I
I"
-0.5 -0.6
-
- _--I_
_L
~
stants found by the method of least squares, to represent this system. His calculated y values are given in column 9, with his differences between experimental and calculated values in column 10 Comparison of columns 8 and 10 shows that the empirical method presented here gives at least a5 good a correlation with three easily and graphical11 derived constants, as the theoretical method with four constants, derived by the time-consuming method of least squares. Attempts to use the Duhem-Margules equation with only three constants did not lead, according to Rosanoff, to satisfactor> results. Another method of representing z-y values, also based on thermodynamics, is the van Laar equation To offer a comparison of the van Laar method with the method presented hereand to show that the latter
i: -
---
I -
The next step is to derive the constants A , B, and C, from the constants of the straight line of Figure 1. The simplest method is to use the slope, m, of the straight line, and its intersection with the '>axis, Y , together with x1 and a t ,by using the equations:
-
011
____
-0.7
a
5 - Xi - a i , and values, respectively. ff -
A plot of the latter x - x1 values versus z shows that they fall on a line ff - cy1 = -2.775 z - 0.8325, from which the values of the constants are found to be: A = 5.520, B = 0.3000, and c = 0.0406. column 7 gives the 1/ values calculated with these constants. while column 8 shows. t,hr differences between experimental and calculated values in units of the fourth decimal. a
-i~--~~~h
-0.4
Vol. 43, No. 8
011
measured by Bogart and Brunjes (1). Table XTT give4 the values, arranged as before. The x value of 0.0160 was selected as zl, without any particulhr reason. The plot shows the points badly scattered and thc drawing of the line through them leavea much to arbitrariness x - 21 The line -= -0.69 x - 0.0176 connects them reasonttblj a
(4)
(5) 1 c = --
m
cy'
In this example m = -0.542 and Y = -0.01 as read from Figure 1. Substitution in Equations 4, 5, and 6 gives A = 1.1662, B = 0.01845, and C = 0.7980. From these values, the a versus x curve could be calculated. The author's interest, however, lies in deriving a direct equation for the y versus x curve. This is done by combining Equation 3 with Equation 2 to give Equation 1.
Figure 2 shows the x-y curve, calculated from Equation I with the numerical values of A , B , and C, as given above. The experimental points are also shown, and the agreement is satisfactory.
- a1
well, leading to the constants: A = 0.147, B = 0.0255, C = 0.348 The y values calculated with these constants are given in colunm 7 of Table 111. The authors report their results by means of the van Laar equation with A = 0.298 and B = 1.525. According to the theory, B is the logarithm of the terminal activity of phenol, corresponding to a value of y = 33.5. The terminal activity of phenol cab also be derived very simplv from A, B, and C, of the preseni method. The slope of the x-y curve, at x = 0, which is equal t o thc bA terminal relative volatility, is - = 2.008. The vapor pressure B of phenol a t 100 O C., calculated from the formula ( 4 ) : (-0.05223) (49644) ~ ~ g ~ o ~ r=n r n . 273.1 t
+
equals 43.39 mm. then
+
8.587
The terminal activity of phenol in watw ili 2.008 = 35.2
APPLICATIONS O F ALGEBRAIC METHOD
While methods based on thermodynamical principles become even more cumbersome when dealing with systems containing components associating in the gas phase, the method presented
This value is in reasonable agreement with Bogart and Brunjes' value (1). The composition of the azeotrope is found from z=y=- to be 0.01907.
c+1
INDUSTRIAL AND ENGINEERING CHEMISTRY
August 1951
1769
Operating lines, tangent to a given point of the curve, to determine minimum reflux conditions etc., are found algebraically. The intersection of the x-y curve with the diagonal, giving the composition of the azeotrope, is found as
Equation 1 is especially valuable in calculating McCabe-Thiele diagrams. The process of going from an x on the operating line to the corresponding y on the curve, and from there to the corresponding x on the operating line, etc., is a simple and exact algebraic procedure carried out by substituting the numerical values of x and y into the equation of the x-y curve and that of the o p erating line, respectively. Contrary to the graphical method, it can be performed with equal accuracy over the whole range, including very high and very low values of x. In addition to these examples, the equation lends itself easily to many other algebraic equations. The method described here was applied to many systems, some taken from the literature, some measured in this laboratory. I n many cases, especially where the x-y curve was measured at constant temperature, or where, when measured a t constant pressure, the temperature changes were not too great, the straight 1
line of the
x - 51
versus z plot extended over the whole range of 2.
a
In many other cases, especially where the temperature varied Carbon Tetrachloride-Methanol System at 35 O C. considerably, the experimental points deviated considerably from the straight line. These deviations were found, in general, to be An attempt to calculate the y points for the experimental z largest toward that end a t which the largest variation of temperapoints with the constants of the van Laar equation and the vapor ture occurred. In such cases the equation given here can be pressure of phenol calculated by the formula abwe, did not lead used over the range over which the experimental points are repto a good result. It seems that the authors used a vapor pressure resented by the straight line. Outside of that range, a different of phenol below that calculated, presumably about 41.5 mm. The straight line, giving a different set of constants for Equation 1 y points, calculated with this value, are given in column 9 of can be drawn. The break in the x-y curve caused by this proTable 111. A comparison of the differences between the expericedure, although objectionable from the theoretical point of view, mental y points and those calculated by the two methods, colis practically always negligible compared with the experimental umns 8 and 10, shows the method presented here to be at least accuracies. equivalent to van Laar's for the present purpose. NOMENCLATURE I n applying this method to an unknown system, it may occur Figure 2.
that the straight line through the
5 2 1
-
= relative volatility = activity coefficient
points is, within the fim-
a1
y
That indicates that, within the limits Z -A of error, the a versus x curve is a straight line, CY In
= mole fraction in vapor phase A , B , C = constants
that case, the equation:
x1
=
LY,
= one numerical value of a,corresponding to x1
its of error, horizontal.
I :
'
z(A =
z(A
- z)
- X)
(7)
+ Y(l - x)
can be used instead of Equation 1 or the constants of the equation CY = ax b can be derived directly from the Q versus x plot. With the help of Equation 1, many operations normally done graphically can be done algebraically. The slope of the curve a t any point is found by differentiation. The terminal slopes, in particular, which are of interest in thermodynamical calculations, are
=
CA -and B
1 = C Y ~ = I
G -+I ) , respectively
e
mole fraction in liquid phase
y
one numerical value of x, selected as basis
m = slope of the
- 21 - a1 versus x line
5
a
SYSTEM AT 760 MM. TABLE 11. ACETICACID-BENZENE
+
os=0
.
Y
a
0.1496 0.2248 0.2579 0.3141 0.3557 0.4224 0.5218 0.6118 0.6851
0.3198 0,2410 0.2134 0.1956 0.1837 0.1741 0.1590 0.1542 0.1505
-
2,
n
a
- al:
0 -0 -0 -0 -0 -0 -0 -0 -0
0788 1064 1242 1361 1457 1608 1656 1693
z
-
Y 21
a
-2:246 - 2 488 - 2 784 - 2 905 - 3 108 - 3 221 - 3 357 -3.428
Prahl's Values Calcd. Diff. 0 1496 0,2229 0.2603 0.3137 0 3562 0.4201 0.5244 0.6121 0 6851
0
- 19
+- 24 4
k2: f26 + 3 0
Rosanoff's Values (7) Calcd. Di5. 0.1497
0,2220
0.2604 0.3136 0.3561 0 4197 0,5239 0 6083 0 6849
+ 1 - 28 +25 - 5
2+21274 -35 - 2
SYSTEM AT 760 MM. TABLE111. PHENOL-WATER z
Y
a
0,0019 0.0056 0.0111 0.0160 0.0198 0.0220
0,0033 0.0089 0,0144 0.0176 0.0195
1.739 1.595 1.302 1.102 0.985 0.907
0.0200
z
-
z G,
-0,0141 -0.0104 -0,0049
., .. ..
fO.0038 +0,.0060
a
-
(11
0.637 0.493
0.200
.....
-0.117 -0.195
a
a - a1 -0.0221 -0.0211 -0.0245
... ...
-0,0325 -0,0308
Y Bogart and Prahl's Values Brunjes' Values ( I ) Calcd. Diff. Calcd. Diff. 0.0035
f2
0.0088
-1
0.0148 0,0176 0,0193 0.0201
-1 0 -2
+1
0.0033 0.0084 0.0140 0.0174 0.0193 0.0202
0 -5 -4 -2 -2
+2
INDUSTRIAL AND ENGINEERING CHEMISTRY
1770
Y
intersection of the
=
x01
-
21 011
x ~
01
-
a1
versus
line with the
axis LITERATURE CITED
(1) Bogart, M.J. P., and Brunjes, A. S., Chem. Eng. Piogress, 44, 95(2)
104 (February 1948). Carlson, H. C., and Colburn, A. P., IND.ENG.CHEM.,34, 581-9 (1942).
Vol. 43, No. 8
(3) Clark, A. hl., Trans. F w a d a y SOC.,41, 718-37 (1945). (4) International Critical Tables, Vol. 3, p. 221, New York, McGramHill Book Co., 1928. (5) Kretschmer, C. B., and \Viebe, R., J . Am. Chem. SOC.,71, 1i93-7 (1949). (6) Ihid., pp. 3176-9. (7) Rosanoff, M. A., and Easley, C. W., Ibid., 31, 953-87 (1909). (8) Scatchard, G . , Wood, S. E., and Mochel, J. M., Ihid., 68, 1960-3 (1946). RECEIVED April 14, 1950.
Rosin-Modified Phenolic Resins P. 0. POWERS1 Battelle Mernorial Institute, Colunabus, Ohio
T
Rosin has proved LO be an effective fluxing agent for HE m a n y d e s i r a b l e It is unfortunate that the phenol-formaldehyde condensates. Recent work has properties of phenol-forname “pimaric acid” was shown that rosin consists of several hydroaromatic acids, maldehyde resins early sugchosen for both the dextro some of which may not react with phenolic resins. Algested their use in drying and levo varieties, for they though direct proof has not yet been given, there is much oils; Baekeland ( 1 )suggested are not geometrical isomers. evidence of chemical combination between the phenolic Dextropimaric acid has been the use of an alcoholic soluresin and the rosin acid. A chroman structure of the tion of a low-stage phenolshown by Harris to differ addition product has been suggested. from abietic acid ( 6 ) in havformaldehyde resin as a proA certain amount of crohs linking between the phenoltective coating. These mateing an unsaturated group in formaldehyde condensate and two or more molecules of rials are used as baking varthe side chain. Isodextropiresin acid is required to achieie the high melting point of nishes to some extent, but maric acid differs only in the the resulting resins. This melting point is also associated their brittleness and dark r e l a t i v e positions of the with the hardness and rapid drying of finishes containing color seriously limit their methyl and vinyl groups (9). these resins. field of application. CresolIt is apparently not known -Methods of preparing various types of phenolic resiiis formaldehyde r e s i n s w e r e whether the dextropimaric are considered in this review, as are the properties of the acids add readily to phenolused with drying oils, particucommercial materials. Methods of incorporating these formaldehyde condensates. larly with tung oil, but resins in various varnishes are also outlined. these resins were also rather When rosin is heated to 250’ 0 . (Figure 2) or above, dark in color. abietic acid is isomerized I n 1916 Berend (2’) suggested the use of rosin to flux phenol-formaldehyde condenoates. to a mixture of acidb, originally termed pyroabietict :wid, The resulting resins were first produced in Germany and known which has been shown Lo cwnsist of dehydro-, dihydro-, :md in the trade as Albertols; about 1926, similar resins were produced tetrahydroabietic acid. The, yeaction is a disproportionation and in the United States. Their introduction made possible thtx can be accelerated by the addition of dehydrogenation aataI\-qts. Because all these acids are much less readily oxidized than h i e t i c “Phour” varnishes. Their use has steadily increased, and in 1949 over 2O,OOO,OOOpounds were consumed here. They have acid, the iesulting products are considerably more stable. The been widely adopted for paints, varnishes, and printing inks. acid is partly decarboxylated to a hydrocarbon. Gum and wood Other natural resins, particularly gum Congo, have also been rosin (9) also contain about 5% each of dehydroabietic and dihyemployed as modifying agents for hard resins. These resins are droabietic acids. It has apparently not been established whether the pyroabietic acids react with phenolic resins, but such acids similar in properties to the rosin-modified matrrialfi hut are lem are seldom, if ever, used in the preparation of commercial resins. widely used. HOSIN ACIDS However, some of the resin acids in the rosin are probably changed to pyroabietic acids during the process of manufacture. The structure of the acids contained in rosin ha6 been greatly The content of individual acids in various commercial romx clarified in the past 20 years, and a fairly accurate estimate of the ill vary somewhat with the hktory of the sample. Oleoresin amount of the various resin acids present in gum and wood ropin is usually contains about 35% levopimaric acid, while the h i s h e d now available (9). Abietic acid is one of the principal acids, and wood or gum rosin contains none. The abietic acid content of is the most readily isolated (Figure 1); the term has often been wood rosin, however, is somewhat higher. I n wood and gum used to signify all rosin acids. Methods for its separation have rosin, the content of maleic-reactive acids is about the aame, or been established. Levopimaric acid has a similar structure and 50 to 55% of the acids prewnt; about 5% of unsaponibble metis readily transformed to abietic acid by heating. Levopimaric ter is also present. It has not been established whether acid8 acid reacts with maleic anhydride at ordinary temperatures, other than abietic acid condense with phenol-formaldehyde conwhereas addition t o abietic acid occurs only on heating above densates. The reactivity of the fatty acids with phenolformalde150’ C. The same adduct is formed by both acids. Abietic acid hyde condensates parallels their reactivity with maleic anhydride. presumably isomerizes to levopimaric structure before addition to Rosin can be polymerized by the use of acid catalysts, such as maleic anhydride occurs. Neoabietic acid has recently been zinc chloride, sulfuric acid, and boron fluoride, to give a dimer of found in rosin by Harris (9-11) in the presenceofacids,it,likelevoabietic acid. The structure of this material has not been estabpimaric, isomerizes to abietic acid. The three acids described lished. However, it contains considerably less unsaturation than here have a conjugated diene structure; the other acids found in rosin itself and might be considered a suitable material for pherosin do not. nolic resins. Apparently it has not been so used to any grcat ex1 Present address, Pennsylvania Industrial Chemical Corg., Clairton, Pa.
