I
CLINE
BLACK
Shell Development Co., Emeryville, Calif.
Phase Equilibria in Binary and Multicomponent Systems Modified v a n Laar-Type Equation This simplified approach to predicting phase equilibria cuts a clean path through the usual tangle of data and calculations
provide an approximation for binary systems of associating substances. Modifled van LaarType Equation I n early attempts to explain the nonideality of solutions all deviations from the ideal were attributed either to physical interactions between molecules or to molecular association. Today it is well known that physical interaction, molecular association in the pure liquids, and interassociation between unlike molecules are all contributing factors. The characteristic effect of association on the activity coefficient-composition relation is well known. Association of like molecules of a pure liquid increases both the magnitude of the activity coefficient and its slope at low concentrations. Used empirically, the van Laar equations adapt easily to the first but not to the latter effect. Either the association effects must be taken into account on a theoretical basis or they must be obtained empirically by a modification in the equations. Interassociation between unlike molecules in solution also influences the activity’ coefficients. The effect is to decrease the activity coefficient and to reduce, or even reverse, the slope a t low concentrations. Qualitatively, association and interassociation are opposing effects. In some solutions in which both are present, they compensate to such an extent that the van Laar equations still provide an adequate representation of the activity coefficients. However, this is not always the case-usually there is a dominant effect on the shape of the activity coefficient us. composition curve because of association, interassociation, or the over-all influence of both. T o provide simpler relations with a minimum number of coefficients no attempt is made to separate the magnitude of the contributions due to the individual effects of physical interaction, association, and interassociation. I t is sufficient to provide a relation capable
REPRESENTATION
.
of phase equilibria with activity coefficients immediately presents three important problems: effects of vapor phase imperfections on phase equilibria and the influences of liquid composition and temperature on the activity coefficients. I n dealing with the second of these problems, it is desirable to represent the activity coefficient algebraically as a function of liquid composition. T o be of most practical value, the equations should require a minimum number of coefficients, readily derivable from a few simple experimental measurements, and requiring not more than three coefficients for a binary system. Multicomponent equations should make full use of these binary coefficients, which in the ordinary case should also be sufficient to predict multicomponent equilibria. Equations of the,van Laar type (4, 34, 36, 37) and empirical series expansions of the Margules type ( 7 , 4, 22, 23, 27, 29) have been frequently discussed in recent years. Both types have been applied to nonassociating solutions with considerable success. For solutions of associating components the van Laar equations are generally unsatisfactory; they are adequate for only special cases or conditions. Similarly, with two coefficiehts the Margules relations are also inadequate for most associating systems and particularly binary systems in which the activity coefficients for the two components are nonsymmetrical. I n such cases the van Laar-type equation is usually superior. However, the Margules-type equations have the advantage that more terms can be taken to improve the representation. Often four coefficients are required to
of reproducing the over-all influence of these factors only on the shape of the curve for the activity coefficient, or for the excess free energy, us. composition. The combined magnitude of the several contributing effects is determined empirically from the data. The excess free energy F E per mole has been defined by Scatchard (28) as the excess above the ideal value per mole. The partial molal excess free energy for a component i, in terms of the activity coefficient y z , is
Ffr = RT In y z
(1)
The quantity Q is defined as Q = FE/2.3 RT
(2)
Then, with the aid of Equation 1,
4
= log Y*
(3)
The thermodynamic relation for a partial molal quantity furnishes, in view of Equation 3, log y P = Q
+ bQ/bx%- Z x , b Q / b x ,
(4)
An empirical function Qe is superimposed on the van Laar-type relation Qu so that
Q
=
Qv
f
Qe
(5)
The function Q e is an additive correction to the van Laar function, Qu, to account for the influence of association, interassociation, or both, only on the shape of Q us. composition. The main contribution of association or interassociation to the magnitude of Q is included in the function Q. Indeed. for equimolal mixtures Q v represents the combined magnitude of all contributing factors. For component z at infinite dilution in liquid j the common logarithm of the activity coefficient is denoted by A,,, the contribution of the empirical term by c,, and the contribution of the van Laartype term, also a t infinite dilution, by ai, =
A,, -
cb,
(6)
The subscripts of c are interchangeable, as c,, = c , ~ . VQL. 50,
NO. 3
MARCH 1958
403
Further, a molecular interaction coefficient hii is defined by
in which q, is an unspecified molecular property of component i. For this quantity the denotation of Wohl (36) is used. I n van Laar's original derivations this quantity is identified with the covolume b,. In the treatments of Scatchard and Hildebrand q1 is identified with the molal volume, Vi. Langmuir relates the quantity to molecular surfaces and cross sections. The expression q, is made an empirical quantity, and the interaction coefficient is made independent of the order of the subscripts-namely, h:, = h i
(8)
Again, the nomenclature of Wohl is used in defining z, to be 2%=
x,q,/q
(9)
in which q is defined as q = ?x74,
a:,/a:,
= d q ,
(11)
which leads to the following relation between these ratios in a ternary system RI* = R12R23
a$. Rj, x ~ x , / ~ xR,2 ;
log yi = aQ,,/axi
+ bQe/bxi - 34.
(15)
The indicated operations are performed in Equation 15, and with the aid of Equations 13 and 14 the logarithm of the activity coefficient is obtained according to Equations 16-18. The denotation j k g indicates that the variables j and k assume every value except j = k while RM indicates that R and
iM assume every value except S,including R = iM,which need not be excluded, as it becomes zero. The logarithm of the activity coefficient is given according to log yi = qi [ Z hzz:
(4
(12)
Having defined the quantities used in subsequent relations, we return to Equation 5. The first term, Q,, is represented with the aid of the van Laar-type relation,
0.52
The quantities Q , and Q e are homogeneous functions of x of the first and fourth degrees, respectively. With the aid of Euler's theorem, ZxidQ,/dxj = Q u and ZxibQ,/dxi = 44,. Substituting Equation 5 into 4 and remembering the above relations we get
#
($0)
The ratios R,, are defined according to R,, =
the respective classes is denoted by X, and by X A f . The second term of Equation 5, Q., is represented with the aid of the empirical function
+
+ 0.5 Z 3k#
-
&)I,
zjzk
+ Es,
(16)
where
For reasons made known later, components are distinguished according to respective classes. A homologous series is considered to constitute a class. Now n classes of substances 1, 2, 3, 4, . . . n are considered, each of which may be made up of a number of componentse.g., class 1 may consist of components l a , l b , IC, I d , etc. The general class variables, R and M , distinguish any two classes, while r and m are variables denoting any component within classes R and M , respectively. Let S refer to a particular class, and as usual i, j , and k are general variables which can represent any component of any class. Compositions x, and x, refer to the mole fractions of the individual components in classes R and M , respectively, and the summation of compositions within
404
( n / 2 ) ( n - 1) (Minimum)
Special Conditions a n d Restricted Relation. The original equations were derived by van Laar with the aid of van der Waals equation of state. Based on these relations, the interaction constant ht. is related to the van der Waals constants according to =
(bi/qi)(ap.s/bi -
~p,'/b,)~/2.T 3 R (19 )
This shows a split of the interaction constant into terms characteristic of each component. Alternatively, critical constants can be substituted for the van der Waals constants in Equation 19. Scratchard and Hildebrand split the interaction constant according to
(13)
qi/q2.
(n' - 1) (Maximum)
Usually all the binary systems, made up from the n-components, do not require a third coefficient cij. Accordingly, the number of coefficients required to represent an n-component system is usually considerably less than the maximum indicated above. The minimum number of coefficients is obtained when all c i i s are zero and the binaries are each symmetrical. This is expressed according to
h$
j
Each of the running variables i and i assumes every value from 1 to n in an n-component system. The case where i = j is not excluded as h$ is zero. As h:i = h,2, each term is counted twice, thus the coefficient 0.5. Similarly, u:< is zero and a$.Rjz = u:RGz and Riz =
bon fractions or oils in which a number of different classes are present each of which contains numerous members of the same homologous series. The coefficients of Equation 18, the aij)s, the aid's, and the cii)s, are the coefficients of the individual binary systems. Equation 12 reduces the maximum number of independent coefficients for a ternary system from nine to eight. The minimum number is three. Accordingly, the maximum number of independent coefficients for an n-component system is
h:
Equations 16 to 18 are general expressions for the activity coefficient of component i in a multicomponent solution. Equation 18 furnishes the more practical nomenclature. If the c coefficients in the function Esi are zero the equations reduce to the van Laar-type relations. Such equations have been presented by Wohl (36). If each component of a multicomponent mixture represents a different class of substances. mole fraction:. X s , XR, and X , reduce to values x 8 , x,, and x,, respectively. If a class consists of a "fraction" made u p from many members of the same homologous series, this class can be represented by an arbitrary number of suitable chosen components without changing the value of the E function calculated for members of the other classes in the mixture. This is important in multicomponent mixtures of many components, such as hydrocar-
INDUSTRIAL AND ENGINEERING CHEMISTRY
(Si - 62)'/2.3RT
(20)
in which the characteristic properties are called "solubility parameters." These are approximated in terms of the cohesive energy density. The characteristic properties need not be interpreted in terms of the cohesive energy density or the van der Waals coefficients. They can be assumed to be empirical coefficients and written as hR = ( h i - hj)'
(21
Thus, hi and hi are unspecified properties of the pure components. The sign of h t j is determined by the relative magnitude of the property h for components i and j . If hi < h , the sign of hij is negative; if hi > h j the sign of hij is positive. The components are numbered such that an increase in h itself also means an increase in the subscript i. Thus, hij is always negative and h,i is always positive if j > i. Similarly, if the positive root is always taken for q p . 5 , Equation 7 leads to
PHASE EQUILIBRIA a,,
< 0 (negative) and a,;
and to moderate pressures in general, is
> 0 (positive)
(22)
For three components the split interaction coefficient furnishes the restriction hia = hiz
+ hza +
(h?l
+ h:k - h,%) 2(h,
=
- h,)(h, - hk)
= 2h,,h,1,
(24)
This is the new coefficient for the cross term. Because the term within the brackets of Equation 16 is now a perfect square, the equation simplifies to log Y $ =
q1
[?h,?zjl2
+ Ea.
(25)
in which Es, is expressed as before. With the aid of Equations 7, 9, and 10, qt, h,,, and z , are eliminated from Equation 25 to get log Yz =
[L: a,,
R?2
x,/q.lc
I
Rkz12
+ Es
(26)
Similarly, Equation 23 becomes ala =
a12
- a23Ry;S
(27)
Equations 26 and 18 are identical for binary systems. For three or more components, Equation 26 is the special case of the more general Equation 18 which results if Equations 27 as well as 12 are satisfied. If the G , ~ ’ sare each equal to zero, Equation 26 reduces to a special van Laar-type relation. The equation proposed by White (34) is of the same general form except that the coefficients are different and they include a temperature factor. The maximum number of independent coefficients required in order to apply Equation 26 to an n-component system is (n/2)(n
+ 3) - 2
(Maximum)
The minimum number when all c,,’s are zero and the binaries are all symmetrical is (n
-
+
in which
(23)
Returning to Equation 16 the coefficient of the cross term (h% h;& /fk) is examined in view of the split interaction coefficients. Substituting Equation 21 into the expression,
1 ) (Minimum)
Equation 26 has important advantages over Equation 18 when it can be safely applied in multicomponent calculations,
=
(bi
- V,‘ - att;/RT)/2.3RT
GP, = (a,Fp)’3.5
Vapor Phase Imperfections and Effect of Pressure on Vapor-Liquid Equilibria. The effects of vapor imperfection and of pressure on the liquid phase are combined in the “imperfectionpressure’: coefficient Oiwhich is calculated according to a method reported elsewhere (2). The equation, applicable to low pressure differences (P - Pp)
- (a,:;)
&PI
0.5
and
-
= (a,~:)0.5 -
(a,EP)o.6
Because 7, G& and 6; depend only upon the temperature and the properties of the pure substances, they can be plotted as functions of temperature. Such plots conveniently provide values a t any temperature. If the differences denoted by G$ and (7; are small and insignificant relative to the other terms, the composition terms can be neglected. The molal volumes of the liquid Vi can be estimated at any temperature from the volume a t one temperature with the aid of Watson’s method (33). Usually a density is available a t some temperature. The liquid volumes of hydrqcarbons are readily estimated from the method of Kurtz and Lipkin (72) or Kurtz and Sankin (73).
Correlation of Binary Vapor-Liquid Equilibria. Complete vapor-liquid equilibria in terms of the liquid composition x, the vapor composition Y,, the total pressure, P, and the temperature, T, involve an over determination. The pressure-temperature relationship a t liquid composition xi and the vapor composition a t the corresponding x2 and T , or P, provides a double check on the consistency of the data. The two are related through the activity coefficient [log 7, = log ( Y z P / x , P f ) log e,] according to
+
P
= ZY,
X$
Pi“/&
(29)
and alz = y1 PP ez/yz P:
el
= ylxz/xl~z
(30)
in which 0112 denotes the volatility of component 1 relative to component 2. An adequate correlation of the activity coefficients must simultaneously satisfy both Equations 29 and 30. I t has been found most advantageous to examine experimentally derived activity coefficients Y,
Phase Equilibria in Binary Systems
+
loge, = (P - p:)[T, ( Z Y GW ~ 2.3R2T2 ( Z Y , &,)2/2.3R2T2] ( 2 8 )
= Y,Pe,/xi Pp
(31 1
on the two plots: Plot 1. (log
y1)O.S
us. (log y1)0.5
and Plot 2.
log (yl/y>) us.
x1
Plot 1 is new and to the author’s knowledge has never been proposed before in the literature. The second, Plot 2, was proposed earlier (8, 27) and has been used extensively in recent years.
For isothermal data, consistency of the experimentally derived values on Plot 1 is an indication of high quality in the total pressure measurements. For isobaric data, consistency on this plot is an indication of consistency in the boiling-point measurements. The second plot involves only the x - Y - T data. The two plots are applied later (Figures 3 to 10) to correlate experimental data. For isothermal conditions, the GibbsDuhem relation requires (8, 27) that
J1
(log
2)
dxl = 0
(32)
This area condition must be satisfied in Plot 2 if the data are thermodynamically consistent. Scattering of the experimentally derived values in this plot indicates inaccuracies in the compositions which may be reflected from the analytical procedures. Consistency of data in Plot 2 with simultaneous irregularities in Plot 1 reflects the quality of the temperature-pressure measurements. For isobaric data, the area condition still applies if the temperature range of the data is small enough or if the effect of temperature on the activity coefficients can be neglected. The activity coefficients are given according to Equations 18 or 26. The compositions are eliminated by simply transforming the equations to furnish a relation between the activity coefficients for the two components-namely, .(log yl
- E1)0.5 = a 12 (a1z/az1)(log yz - EPP.5 ( 3 3 )
As E1
= E? = zero a t XI = x ? = 0.5, Equation 33 gives the straight line tangent to the curve through the data in Plot 1 a t the point where x = 0.5. If cij is zero, E1 and E2 are zero over the whole composition range, and Equation 33 predicts a linear relation for (log y1)O.b us. (log yz)OJ. In the general case where cij is not equal to zero, the value of cij is readily estimated by drawing in the straight line tangent at x = 0.5 and extending it to intersect each coordinate in the Plot 1. The difference between the square of each intercept and the corresponding value for the intercept of a curve drawn through the data furnishes an approximation for c12 and 621. Because c12 = CZI, the slope of the straight-line tangent is adjusted until this equality is satisfied. The consistency of the results is checked on both Plots 1 and 2. The value of 612 so obtained is used to calculate the values for and ug, from the experimental data according to
VOL. 50, NO. 3
MARCH 1958
405
Table I. Values for Function E / c in Binary Systems" E/c
3:
0.00 0.01 0.02 0.03 0.04
1.00000 0.90287 0.81135 0.72525 0.64438 0.56858 0.05 0.06 0.49764 0.07 0.43141 0.08 0.36971 0.09 0.31236 0.10 0.25920 0.11 0.21006 0.12 0.16479 0.13 0.12322 0.14 0.08520 0.15 0.05058 0.16 0.01919 0.16667 0.00000 0.17 - 0.00909 0.18 -0.03443 -0.05695 0.19 -0.07680 0.20 -0.09411 0.21 0.22 - 0.10903 -0.12166 0.23 -0.13715 0.24 0.25 - 0.14063
E/c
- 0.14063 -0,14719 -0.15198 -0,15511 - 0.15667 - 0.15680 -0.15559 -0.15315 - 0.14957 -0,14497 -0.13943 -0.13304 -0.12590 -0.11809 - 0.10970 - 0.10080 - 0.09148 -0.08181 - 0.07187 -0.06172 -0.05143 -0,04106 - 0.03067 -0.02033 - 0.01009 0 * 00000 Nj)(l
-
Tz)
and Oil
z
X
0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50
0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75
Elc
3:
0.00000 0.00989 0.01954 0.02889 0.03792 0.04658 0.05483 0.06264 0.07000 0.07686 0.08320 0.08901 0.09426 0.09895 0.10306 0.10658 0.10950 0.11182 0.11354 0.11467 0.11520 0.11515 0.11453 0.11334 0.11162 0.10938
0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
E/c 0.10938 0.10663 0.10341 0.09974 0.09566 0.09120 0.08639 0.08129 0.07591 0.07033 0.06458 0.05871 0.05278 0.04684 0.04096 0.03520 0.02962 0.02430 0.01930 0.01470 0.01058 0.00701 0.00408 0.00187 0.00048 0.00000
+ 23:jl. Based on Equations 18 or 26
=
(log y
2 -
E?)[1
+
Y1
(log
y1
-
(log
*/I
- E1)P.'
= 1/01?
+
E1!/x2 (log ";n - E ? ) ] * (351
Unreliable experimental data points, as judged from their consistencies on the two plots, are eliminated completely or given less weight in the final selection of values for & and &. If necessary c12 can be adjusted higher or lower than the value obtained in the first approximation to eliminate trends in the derived coefficients with composition. T o assist in calculating the function E, the values for E/c for each 0.01 unit in mole fraction are given in Table I. I t is usually advisable to draw also the 45' line in Plot 1 and find its inrersection nith the curve through the experimental data. This point represents the value of (log y l ) o . j and (log y2)oJ for the composition at which log (yljyz) is zero. This composition is determined from a curve through the data in Plot 2. Having already estimated a value for 612, these data are used in Equations 34 and 35 to furnish the corresponding values for and a&. Based on Equations 18 or 26, the function log(y,ly,) is given by log ( Y d Y a ) = Q:,a~,[ai,Xi -
f122x:]/
+ 0 i , x 2 ] ~ + EI- Ea
(36) A third correlation plot which is especially useful if one component is relatively nonvolatile is [Q?zXi
Plot 3. (log yi)-".j
us. X i / ( l
-
Xi)
Such plots have been used before (34) to derive the coefficients for the van Laar equations which give straight lines in these plots.
which is a straight line through the data at the points xJ(1 - x,) = 0 . 2 and x, (1 - x,) = 1 in Plot 3. The third coefficient c12 is calculated from the intersection of this straight line with the axis at ~ / ( l- x,) = 0 and the intercept of the curve through the data at the same value of xt,'(l - 2 % ) . If the data curve intersects the axis below the straight line intercept, the sign of CIP is positive. If the data intercept is above the straight line intercept the sign is negative. Thus, from exact total pressure measurements in a system with one volatile and one relatively nonvolatile component, all three coefficients can be estimated. Usually a second or third approximation is required. This is done by first estimating the activity coefficient for the volatile component 1 from the relation =
(P - y 2 x 2 p ~ / e 2 ) / ( x 1 P ~ j e l(38) )
assuming 7 2 to be unity. After a first approximation of the coefficients a&, u:~, and c12 has been obtained, as described above, a second approximation for y? is obtained using these coefficients in Equation 18 or 26. The procedure is repeated until no significant changes result upon reoeating the calculation. Influence of Pressure on Azeotropic Composition. The relation between The coefficients of Equation 18 or 26 and the azeotropic composition is derived from the condition log
( Y ~ / - /= ~ 10s ) ~ (P," ~ el/P:
&Ini(39)
The right-hand side of the equation is evaluated with the aid of Equation 36 to give
For a given value of the ratio ( c l ~ , ~ ' a ~ l ) the left-hand side of the equation can be expressed in terms of x 1 and R I ~ . Figure 1 is a graphical representation with (c12/u;.) equal to zero. Similarly, other graphical representations can be developed with fixed values for the ratio (c12/'&). Figure 1 show^ how the value of R12 influences the azeotropic composition X I . If c12 = 0 and R I Z= 1, the azeotropic composition is a linear function of the left-hand side of Equation 40-namely)
Examination of Equation 40 shows what change should be expected in the azeotropic composition as the total pressure changes. An increase in iota1 pressure implies a corresponding increase in the temperature. The implications of Equation 40 can be examined with respect to increasing temperature. The changes in u;., R12, ( c ~ ~ / ' a ~ and ,). in the ratio (PiO1/P$9,) brought about as temperature increases must be known in order to predict the change in azeotropic composition with increasing pressure. The coefficients may be available either empirically from data at several temperatures or from correlations. The change in the ratio P!&) as temperature increases may be either reinforced or opposed by the corresponding changes in the coefficients &, R I ? , and 612. Usually R I Z moves closer to unity as the temperature increases. For nonassociating solutions decreases as the temperature is increased, but this is not ahvays the case in associating solutions a t loiv temperatures. Partial Miscibility. Algebraic expressions for the activity coefficient, such as Equation 26, furnish a continuous representation over the entire composition range. In cases of partial miscibility, it is important to predict the mutual solubilities and to describe the equilibria over both the miscible and partially miscible regions. Figure 2 shows the result of incorrectly calculating vapor-liquid equilibria beyond the proper solubility limits and into the partially miscible region. Here the vapor-liquid equilibria have been calculated with the aid of equations for the activity coefficient ignoring the solubility limits, X I = 0.0474 and y l = 0.705. The dashed lines represent the fictitious values obtained in the partially miscible region. The vapor-liquid equilibria are completely determined by
,PHASE EQUILIBRIA 1.0
0.9
0. a
0.7
X
0. 6
+--REGION I
I
I
I
-
I
0. 6 yI MOLE FRACTION COMPONENT 1
0.2
*l
0.5
OF PARTIAL MISCIBILITY __I
,
0.4
1.0
0.8
Figure 2. Significance of partial miscibility in calculating va por-liquid equilibria
0.4
4 Figure 1.
Azeotropic composition as a function of
[log (PPz/P%)l /aZl For systems forming minimum boiling azeotropes more volatile component and R = afa/azl
0.3
is 1
For systems forming maximum boiling azeotropes less volatile component is 1 and R = ag1/af2
calculations in the miscible regions up to the solubility limits.
is related to the total composition according to
Calculating Relative Volatilities in Partially Miscible Region. The relative
or,i =
volatility in the region of two liquid phases depends on the activity coefficients corresponding to the equi1ibriu.m solubilities, and on the total composition or the ratio of phases. A mixture of a polar and a nonpolar liquid which separates into two liquid phases can be considered. The solubility of the nonpolar substance in the polar phase is denoted by x, the solubility of the polar substance in the nonpolar phase by y , and the corresponding activity coefficients by y and I’, respectively. The relative volatility for components i and j
(%)
Xi
(42)
The relative volatility becomes (rzP$3,/
r,P,”&)at the composition corresponding to
100% polar phase
and
to
rjP:0,) a t 100% nonpolar phase.
(l?&%,/
procedure is to plot y1 x1 us. x1 and y1 us. ~ 2 x 2 . If partial miscibility occurs, the curve in the second plot will intersect itself. This point of intersection gives the activity, 71x1 in one phase and Fly1 in the other phase a t which separation occurs. Locating this value of the activity ~ 1 x on 1 the first plot gives the mutual solubilities x1 andy,. Scatchard (26) has discussed such a method as well as the earlier methods of Gibhs ( 6 ) and of Seltz (32). Calculating Coefficients for Equations 18 or 26 from Mutual Solubilities. Mutual solubility data combined with a knowledge of the coefficient cI2 are sufficient data to provide values of the coefficients a& and ai1. For the equiXI
Predicting Binary Mutual Solubilities from Equations for Activity Coefficients. The mutual solubilities can be predicted from the equations once the coefficients are known. A graphical
Figure 4.
0.1 2
Propylene ( 1 )-propane (2)
Temperoture = 37.78’ C. 0. Data from ( 2 0 ) a:, = 0.0155, ail
0 ,
0.08
-
0.04
-
-.
= 0.0197
s
4
v
I
0 0
0.04
0.08
0.12
0.16
(log V2)O.S
Figure 3.
Propylene (1)-propane (2)
Temperature = 37.76’ C. 0. Data from ( 2 0 ) a,: = O.O155,a~, = 0.0197
-0.01
-
-.
-0 0 2 0
0 2
O b
0.4
0 8
1 0
XI
VOL. 50, NO. 3
MARCH 1958
407
Figure 5. Ethanol (1)-2,2,4-trimethylpentane (2) Temperature = 50' C. 0. Data from (10) Tangent according to Equation 33
- - - -. -.
a:2 =
0.990,
= 0.870,c12 = 0.270
-1.2
i
CL
.
l
0.2
Figure 6.
-2
[log (YllXl) E1
- tll/[(l
+
+
(1 and Z = [log (gl/xd
(z) z]
RizX1/Xz)-2
+ +
-
RlZYl/Y2)-21
E1
(43)
(44)
- ell/
[log ( X Z / J Z )- (E2 - en)] (45) The symbol c, denotes the function E in the more polar phase, while E , denotes the corresponding quantity in the less polar phase, according to Equation 17. If c12 is equal to zero, Equations 43, 44, and 45 reduce to the van Laar relations given earlier by Carlson and Colburn (4). Applications. Several examples are given to show the application of The equations and methods to the correlation of activity coefficients. PROPYLENE(1)-PROPAUE(2). The binary propylene-propane is a system in which no effects exist due to association or interassociation. Activity coefficients were calculated from the experimental data of Reamer and Sage (20) with the aid of Equation 31. The data are correlated in Figures 3 and 4 which correspond to plots No. 1 and No. 2, respectively. The pressure-temperature measurements appear to be of higher quality than the x - Y determinations. Accordingly, Figure 4 which reflects the quality of the x - Y data, has been given less weight than Figure 3 in the
408
I
0. 6
Temperature = 50' C. 0. Data from ( 7 0 ) , a : , = 0.990,ail =
librium condition That the activity of a component is the same in each phase, the following is obtained with the aid of Equation 26
=
l
0.4
correlation of the data, illustrating the importance of using both plots No. 1 and No. 2 in the correlation of even the simplest of systems. Taken alone, several different sets of coefficients which would not be consistent with the data in Figure 3 might adequately represent the scattered points in Figure 4. Used together the arbitrariness is reduced significantly. Thermodynamically consistent vapor-liquid equilibria are calculated with the coefficients a:* = 0.0155, aE1 = 0.0197, and c1z = 0 E T H A N 0L ( 1) -2.2,4 -TR IME T HY LP E N TANE(~). In the binary ethanol-2,2,4trimethylpentane, ethanol is highly associated in the liquid, and no significant effects exist because of interassociation between the unlike components. .4ctivity coefficients a t 50" C. were calculated from the good experimental data of Kretschmer, Nowakowska, and Wiebe (10). These results are plotted in Figures 5 and 6. The dashed line in Figure 5 is the straight line tangent through the data at x = 0.5, according to Equation 33. From the intercepts of this line with the axes and the corresponding intercepts for a consistent line through the data, the third coefficient c12 = 0.27 is estimated. Using this value of c12 in Equations 34 and 35 the coefficients u& and u& are calculated from the experimental data points. The points a t concentrations below XI = 0.05 and above x1 = 0.95 were excluded because inaccuracies in these points make the area condition in Figure 6 slightly in error. The value of log 71 at y1 = y2 was also taken from the 45' line in Figure 5 at the concentration
INDUSTRIAL AND ENGINEERING CHEMISTRY
-
,
0 .'8
,
1 .! 0
Ethanol (1)-2,2,4-trimethylpentane (2)
-
a:,
,
0.870,C I Z = 0.270
= 0.478 in Figure 6, and these values were also used to calculate a:z and a&. .4n average set of values: a:, = 0.990, a i l = 0.870, and c12 = 0.270 was selected to represent the binary according to Equation 26. These results are shown as full lines in Figures 5 and 6. The system ethanol-2,2,4-trimethylpentane is extremely nonideal in the liquid phase, yet it remains completely miscible over the whole concentration range at 50" C. An attempt to correlate this system with a two-constant equation for the activity coefficients, gives not only a very poor representation of the data but it incorrectly predicts two liquid phases as well. Equation 26 furnishes a reasonably good representation of the system with coefficients given above. METHANOL(I)-BENZENE(2). The binary methanol-benzene is a system in which one component, methanol, is highly associated in the pure liquid and in which a moderately weak interassociation exists (9) between the unlike components, methanol and benzene. The high quality data of Scatchard (29, 37) for this system at 55" C. have been used. The activity coefficients derived from these data are plotted in Figures 7 and 8. The full lines are the correlated results, according to Equation 26, using the coefficients: = 0.920, a i l = 0.710, and c12 = 0.120 Even though some interassociation exists in the system, the dominant effect on the y I us. xi relation is that of the association of the methanol. The representation by means of Equation 26 is quite satisfactory. XI
PHASE EQUILIBRIA 1.2
0. a
0.4
-
-1
-r_
0
-0.4
-0.8
Figure 7. ‘ Methanol ( 1 )-benzene (2) I
Temperature = 55’ C. 0. Data from (31) A. Data from (29) a:* = 0.920, a : , = 0.710,
-.
c12
CHLOROFORM(1)-ETHANOL(2). Strong interassociation due to hydrogen bonding exists between the unlike molecules in mixtures of chloroform and ethanol. In addition, the ethanol molecules are highly associated in the liquid. Activity coefficients have been calculated from the very consistent data of Scatchard and Raymond (28) at 55’ C. They are plotted as points in Figures 9 and 10 where they are compared with the correlated results shown as full lines. In Figure 9, negative values for log 7 2 are taken equal to zero, and the corresponding marks are given on the axis. The dashed line in Figure 9 is the straight line tangent, according to Equation 33, used to estimate the value
= 0.12
Figure 8.
0. a
0. b
0 4
0 2
I
Methanol ( 1)-benzene (2)
Temperature = 55’ C. 0. Data from (31) A. Data from (29) a:a = 0.990,ail = 0.870, el2 = 0.120
-.
of the third coefficient c12. The coefficients, = 0.3121, agl = 0.7184, and c12 = -0.053 when substituted into Equation 26, provide a good representation of the data. ACETONE(l)-wATER(2). Both association and interassociation exist in solutions of acetone and water. Experimental vapor-liquid equilibria are available from several sources (74, 76-78) for this binary system. The activity coefficients calculated from the data a t 760 mm. of mercury were correlated in the usual way. Because the data points from the several sources scatter
significantly a t low concentrations of acetone, the small influence of increasing temperature is not readily apparent. The influence of temperature on the coefficients is known from correlations a t several pressures. Only slight improvement is obtained by including the temperature influence, so only the result obtained with the aid of a single set of coefficients a t 55’ to 60’ C. is compared with the experimental data in Figure 11. The “imperfection-pressure coefficients” 01 and 0 2 were calculated from data .for acetone and water given in another paper (Z), and the ratios Pp/O1 and PzO/Oz were plotted as a function of temperature. The boiling points corresponding to total pressures of 760 mm.
Figure 9. Chloroform (1)-ethanol (2) Temperature = 55’ C. 0. Data from (28) Tangent according to Equation 33 a?, = 0.3121, agl = 0.7184, C I Z = -0.053
- - - -.
-.
I 0
,
I
I
I
0 2
0 4
0 6
0 8
Figure 10. 0 0
I
I
0 2
0 4
(1% Y Z P
’
0.6
0 8
Chloroform (1)-ethanol (2)
Temperature = 55’ C. 0. Data from (28) a?2 = 0.3121,
-.
ail
= 0.7184,
VOL. 50, NO. 3
c12
= -0.053
MARCH 1958
409
Table II.
Coefficients for Typical Binaries" Range of % Deviation, Calcd. from Exptl.
t,
c.
Component(1)
Component(2)
37.78 50.0
Propane Ethanol
55.0 55.0 55.0
Methanol Chloroform Methanol
55.0
Carbon tetrachloride Acetone
Propylene 2,2,4-Trimethylpentane Benzene Ethanol Carbon tetrachloride Benzene
O
Aiz 0.0155 1.260
0.0197 1.140
0 0.270
1.040 0.2591 1.146
0.830 0.6654 0.911
0.120 -0.053 0.150
0.0440
0.0444
55 to Water 1 ..050 60 65.5 Water Furfurali 0.940 a a,; = A12 - el* and ail = An1 - e21 for Equations 18 or 26.
A21
Y'S
e12
d Y 2
0 to 0.6 9 pts. 0 to *3b
10 pts. 0 to *2c 23 pts. 0 to *3 10 pts. 0 to Zt3.5f
0 to 1 1 . 3 9 pts. 0 to 1 6 *
13 pts. 0 to 3 ~ 4 ~ 22 pts. 0 to 1 4 e 10 pts. 0 to *7.7'
. .a ... ...
0
...r
I
0.760
0.100
1.669
0.180
... .
h
h
* *f
j
Excludes points below 51 = 0.05 and above zi = 0.96. c Four points with substantially higher deviations, one positive and three negative all at low methanol concentrations. Excludes one point which deviates by about 10%. One point deviates by approximately 5%. 1 Five points exceed this. 0 Estimated Assumed E' = 0.026 from data at 40° arid 50' C. and constant pressure data 76" to 80' C. *' Deviations not estimated--see Figure 11. [see (2) for critical constants and E' values]. j Based on total pressure data. of mercury were obtained by trial calculations. vlrATER(1)-FURFURAL(2). Total pressure data for water-furfural mixtures a t 65.56' C. have been reported (79) bv Pearce and Gerster. Mutual solubilities are available from several sources (5, 7, 15, 25). These data are used to derive the activity coefficients and also the coefficients in Equation 26 for the binary system. Because the solutions involve both association of like molecules and interassociation of the unlike ones, it is expected that three ciefficients will be required. One procedure for deriving the coefficients consists of the following steps: First, with c12 equal to zero, the mutual solubilities are substituted into Equations 43, 44, and 45, and preliminary values for & and uil are calculated. These are used in Equation 26 to calculate activity coefficients and a total pressure at any concentration below X I = 0.10 and above X I = 0.90. If the calculated total pressure is less than the experimental, ci2 has a positive value. If it is higher than the experimental, c12 is negative. Second, using an estimated value for c12, new values for a& and are calculated from Equations 43. 44, and 45. With the aid of these coefficients and
Equation 26, the activity coefficients for water in the aqueous phase and for furfural in the furfural phase are calculated a t liquid compositions corresponding to the total pressure data. These are substituted into Equation 38, and its corresponding expression for component 2 and the activity coefficients are derived from the total pressure data. Finally, the resulting activity coefficients are plotted in the usual correlation plots, and the three coefficients a&, u&, and c 1 ~ are approximated. If these differ significantly from the values used in the second step, the second step is repeated until no significant change results upon going through the final step. Usually two approximations are sufficient. The predicted vapor curves and the calculated total pressures are shown in Figure 12 where the pressures are compared with the original experimental data. Coefficients for the sample binary systems discussed above are summarized in Table 11. Comparison of Modified van Laar E q u a t i o n with Margules Power Series. Systems in which the activity coefficients for the two components are very nonsymmetrical are usually represented better by the van Laar equations than b>
the hfarqules power series with the same number of coefficients. This is readily apparent from the results of Wilson and Simons (35) on the system isopropyl alcohol-water. Here the van Laar equation, which corresponds to Equations 18 or 26 with c12 equal to zero, represents the system as well as the Margules power series (22) with three coefficients. Similarly, the system ethanol-2,2,4-trimethylpentane is represented as well with Equations 18 or 26 with the three coefficients given in Table I1 as with the Margules power series with four coefficients. A few systems are compared in Table 111. Phase Equilibria in Multicomponent Systems Full use should be made of binary data in the calculation of phase equilibria in multicomponent systems. From a practical point of view the calculation method has its greatest value for systems in which no coefficients depending on ternary or multicomponent data are required. The calculation method loses its main advantages if the data required for its application become excessive. Accordingly, the equations derived in the first section do not include terms depending on coefficients other than
1
I
Figure 1 1 . Acetone (1)-water (2)
I 1
d 0.10
Pressure = 760 mm. of mercury 0. Data from (17) A. Data from ( 1 4 ) Data from (16, 1 8 ) Calculated with Equation 26: a:, = 0.950,
0
0. -.
u i 1 = 0.660,c12
/
C
0
Figure 12.
0,100
3. 6
0.2
0.8
1.0
Water (1)-furfural (2)
Temperature = 65.56' C.
0. Data from (79) n
x
41 0
INDUSTRIAL AND ENGINEERING CHEMISTRY
7
1 ,
r
8
c -
,
Calculated with Equation 26: a:,
1.489, ci2
0 180
=
0.760,aEl =
P H A S E EQUILIBRIA those derived from binary data. The applications shown here also correspond to these limitations. If experimental data are available for the binaries comprising the components of the mixture the data are correlated, and the coefficients for each binary are established. The components are numbered in increasing order of h, or alternatively, to satisfy the sign conditions in view of Equation 27. Usually the most polar substance has the highest number and the least polar substance the lowest. Several sets of binary coefficients are examined to determine how closely each of the three combinations of binaries satisfies Equations 12 and 27. If both relations are satisfied within the accuracy of the data for the binaries, Equation 26 can be used. If only the first relation is adequately satisfied, equation 18 is tobeused. If neither relation is satisfied, either the system cannot be calculated from the binary coefficients alone, or some of the binary data are not sufficiently well established. Combinations of three binaries in which two of the three substances are very similar-e.g., two saturated hydrocarbons, two alcohols, two ketones, two members of a homologous series-ordinarily satisfy both Equations 12 and 27. Accordingly, Equation 26 is applicable to such ternaries. In the separation of multicomponent hydrocarbon mixtures with a solvent in separation processes such as extractive distillation and extraction, this leads to a considerable reduction in the number of data required to calculate the multicomponent systems. Vapor-Liquid Equilibria. Application of the equations to multicomponent calculations is illustrated below with the ternary system carbon tetrachloridebenzene-methanol. Excellent vaporliquid equilibria are available from the work of Scatchard and Ticknor (29) for this system. In addition, the corresponding data a t 55" C. have been reported also for carbon tetrachloridemethanol (Zg, 37) and for benzenemethanol (29, 3 7 ) . Data for the third binary a t temperatures above and below 55' C. are alqo available (3, 24, 29, 30, 38). The binary system benzene-methanol has been correlated in an earlier section (Figures 7 and 8). The other binaries were correlated in a similar manner. The resulting coefficients are = 0,761 a i l = 0.996 = 0.120 c13 = 0.150
= 0.04400 a i 8 = 0.710
t,
Component
50
Ethanol
c.
0
(1)
= 0
623
These coefficients satisfy Equation 12 but do not adequately satisfy Equation 27. Accordingly, Equation 18 is used
'
2,2,4-Trimethylpentanea 35 Ethanol Methylcyclohexaneb 80 to Isopropyl WaterC 100 'alcohol a
0.990
0.870
0.270 0.950
-0.044
0.174
-0.100
1.020
0.875
0.185
0.920 . - 0 . 1 0 4
0.162
-0.075
1.000
0.483
0
0.647
0.076
-0.206
0
Reference (IO). Reference [ I I , 231. Reference (36). Table IV. Calculated Vapor-Liquid Equilibria for Carbon Tetrachloride(1)-Benzene(2)-Methano1(3)" 51
28
0.1960 0.3961 0.3963 0.5922 0.3230 0.2134 0.1115 0.0814
0.1880 0.1983 0.1982 0.1945 0.3590 0.5557 0.7515 0.8433
log 7 2
log Y l 0.0619 0.0660 0.0644 0.0606 0.1501 0.2912 0.4942 0.6147
0.0550 0.0675 0.0658 0.0689 0.1493 0.2787 0.4533 0.5700
lo@;Y3 0.5423 0.5465 0.5329 0.5497 0.3181 0.1536 0.0533 0.0231
Yl 0.131 0.253 0.257 0.371 0.245 0.223 0.189 0.188
Ys 0.516 0.521 0.513 0.511 0.545 0.574 0.627 0.676
P, atm. 0.8708 0.9214 0.9044 0.9293 0.9426 0.9496 0.9321 0.9033
0.005 -0.014 -0.010 0 * 000 -0.007 -0,006 -0.007 -0.008
a Temperature, 5 5 O C.; 71 = -0.C2049, 72 = -0.01993, 73 = -0.0227, Py = 0.49605, Pi = 0.43039, Pg = 0.67921; calculated with the aid of Equation 18; data from (8s).
to calculate the activity coefficients in the ternary mixtures. Calculated results are compared with experimentally derived total pressures in Table IV. Similarly, vapor compositions predicted from binary data are compared with the corresponding experimental values in Figure 13. Liquid-Liquid Equilibria. Calculation of liquid-liquid equilibria in ternary and multicomponent systems can be done in a manner similar to that for the binaries. The principles involved in calculating a ternary system are described in Figure 14. I n the triangular diagram, lines of constant activity for component 2 , ~ ~ or x zI'2y2, are derived from Equation 26 over the region of the ternary system
where the solubility curve is expected to be. (If the c,ts are not equal to zero this involves a trial calculation.) Activities y,xi's or T2y2's, are calculated for components.1 and 3 at compositions along the lines of constant activity for component 2. The resulting activities are plotted as indicated in Figure 14, center. From this, lines of constant activity for the x and y phases furnish values of y3 and x 3 corresponding to given values for a3 and for al. A plot of these values, y 3 cs. x3 as indicated in the lower graph of Figure 14, furnishes 2
a,
CONST.
LINE a2 CONST.
1
3
r; 0 . 4 0.3
0.2
0. I
0
0 1
0 2
0.3
0.4
0 5
0 b
0.7
YEXPT
= 0.04444 ai2 = 0.920 GI2
Table 111. Comparison with Margules Power Series Component Equations 18 or 26 Margules Power Series (2) UZl c12 B C D E
Figure 13. Calculated vs. experimental vapor compositions for carbon tetrachloride-benzene-methanol
o, carbon +e+rachloride 0. A. Benzene
f.
Me+hanol 55' c.
x3
Figure 14. Liquid-liquid equilibria in ternary systems VOL. 50, NO. 3
MARCH 1958
41 1
literature Cited Table V. Calculated Liquid-Liquid Equilibria [Hypothetical system of paraffinic hydrocarbon(l), aromatic hydrocarhon(2), and polar solvent(3)] Solvent Phase Hydrocarbon Phase 51
52
0.0348 0.0394 0.0425 0.0459 0.0512 0.0569 0.0669 0.0789
0.00 0.0600 0.1000 0.1403 0.2005 0.2610 0.3536 0.4412
xa 0.9652 0.9006 0.8575 0.8138 0.7483 0.6821 0.5795 0.4799
two curves the intersection of which gives the equilibrium concentrations for component 3 in the two liquid phases. Locating these points on the lines of constant activity for component 2 in the triangular plot a t the top of Figure 14 gives the tie line and two points on the solubility curve. The ternary liquid-liquid equilibria are given for an hypothetical system consisting of a paraffinic hydrocarbon (I), an aromatic hydrocarbon(2), and a polar solvent(3). The equilibria were calculated with the aid of Equation 26 using the coefficients af2 =
0.175
a i l = 0.114 c12 = 0
= 0.3774 ai;, = 0.7355 628 = 0 a:,
= 1.592 azl = 1.966 6 1 3 = 0.200 a:,
Ul 0.9873 0,8825 0.8140 0,7472 0.6508 0.5583 0.4281 0.3195
qi
R Rij
T
x, x, x8
X, X,v
X, xi
Xi
The results are summarized in Table V. Of the nine coefficients above only seven are independent and these are sufficient for calculating the ternary equilibria. Numerous ternary diagrams have been derived in this way for selected systems although the method is slow. I t is presented here to illustrate the principles involved.
y,
Nomenclature
Pi
a,
= van der Waals attraction con-
stant for component i b , = van der Waals constant, covolume for component i ai, = (AtI - c z f ) , a coefficient in Equations 18 and 26 AiJ = common logarithm of activity coefficient for component i at infinite dilution in liquid j c i j = coefficient of empirical function representing dominant effect of association or interassociation (see Equations 14, 16, and 17) Et = defined by Equations 16 and 17; refers to less polar phase when two liquid phases exist et = same as E, except it refers to more polar phase when two liquid phases exist FE = excess free energy per mole FF = partial molal excess free energv h i j = a & / q i , a molecular interaction coefficient h, = split interaction constant-an unspecified molecular property n = number of components P = total pressure Pg = vapor pressure Q = FE/2.3RT
4 12
Yi
2,
Z
Y2
0.00 0.1003 0.1649 0.2274 0.3159 0.3982 0,5069 0.5850
1/a
0.0127 0.0172 0.0211 0.0254 0.0333 0.0435 0.0650 0.0955
an unspecified molecular property of component i = universal gas constant = dj/a?i = absolute temperature = mole fraction of any component in class R = mole fraction of any component in class M = mole fraction of any component in particular class S = summation Ex,-mole fraction of class R = summation Ex,-mole fraction of class iM = summation 2x,-mole fraction of class S = liquid composition, mole fraction of any component i in more polar phase = total liquid composition, mole fraction component i, in region of two liquid phases = vapor composition, mole fraction component i = liquid composition, mole fraction component i in less polar phase = Xtqi/zi,qi = defined by Equations 43, 44, and
yi
to *
4” -
lo
ei
volatility of component I relative to component 2 = liquid phase activity coefficient for i in the less polar liquid phase = liquid phase activity coefficient for component i = limiting value at zero pressure of the molecular attraction coefficient defined in (2) = nonpolar part of the molecular attraction coefficient at zero pressure = polar part of the molecular attraction coefficient at zero pressure = “imperfection-pressure coefficient” giving the combined effects of the vapor imperfections and of pressure on vapor-liquid equilibria =
Acknowledgment The author expresses his appreciation to Mott Souders, Clarence L. Dunn, and Otto Redlich for their cooperation and helpful suggestions, to Flora Black for assistance in the preparation of the manuscript, and to Doris Lidtke and Emily King for assistance in calculating some of the examples.
INDUSTRIAL AND ENGINEERING CHEMISTRY
(8)
=
A. -i 0112
(7)
(9)
(10)
Benedict, M., Johnson, C. A,, Solomon, E., Rubin, L. C.,Trans. Am. Inst. Chem. Engrs. 41, 371 (1945). Black, C., IND.ENC. CHEM.50, 391 (1958). Campbell, A. h-.,Dulmage, W. J., J . Am. Chem. Soc. 70, 1723 (1948). Carlson, H. C., Colburn, A. P., IND. END.CHEM.34, 581 (1942). Evans, \Y. V., Aylesworth, M. B., Ibid., 18, 24 (1926). Gibbs, J. W., “Collected Works,” p. 118, Longmans, Green, New York. 1906. Griswoid, J., Klecka, hl., West, R., Chem. Eng. Progr. 44, 839 (1948). Herinaton, E. F. G., Nature 160, 610 (1947); Research 3, 40 (1950): Jones, L. H., Badger, R . H., J . Am. Chem. SOC.73, 3132 (1951). Kretschmer. C. B.. Nowakowska. J.. Wiebe. R.. Ibid..‘ 70. 1785 (1948).’ Kretschmer,’C., Wiebk, R., ibid., 71, 3176 (1949). Kurtz, S. S., Jr., Lipkin, M. R., IND.ENG.CHEM.33, 779 (1941). Kurtz, S. S., Jr., Sankin, A., Zbid., 46, 2186 (1954). Luckenbill. D. B.. Dunn, C. L., MiIIar, R. W., Shell Development Co., Emeryville, Calif., unpublished data. Mains, G. H., Chem. t 3 Met. Eng. 26, 779 (1922). Othmer, D. F., Benenati, R. F., IND. ENC.CHEM.37, 299 (1945). Othmer, D. F., Chudgar, M. M., Levy, S. L., Ibid., 44, 1872 (1952). Othmer. D. F.. Morlev. F. R.. Zbid..
(19) Pear&, E.’ J., Gerster, J. A., Ibid., 42, 1418 (1950). (20) Reamer, H. H., Sage, B. H., Zbid., 43, 1628 (1951). (21) Redlich, O.,Kister, A. T., Zbid., 40, 345 (1948). (22) Ibid., p.341 (1948). (23) Redlich, O., Kister, A. T., Turnquist, C. E., Chem. Eng. Progr., Symposium Ser. 2, 48, 49 (1952). (24) Rosanoff, M. A., Bacon, C. W., Schultze, J. F. W., J . Am. Chem. Soc. 36, 1803, 1993 (1914). (25) Rothmund, V., Z. physik. Chem. 26, 433 (1898). (26) Scatchard, G., J . Am. Chem. SOC.62, 2426 (1940). (27) Scatchard, G., Chem. Revs. 44, 7 (1949’1. (28) Scktchard, G., Raymond, C. L., J. Am. Chem. SOG.60, 1278 (1938). (29) Scatchard, G., Ticknor, L. B., Ibid., 74, 3724 (1952). (30) Scatchard, G., Wood, S. E., Mochel, J. M.. Zbid.. 62. 712 (1940). . . Ibid., 68, 1957(1$46). ’ (31) Ibzd., Seltz, H., Ibid., 56, 307 (1934); 57, (32) 191 11935). 391 . _ (1935). M., IND.ENC. CHEM. (33) Watson, Watso-., K. 35, 398 (1943). (34) White, R. R., Trans. Am. Znst. Cliem. Engrs. 41, 539 (1945). (I (35) Wilson. Wilson, A., A.. Simons, Simon: E. L., IND.ENG. CHEM. ~IY~L). CHEM.44. 2214 11952). (36) Wohl, Kurt, Trans. Am. Znst. Chem. Engrs. 42, 215 (1946). (37) Yu. Kuo, Tsung, Coull, James, Chem. Eng. Progr., Symposium Ser. 2, 48, 38-45 (1952). (38) Zawidski, J. V., Z . phjsik. Chem. 35, 129 (1900). RECEIVED for review March 21, 1956 ACCEPTED July 18, 1957 Division of Industrial ana Engineering Chemistry, 129th Meeting, ACS. Dallas, Tex., April 1956.
