I
G. 1. PIEROTTI, C.
H.
DEAL, and E. L. DERR
Shell Development Co., Emeryville, Calif.
Activity Coefficients and Molecular Structure When selecting separation processes, the simple relations presented here, among solution nonidealities of structurally related systems, should be useful D E V I A T I O N S FROM IDEAL behavior in solutions of nonelectrolytes are frequently important in design and operation of separation processes (7, 8). Although a qualitative understanding of the more important solutiop effects is available (3, 9, 70, 12, 75, 76, 78, 79),this is of limited use when quantitative estimates are needed. Particularly for mixtures containing one or more polar components, solution effects cannot be quantitatively predicted from theory, and direct experimental data are ordinarily required. T h e number of such data needed to describe behavior has commonly been minimized by using empirical or semiempirical but thermodynamically consistent correlations between activity coefficients and concentration within a given system (2, 4, 7, 77). The possibilities of minimizing data requirements by developing empirical or semiempirical correlations between solution effects and molecular structure has, however, received but little attention. Butler and others ( 6 ) showed that the partial molal excess free energy of alcohol and other aliphatic homologs in aqueous solution at infinite dilution increased linearly with their number of carbon atoms. Broensted and Koefoed (5) showed that such energy of binary paraffin mixtures, covering a moderately wide molecular weight range could be correlated with a simple expression involving the square of carbon number differences. Significant extensions of these or similar approaches have not appeared. In the present work, phase equilibria data for many structurally related sets of binary systems have been obtained either experimentally or from the literature. These data have been reduced to activity coefficients of the solute at infinite dilution in the solvent, and correlated on the basis of systematic effects on the logarithm for infinite dilution activity co~~
1
A supplement covering the experimental part of this work and detailed comparison of correlated and experimental activity coefficients has been filed with the American Documentation Institute (13). This includes valuable new data for 275 binary systems.
i
efficients (excess partial molal free energy) caused by systematic changes in solute and/or solvent structure. The resulting expressions are surprisingly accurate for predicting infinite dilution activity coefficients from minimal data. Correlation of Activity Coefficients with Solute-Solvent Structure
T h e logarithms of activity coeffcients for members of solute families a t infinite dilution fall into simple correlation patterns. They are in turn susceptible to a simple, semiempirical interpretation based on molecular interactions. For methylene homologs, R1X1, a t infinite dilution in methylene homologs, R2X2, the pattern is described by: FlE/2.3RT = log yl0 = Cdnl
+ D(n1 -
A1.2
n2I2
+ Bznl/nz + + Fz/n2 (1)
nl,
n2
A1,2
Bz
C1
D
solute a t infinite dilution in the solvent in excess over the Raoult’s Law value (pure liquid a t its saturation pressure as standard state) = activity coefficient of component RlXl at infinite dilution in component R z 2 = numbers of carbon atoms in hydrocarbon radicals R1 and R2, respectively = coefficient which depends on nature of solute and solvent functional groups, XI and X2 = coefficient which depends only on nature of solvent functional group, X z = coefficient which depends only on solute functional group, x1
= coefficient independent
XI and Xz
Ft
Correlation Term
of both
= coefficient which essentially de-
pends o n l y s n nature of the solvent functional group, Xz This relation, although similar to that of Butler (6),is extended and refined. For simple interpretation it is assumed that excess free energy or log y o can be treated as a sum of contributions from individual interactions between pairs of structural groups in the solute and solvent molecules which depend solely on the number, type, and configuration of the groups in the respective structures. T o retain simplicity, the more fundamental
Form
Predominant Contributing Interactions
XrX1, XrX2, Xz-xz
A
Ai,z
B
F
Xz-Xz, Ri-Xz Bznllnz Cdnl Xt-Xl, Rl-Xi D(n1 - n ~ ) ~ Ri-R;, RI-Rn, Re-R2 Xi-Xz; Rz-XZ, Fdnz
[El
[Eln2lml
D
= partial molal free energy of
yl0
Interpretation of Terms in Equation 1
C
where &E
approach which treats excess heats and entropies of solution separately has not been used here. .If RI and Rz represent the sum of methyl and methylene groups in the solute and solvent molecules, the individual interactions between R1X 1 and its standard state environment are those between the groups X1-X1, XI-Rl and R1-R1. Those between the solute molecule and its solvent environment are X I - X ~ X2-R1, , X1-R2, and R1-R2. The competing solvent interactions are X Z - X ~ , X2-R2, and R2-R2.
xrx2
Rz-XI
Constant Al.2 may be considered as the limiting sum of interactions for zero carbon-numbered members of the homologous solute and solvent series. Thus, it may include contributions from each of the interacting combinations, but undoubtedly it is associated with XI-Xl, X I - X ~and , X2-X2 interactions. CI(l / n ~ ) accounts for the change in the number of X1-X1 interactions which occur as the solute carbon number increases and the polar groups are, in effect, diluted by the methylene groups of R1. F z ( l / n z ) similarly accounts for the change in the number of the X 2 - x ~interactions, because they are diluted by a n increase in the methylene groups of R z . Bz(nl/nz) or B term accounts for the increase in the number of x 2 - X ~interactions which must be broken and for the increase in the number of R I - X ~interactions which occur as the solute molecule increases in carbon number. As in the F term, these contributions are inversely proportional to the solvent carbon number because X2 groups are diluted as the number of methylene groups in R2 increases. D(nlnz)z accounts for changes in Rl-Rl, R1-R2 and R2-Rz interactions with carbon number. The form of this term has been VOL. 51, NO. 1
JANUARY 1959
95
taken directly from the Broensted and Koefoed and the van der Waals work with paraffin systems (5, 20). Contributions proportional to nz/nl which might arise from RpX1 interactions are not necessary in many systems and have not been generally used. This analytical form just described becomes a simpler expression in special cases because effects described by one or more terms may either not occur or remain constant. Thus, when describing 7 ” ’ s for a homologous series in a given solvent, Fz(l/nz) and Bz(lln2) remain constant and Equation 1 reduces to, log 71’= K
+ Bnl + C I ( ~ / ~+I ) D(n1 -
n2Y
(2)
where constant K corresponds to the sum of a n A and an F term and B corresponds to Bs/nz. When nl or n2 are zero-(e.g., water as a solute or solvent)-terms normally reflecting the influence of the respective alkyl groups are absorbed in group constant K . Although Equation 1 has been developed largely for straight chain R groups, moderate branching in R’s not in the immediate vicinity of the functional X groups results in only slightly different yo’s. Correlations for Methylene Series Containing Hetero Groups
Correlations for homologous series containing hetero groups are presented in Table I in a manner which illustrates the various sets of the terms discussed previously. The appropriate constants at discrete temperatures (25”. 60°, and looo C.) are listed. Either y o values or correlation constants a t intermediate or moderately extrapolated temperatures can be obtained by customary plotting procedures. Correlations for normal alkyl homologous series with terminal functional groups as solutes in water solvent illustrate the use of A-. B-, and C-type terms only. Correlations for series where the functional group is not in the terminal position illustrate how minor modification in C-type terms can account for changes in structural organization. Correlations for series in paraffin solvents illustrate use of A-, C-, and D-type terms. Correlations for water in homologous series of solvents illustrate the use of A- and F-type only. Correlations for paraffins in ketone homologs as solvents illustrate A-, B-, D-,and F-type and, in addition, some of the limitations of the simple ideas used here. Homologs in Water. Correlations listed in Table I1 do not involve those terms depending on n z and the representation uses only A-, B-, and C-type terms. Where the solute functional group is not terminal, a modified but closely related form of the C term which ac-
96
INDUSTRIAL AND ENGINEERING CHEMISTRY
counts for this changed configuration is used. In this form, each alkyl group originating at the functional group contributes in much the same way as the alkyl attached to a terminal functional group. The term becomes C1(l/nl’ l/nl”) for a secondary grouping and C1(l/nl’ l/nl” l/nl”’) for tertiary groupings, where primed n’s are carbon numbers of the respective branches counted from the polar grouping. Thus, for tert-butyl alcohol, the central carbon nl’ = is counted in each branch-Le., n1” = n1’” = 2-even though the total carbon number is four. With these relations, each kind of branching is considered to form a homologous series with its associated values for the constant term K and C1-coefficient. The C1 coefficient, is not solvent independent for such extremes as water and paraffins. A more satisfactory correlation is obtained, a t least for alcohols, by using terms such as Cl(l/n1 - l ) , Cl[(l/nj’ 1) ( l / n l ” - I ) ] and Cl[(l/nl’ - 1) (l/nl” - 1 ) (l/n”’ - I)] for primary, secondary, and tertiary alcohols, respectively. With this form, all alcohols may be treated as a single homologous series with little sacrifice in accuracyi.e., primary, secondary, and tertiary alcohols, can be described lvith a single constant term and a single C1 coefficient. Here, the C1 coefficient is solvent independent for the water and paraffin solvent extremes (compare coefficients for alcohols in paraffins). A second modification of the C term is used in the acetal correlation. With acetals of the structure R’”(OR’’)OR’, the unique configurational position of R”’ can be taken into account by a C term such as C ~ ( l / n ~ ’ l / n l ” 2/n1’”). A third modification of the C term is used in the correlation of alkyl-substituted cyclic compounds. Here, the cyclic nucleus is treated as though it were a functional group and an expression Cl/(nl- 4) is suitable. Homologs in Paraffins. Correlation of alcohols in paraffins illustrares several aspects of the correlation pattern. Because paraffin solvents have no funcLiona1 group, interac5ons associated with B and F terms do not occur, and these terms are omitted. Alkyl groups exist on both solute and solvent molecules, and A : C and D terms are required. The C1 constant is identical with the general correlation of alcohols in water. Given the C1 constant from a correlation of alcohols in some other solvent, only one constant, A I , ? need be determined experimentally for any paraffin solvent. Thus. a t the expense of this one experimental y o value, systems covering alcohols from C1 to higher than CS, and paraffins from C, to a t least (220, may be predicted to sufficient accuracy for most process considerations. Ketones in paraffin solvents represent a situation parallel to alcohols in paraffins
+
+
+
+
+
+
+
+
except that only one form of the C term need be used, because the secondary structure characterizes the ketone group exclusively. The C1 constants employed are not identical to those found most suitable for the ketone in water correlations, although the differences are not large enough to represent a serious departure from the solvent-independent nature of the C1 constant. Water in Alcohols and Ketones. For water in homologs of alcohols and ketones, only A I , * and Fz constants are required. As for the C term, branches in the solvent structure which originate a t the functional group are treated as making separate contributions to the F term. As used in the present correlations, the F term undoubtedly absorbs contributions from interactions other than the solvent structure-dependent X2-X2 and R,-Xz interactions with which it is primarily identified. In such systems as those formed by water in a homologous series of alcohols, the yo value of water decreases as the solvent carbon number increases. This might be expected to result in part from an increase in the number of attractive R?-X1 interactions (E-type, E1 >0) and in part from the dilution effect of R P on X Z - X P interactions (F-type, F P >O). Because both effects would be in the same direction, their net sum can undoubtedly be expressed to some degree of accuracy by an F term alone; howrever, the first effect is solute dependent and the latter is solvent dependent. An F term alone adequately correlates data for a given solute series in a given solvent series, but the F? coefficient obtained is not entirely solute series independent. So long as both solute and solvent series structures contain strongly interacting groups, however, the F? coefficient is essentially solute series independent and may be used from solute series to solute series in a given solvent series \vith no serious error (compare water and ketones in alcoholic solvents) even though it contains E-type contributions. On the other hand, when systems are treated where solute structures contain no strongly interacting groups-Le., no RZ-XI or E-type interactions at all can occur-then a new F? coefficient must be determined for a solvent series (see paraffins in ketones). For the sake of simplicity E terms have been avoided, and Fa coefficients which are not entirely solute series independent have been accepted. Homologs in n-Alcohols. For ketones in alcohols, both solute and solvent structures contain functional and alkyl groups; therefore, the correlation includes all five of the terms mentioned. Except for the A1.2 and B S coefficients, however, all coefficients are available from correlations already presented. Of these two: the B Z coefficient may be
.
-
-.-.
..... ,.
A C T I V I T Y COEFFICIENTS
Table 1.
Correlation Constants for Activity Coefficients
(Infinite dilution of various combinations of homologous series of solutes and solvents")
Solute Series n-Acids
.
Solvent Temp., A . B Series C. Constant Term Constant Water
25 50 100 25 n-Primary Water 60 alcohols 100 25 n-sec-alcohols Water 60 100 25 n-tert-alcohols Water 60 100 Water 25 Alcohols, general 60 100 25 n-Allyl alcohols Water 60 100 25 n-Aldehydes Water 60 100 n-Alkene aldehydes Water 25 60 100 n-Ketones Water 25 60 100 n-Acetals Water 25 60 100 n-Ethers 20 Water 25 Water n-Nitriles 60 100 n-Alkene nitriles 25 Water 60 100 %-Esters Water 20 n-Formates Water 20 n-Monoalkyl chlorides' Water 20 n-Paraffins Water 16 n-Alkyl benzenes Water 25 n-Alcohols Paraffins 25 60 100 n-Ketone s Paraffins 25 60 100 Water n-Alcohols 25 60 100 see-Alcohols 80 n-Ketones
Ketones
n-Alcohols
Aldehydes
n-Alcohols
Esters
n-Alcohols
Acetals
n-Alcohols
Paraffins
Ketones
-1.00 -0.80 -0.620 -0.995 -0.755 -0.420 -1.220 -1.023 -0.870 -1.740 -1.477 -1.291 -0.525 -0.33 -0.15 -1.180 -0.929 -0.650 -0.780 -0.400 -0.03 -0.720 -0.540 -0.298 -1.475 -1.040 -0.621 -2.556 -2.184 -1.780 -0.770 -0.587 -0.368 -0.095 -0.520 -0.323 -0.074 -0.930 -0.585 1.265 0.688 3.554 1.960 1.460 1.070 0.0877 0.016 -0.067 0.760 0.680 0.617 1.208
25
1.857
60 100 25 60 100 25 60 25 60 100 60
1.493 1.231 -0.088 -0.035 -0.035 -0.701 -0.239 0.212 0.055 0.0 -1.10
25
None
60 90
0.622 0.590 +0.517 0.622 0.583 0.517 0.622 0.583 0.517 0.622 0.583 0.517 0.622 0.583 0.517 0.622 0.583 0.517 0.622 0.583 0.517 0.622 0.583 0.517 0.622 0.583 0.517 0.622 0.583 0.517 0.640 0.622 0.583 0.517 0.622 0.583 0.517 0.640 0.640 0.640 0.642 0.622
C
C(l/nd C(l/nd C[l/nl'
+
Constant
0.490 0.290 0.140 0.558 0.460 0.230 1 / n 1 ~ ] 0.170 0.252 0.400 0.170 0.252 0.400 - 1) 0.475 1) 0.39 1) 0.34 0.558 0.460 0.230 0.320 0.210 0.0 0.320 0.210 0.0 0.500 l/nl"] 0.330
[::::" 1' +
C l/nl"
(l/nl' (l/ni" (l/nl'" C(l/nd C(l/nl) C(l/nt)
C[l/nl'
+
C[l/nl' C(l/nt)
1
-- +
?]
C?[:{:::;,
+ llni'']
C(l/nd
+
Term
None
None
None
None
None
None
None
None
None
None
None
None
None
None
None
None
None
None
None None
None None
None
None
0.200
0.486 0.451 0.426 0.195 0.760 0.413 0.00 0.760 0.413 0.00 0.260 0.260 0.073
+
None
Term Constant None
+
None
Constant
None
C(l/nl' l/nl") None C(l/nd None C(l/nl) None None None C[(l/nl - 4)] -0.466 None 0.475 D(nl - nz)2 (l/n1' - 1) (l/nl" 1) 4 - 1 0.390 (l/nl"' - 1) 0.340 0.757 n# 0.680 D(nl C[l/nl' l/nl"] 0.605 None None
-
F
D
Term
-
- 0.00049 - 0.00057 - 0.0006 1 - 0.00049 - 0.00057
- 0.00061
None
None None None None None None None F(l/nz)
-0.630 -0.440 -0.280 F(l/nz' 4- -0.690 -1.019
None
None
B(nl/nz)
0.176 0.138 C[l/ni' f l/nt"] 0.112 0.176 0.138 C(l/nl) 0.176 0.138 C[l/nl' l/nt"] 0.112 0.138 C(l/nl' 4- l/nl" 2/n1"') 0.1821 None
B(nlln2) B(nllnd
0.1145 0.0746
+
+
0.50 - 0.00049 P(l/nz) 0.33 D(nl - n# - 0.00057 0.20 - 0.00061 -0.00049 F(l/nz) 0.520 0.210 - 0.00057 0.260 -0.00049 F(l/nz) 0.240 D(nl - nz)a - 0.00057 0.220 -0.00061 0.451 - 0.00057 -0.00049 D(n1
-
- 0.00057
-0.73 -0.557 -0.630 - 0.440 -0.280 -0.630 -0.440 -0.630 - 0.440 -0.280 -0.440
F(l/n*' f 1/VA")
-0.00061
0.402 0.402 0.401
+ Bz 24r + CInl-1 + D(n1 - nz)2 + FZ 1
General expression, log y o = AI,Z
n2
VOL. 51, NO. 1
i
JANUARY 1959
97
taken from measurements on other homologous series in a single alcohol solvent-e.g., from paraffins in ethyl alcohol. The solute-dependent C1 coefficient is available from the ketones in water correlations; the solvent-dependent Fz coefficient from the water in alcohols correlations; the D coefficient is solute and solvent independent as before. Thus, in the present instance, over 130 systems could be predicted in terms of a single experimental y o necessary for determining the one new coefficient required. Alkylation effects could be accounted for in terms of parallel relationships and the only undetermined interaction effects are those involving direct interaction of functional groups. Experimental data to predict the functional group interaction effects may themselves be somewhat reduced as additional insight is obtained. Aldehydes as solutes in alcohol solvents present a particularly striking example of the general applicability of the present correlation pattern. These systems involve such strong, pseudochemical interactions between solute and solvent functional groups (hemiacetal formation) that y o values are less than unity; yet they are represented by a single new A1.2
constant-Bz, CI, F2, and D are carried over from other systems. Esters in alcohol solvents, as for other polar solutes, are represented by a single new system-characteristic AI,* constant; the other constants are derived from ester-solute, alcohol-solvent systems. In aqueous solutions, formate esters required a special A1.2 or K value different from that of the higher esters, but in the present alcohol correlations, they are incorporated quite satisfactorily. For acetals in alcohols, account is taken of the structure in the C term as mentioned previously, but otherwise the only new constant is the system-characteristic A1,z constant. Paraffins in Ketones. I n the correlation of paraffin solutes in ketone solvents, no C term is required because no functional group is present in the solute structure. The usual A , B, D,and F terms are required. Paraffin-ketone systems covering molecular weight ranges from Ca to C l Sketones and Cg to Czs paraffins are represented in terms of only two new system-dependent constants, one of which is zero. As already indicated, the FZ coefficient is substantially different from that based on the water data (0.402 compared to
Table II.
-1.0). Because in the present case of paraffins in ketones, no R2-X1 or Etype interactions can occur, this case appears to provide the better basis for obtaining a solute independent E', constant. If the present E'z constant is assumed applicable to the water solute case, then E1 coefficients derived from the water in ketone studies represent water data satisfactorily. Temp., O
c.
E1
25
60
100
0.156
0.128
0.100
Thus a single l i 2 coefficient can be used in both cases involving ketonr solvents in spite of great differences in the systems. Complete consistency cannot be obtained, however, because the same E1 values do not satisfy the water in alcohol data and are thus not completely solute dependent. Correlations for Series of Hydrocarbons in Various Solvents
Correlations for hydrocarbon solutes in a number of polar solvents (Tables I1 and 111),treat solute families oftwo types. I n the first type, the families are homologous series analogous to those already
Correlation" Constants for Activity Coefficients
(Various homologous series of hydrocarbons in specific solvents) Solute Dependent C'S
Solvents
Term = Temp.,
c.
Solute Series
(ST)
Heptane
Methyl ethyl ketone
Furfural
Phenol
____
Ethyl alcohol
Triethyl- Diethylene Ethylene ene glycol glycol glycol
Solvent-Dependent B's (Term = Bp np) 25 50 70 90
0.0 0.0 0.0 0.0
0.0455 0.033 0.025 0.019
0.0
0.0
0.0 0.0 0.0
0.0 0.0 0.0
0.335 0.332 0.331 0.330 0.70 0.650 0.581 0.480 0.277
0.0937 0.0878 0.0810 0.0686
0.0625 0.0590 0.0586 0.0581
0.088 0.073 0.065 0.059
...
0.161
...
0.134
0.191 0.179 0.173 0.158
(0.275) 0.249 0.236 0.226
Solute-Solvent Dependent K's (Term = K ) 25 50 70 90 25 50 70 90 25 50 70 90 25 50 70 90 25 50 70 90 25 50
Paraffins
Alkyl cyclohexanes
Alkyl benzenes
Alkyl naphthalenes
Alkyl Tetralins
Alkyl Decalins
70 90
-0.260 -0.220 -0.195 -0.180 -0.466 -0.390 -0.362 -0.350 -0.10 -0.14 -0.173 -0.204 4-0.28 f0.24 +0.21 +o. 19 -0.43 -0.368 -0.355 -0.320
0.18
...
0.131 0.09 0.328 0.243 0.225 0.202 0.53 0.53 0.53 0.53 0.244
...
0.220
... ... 0.356 ...
+ Bpnp+ np c,+ D(nl - n# +2
...
0.240 0.239 0.169 0.141 0.215 0.232 0.179
... 0.217 ... 0.871 ... 0.80
0.916 0.756 0.737 0.771 1.26 1.120 1.020 0.930 0.67 0.55 0.45 0.44 0.46 0.40 0.39
...
0.652 0.528 0.447 0.373 1.54 1.367 1.253 1.166
0.870 0.755 0.690 0.620 1.20 1.040 0.935 0.843 0.694 0.580 0.500 0.420 0.595 0.54 0.497 0.445 0.378 0.364 0.371 0.348 1.411 1.285 1.161 1.078
0.580 0.570 0.590 0.610 1.06 1.01 0.972 0.925 1.011 0.938 0.900 0.862 1.06 1.03 1.02
... ...
...
... ... ... ... ... ...
...
0.72
...
0.68
...
1.46
... 1.25 ...
0.80
...
0.74
...
0.75
... ... 1.00 ...
0.83
0.893
...
1.906
...
1.68
0.875 0.815 0.725 0.72 1.675 1.61 1.550 1.505 1.08 1.00 0.96 0.935 1.00 1.00 0.991
...
I . 208 1.154 1.089
...
2.36 2.22 2.08
...
1.595 1.51 1.43
...
1.01
1.92 1.82 1.765
1.43 1.38 1.33 1.28 2.46 2.25 2.07 2.06
... ... ... ... ... ...
... ...
r(
Expression, log y o = K
98
INDUSTRIAL AND ENGINEERING CHEMISTRY
where for all systems: D = -49 X t o C. = 25
-55 X low6- 5 8 X 50 70
-61 X
90
A C T I V I T Y COEFFICIENTS discussed and include the series formed by n-alkyl substitution of various cyclic compounds. I n the second type, the families of solutes are those formed by unsubstituted cyclic compounds themselves. Correlations for the first type follow closely on the correlations already presented, but with the ring structures treated as functional units. Correlations for the second type are a departure. Contributions from naphthenic and aromatic carbons within the ring structures are considered separately. Moderate branching of alkyl groups or multisubstitution on the cyclic nucleus of the solutes has generally second order effects on yo's, and the correlations also give reasonable estimates of yo's in these cases. T h e correlation relation for paraffins and for alkyl substituted cyclic compounds is a special form of relation (Expression 2) which distinguishes between the number of paraffinic carbons in the solute structure, n,, and the total carbon number, n,. For paraffins and alkyl cyclic compounds the relation is log
Y1°
=
K1
+ B,n, + C1/(n, + 2 ) + - nd2 D(n1
na
?io
= Kc
+ Ban" +
-
1)
I
=c-,
I
ring juncand
I
naphthenic carbons which are in the alpha position to a n aromatic nucleus : n, includes all naphthenic carbons not For example, the counts counted in n,. for butyl Decalin would be n p = 4, n, = 2, and n, = 8, whereas those for butyl Tetralin would be np = 4, na = 8, and n, = 2. I n both cases, nl = 14 a n d r = 2. A consistent combination of Expression 3 and 4 is
+ Ban, + 4C, + n, - TZZ)~] + Bpnp + C d n , + 2 ) + D(n1 - n d z ( 5 )
log 710 = ( ~ / r- I )
Bnnn
[KO
- C1/2 - D(na
where the term in brackets corresponds to K1. Inasmuch as Expression 4 is somewhat less accurate than 3, the combined form, Expression 5 has not been used, and no effort has been made to enforce a n exact correspondence on the yo's predicted by the separate correlations such as Expressions 3 and 4. Paraffin Solutes. For paraffin solutes, although the D term makes but relatively small absolute contributions to the yo's of paraffins in polar solvents, it makes substantial contributions in the heptane solvent. Here, agreement is quite satisfactory; however, the Broensted relation used cannot be expected to be equally accurate in all paraffinparaffin systems. I t appears that so long as nl is greater than n2, the relation provides a reasonably good representation, but as has been generally recognized, the symmetry implted by the relation is not correct. T h e y o value in heptane corresponds to the for n-& Broensted predicted value. The pre-
(3)
+G
(1/r
'
ture naphthenic
The correlating relation for cyclic compounds without alkylation is log
includes =CH--,
(4)
Here, K E ,B,, and B, are solvent-dependent coefficients; C, is solvent independent, but different values for tandem (diphenyllike) and fused (naphthalenelike) structures distinguish between two types of structural organization. r is the number of rings; n, and n, are aromatic and naphthenic carbon numbers, respectively, but are subject to several special counting rules. The number
Table 111.
dicted value for heptane in C32 solvent is about 25% lower than the experimental value of van der Waals (20). K1 for paraffins j s roughly equal to about 7 Bp for a number of solvent cases. Thus, y o values for paraffins in a given solvent can be roughly predicted from a value of B, derived from measurements for alkyl cyclic solutes in the solvent or from a single paraffin solute. Alkyl Cyclic Solutes. It might be expected that the predominant interactions which occur with relatively, large nonpolar cyclic nuclei would differ from those which occur with the small, strongly polar groups discussed previously and that the same kind of correlation could not be used. Identical correlational forms can be used, however, with but little loss in accuracy. Althoughcorrelations for alkyl Tetralins and Decalins are given, they are based on data for only one alkyl member and this in only furfural and diethylene glycol solvents. The correlations, therefore, serve only to establish the value of C1and its independence of the solvent. The C coefficients for alkyl cyclics indicate no clear trend with the nature of the nucleus and thus provide very little basis for their correlative prediction from one to another. They are uniformly small in magnitude (0.2 to 0.4) and generally of negative sign so that estimates in this range could probably be assumed for an untested series without serious uncertainty. This then would allow y o values of a wide range of alkyl cyclics to be estimated according to Expression 5 from a knowledge of the correlation constants of paraffins and cyclics in a given solvent. Unalkylated Cyclic Solutes. The correlations of y o values of unalkylated
Correlation" Constants for Activity Coefficients
[Unalkylated cyclic (aromatic and/or naphthenic) hydrocarbons in specific solvents] Solute Dependent C's [Term = Cr(f - I)]
Solvents Heptane
Methyl ethyl ketone
Furfural
25 70 130
0.2105 0.1668 0.1212
0.1435 0.1142 0.0875
0.1152 0.0836 0.0531
25 70 ,130
0.1874 0.1478 0.1051
0.2079 0.1754 0.1427
0.2178 0.1675 0.1185
Temp. O
c.
Solutes
Condensed Tandem
Phenol
Ethyl alcohol
Triethylene Diethylene glycol glycol
Ethylene glycol
Solvent-Dependent B's 0.1421 0.1054 0.0734
0.2125 0.1575 0.1035
0.181 0.129 0.0767
0.2022 0.1472 0.0996
0.275 0.2195 0.1492
0.3124 0.2406 0.1569
0.3180 0.2545 0.1919
0.4147 0.3516 0.2772
Solvent-Dependent B's 0.2406 0.1810 0.1480
0.2425 0.1753 0.1169
Solute-Solvent Dependent, K's (Term = K ) 25 70 130
Cyclicsc
Expression, log y o =
1.176 0.846 0.544
K
1.849 1.362 0.846
-1.072 -0.886 -0.6305
+ Bena + Bnnn+ CY (l/r
-0.7305 -0.625 -0.504
-
1).
-0.230 -0.080 -4-0.020
-0.383 -0.226 -0.197
* Term = Bano + B,n,.
-0.485 -0.212 +0.47
-0.406 -0.186 +0.095
-0.377 -0.0775 +0.181
-0.154 -0.0174 4-0.229
Naphthenes, aromatics, naphtheno-aromatics.
VOL. 51, NO. 1
JANUARY 1959
99
Table IV.
Accuracy of Correlations yo
sys-
terns
Solutes
11 6
n-Acids (Cl-Cl4) n-Primary Alcohols (CIClO)
Water Water
y o Levels 1.3 t o 5 x 1 0 7 1.5 to 6 X IO4
5
sec-Alcohols (Ca-Cd tert-Alcohols (C4-Cd n-Aldehydes (Cz-Co) n-Ketones (CaC7) n-Acetals (C3-
Water
6.5 to 7.5
Water
10 to 40
Water
4.2 to 6.7
Water
7.8 to 1.8
2 6 6 4 5 2 11 4
4 5 13 12 10 3 14 5 12 3 13 6 9 8
7 5 14 9
6 13 9
7 13 9 3 10 8
4 13 9 3 9 9
100
c
Solvelit 5
n-Ethers (CZ-CS) n-Nitriles (C2Ca) n-Esters (CZ-CS) n-Chlorides (cl-c4) n-Paraffins (CSCS) n-Alkyl benzenes (CS-CIO) Alcohols (Ci-Ca)
Water Water Water Water
1
Av. Max. 6.0 1.8
10 3.7
2.4
6.4
0.6
1.5
0.7
3.7
0.3
1.1
1.3
5.0
5.6
103
5.0
9.6
1.1
3.0
18 to 1 . 5 X
lo2
1.2
6.6
0.3
1.3
8 to 2.9 x 104 10.9 to 39.8
2.4 0.0
10 0.0
1.0 0.0
4.0 0.0
21 5.0
1.8 0.6
12 1.0
40
1.2
2.4
350
6.2
20t03.5 x x io* to 7.3 x
108 103
7.5 3.5
x
105
10.2
104 to 4.2 2
84 13
104
Water Water
103
16 4.5
AFB Deviations, yo
x x
Water
8)
x
Devia-
_-tions, % ’ Av . Max.
x
103 to 2
x
106
n-Paraffins 4 . 2 to 92 (C7-CZO) n-Paraffins Ketones (C31.4 to 7.4 C19) (CS-CZO) Alcohols (CiWater 1.6 to 6.2 ClO) Water Ketones (C34.7 to 21.4 C7) Ketones (CI-ClD) n-Alcohols 1.6 to 138 (C1-Cs) n-Alcohols n-Aldehydes 0.15 to 0.8 (C1-Cs) (CZ-Cl) n-Esters (c2-C~) n-Alcohols 1.8 to 3.2 (Cl-c,) Acetals (Ca-C5) n-Alcohols (Ci) 2.3 to 4.0 %-Paraffins (Cj- Ketones (Ca1.8 to 30 Cm) C7) n-Alkyl cyclics %-Heptane 1.1 to 1.78 (Ce-Cia) Cyclics (cS-C18) n-Heptane 1 - 0 2 to 56 %-Alkyl cyclics Methyl ethyl 1.05 to 6.6 ketone (CS-cl8) Cyclics (CC-Cls) Methyl ethyl 1.05 to 8.2 ketone n-Paraffins (C4- Furfural 7.3 to 2.7 X 108 CZO) n-Alkyl cyclics Furfural 2.0 to 3.4 X IO* (Ce-Cza) Cyclics (Ca-Cia) Furfural 1.8 to 21 n-Paraffins ( C r Phenol 7.8 to 3 X 10% Cao) n-Alkyl cyclics Phenol 1.75 to 88 (CSCc22) Cyclics (CS-CI~) Phenol 1.75 to 27 n-Paraffins (Ca- Ethyl alcohol 6.0 to 1.5 X 102 C20) n-Alkyl cyclics Ethyl alcohol 4.6 to 2.5 X lo2 (c6-CZ8) Cyclics (dS-cI8) Ethyl alcohol 4.5 to 4.4 x 10% n-Paraffins (CZ- Triethylene 25 to 1.8 X IO* glycol ClS) n-Alkyl cyclics Triethylene 3 . 0 t 0 4 . 6 X loa glycol (C6-cZS) Cyclics (C6-Cl8) Triethylene 3.0 to 92 glycol %-Paraffins (C7- Diethylene 43 to 7.7 x 103 glycol c16) n-Alkyl cyclics Diethylene 6 . 0 to 3.6 x 104 glycol (c6-CZ8) Cyclics ( C 6 - c ~ ) Diethylene 6 . 0 t O 1.69 X 108 glycol n-Paraffins (C7- Ethylene gly- 2 . 1 X IO2 to 9.3 X I O 4 Cl6) col n-Alkyl cyclics Ethylene gly13.2 to 2.6 X IO4 (CE-CJS) col Cyclics (CS-CIS) Ethylene gly13.2 to 8.5 X IOa col
INDUSTRIAL AND ENGINEERING CHEMISTRY
98 2.4
40
7.5
1.0
12 4.7
5.0
17
5.7
30
7.1
17
7.5
38
3.7
17
2.2
10
5.1
21
6.4
35
4.8
9.1
3.8
7.8
6.7
33
7.8
40
8.6 1.8
15
5.3 1.1
10 11
5.1
14
... ...
8.0 4.0
26
5.6
5.9
15
... ..* ... ...
11.5
37
3.6
13
7.6
48
2.9
12
6.5 5.1
24 19
5.4 20 1.1 3.9
5.6
29
3.7
12
7.5 4.8
25 15
4.2 2.1
20 6.7
9.0
34
2.5
20
10.0 5.8
53 12
2.9 1.2
29 2.3
10
42
4.6
23
13
42
6.5
23
23
1.4
5.5
10
40
2.2
9.5
13
42
3.5
9.5
17
60
2.4
8.8
22
0.9
3.1
37
2.6
9.6
7.5
4.8 12
8.7
15
15
cyclic compounds afford somewhat less precise representation than the homologous series correlations but cover a considerably greater area with very few data requirements. O n the basis of only three solvent dependent constants, K,, B,, and B,, or only three experimental points per solvent, aromatics, naphthenes, and naphtheno-aromatics in the CB-C18 carbon number range can be predicted with reasonable accuracy once the two solvent independent C, constants have been determined. I n the present correlation, values of C, distinguish only between tandem and fused ring structures. The consistently lower y o values observed for phenanthrene as compared to anthracene indicate that better representation of phenanthrenelike and anthracenelike structures could be achieved with different C, constants. These differences are undoubtedly due to shape factors. Basicity characteristics would augur differences in the reverse direction if solvents were strong acids ( 7 7). I n the present correlation, y o values for anthracene have been given greater weight because experimental data for anthracene are probably more accurate. T h e special counting rules just mentioned give a better basis for predictions than if all naphthenic carbons are counted as n,. Further work with naphtheno-aromatics is, however, called for-e.g., y o values for dicyclohexylbenzene have been obtained which are only 25 to 50y0 as high as those predicted by the present pattern. Accuracy of Correlations Since the correlations presented cover a n extremely broad range of y o ’ s and the data are necessarily of variable accuracy, a realistic idea of the accuracy of the correlations and predictions can only be obtained by a detailed review of the cases treated (73). Rough ideas of this accuracy can, however, be obtained from the summary in Table IV. Here, the average per cent deviation over the several correlation temperatures and the maximum per cent deviation between correlated and experimental yo’s and log yo’s are given for each set of systems. For the 44 sets of systems, the over-all average deviation in yo is about 8% and in log y o about 3%. T h e alkyl cyclic solutes and cyclic solutes are correlated somewhat less accurately on the average than the hetero solutes. A more realistic weighting of the correlations according to values of the experimental data would give somewhat lower over-all deviations. Some further idea of the undue influence of one or two particular solutes within a set on these averages can be obtained by comparing the yo and log y o levels, the average, and the maximum deviations within the sets. For yo’s,
A C T I V I T Y COEFFICIENTS large deviations usually occur at high yo levels where experimental errors are generally greatest. Thus, in the worst case (alkylbenzenes in water) at yo levels up to 106, the experimental yo’s are based upon solubilities of about 10-3 to 10-6 mole fraction and are undoubtedly determined only to 50 to 100% of their values. Here, omission of a single point with a y o of 106 reduces the average deviation in yo from 98 to 33% and the maximum from 350 to 73%. I n the next worst case (acids in water) a t y o levels up to 107, available data are likewise poor. At low levels, an estimate of y o involves vapor-liquid measurements and a correction for dimerization in the vapor; at high levels, it involves a determination of solubility at a level of IO-’ mole fraction and an estimate of the ideal solubility. Here, omission of a single temperature point for tetradecanoic acid (yo 107) reduces the average deviation from 17 to 13% and the maximum deviation from 84 to 50’%. However, large per cent deviations at these high levels are generally of little practical significance. For log yo’s (or excess-free energiq), the influence of errors at high levels does not appear so significant but the per cent deviations at yo levels less than about two become extremely sensitive to the accuracy of the experimental data or the correlations. Because deviations of this sort are of little practical significance, three sets in which the yo represented are small (alkylcyclics in heptane, alkylcyclics in methylethylketone, and cyclics in methylethylketone) have been omitted from Table IV.
-
Application to Separation Processes
d
The above ideas and correlations are valuable for estimating the phase equilibria exploited in many separation processes. Their value lies not only in reducing the experimental effort required for screening processes but also in providing a systematic and fairly complete picture of these phase equilibria. Their use follows logically along the conventional lines ; however, several examples and general comments regarding their use are informative.
Equilibria Liquid-Vapor. In simple binary systems, the concentration dependency of the y values can frequently be estimated with fair precision by the conventional use of the thermodynamically con-’ sistent van Laar or Margules relations. The two yo’s which apply to a given binary provide two constants for these relations. These, in conjunction with vapor pressures of the two components, allow estimates of the volatilities of both
components throughout the concentration range. I n more complicated systems, particularly those in which one or both components are associated (76), the concentration dependency cannot be estimated very accurately from the two yo values alone, and additional data must be brought to bear. Even in these cases, however, an interpolation based only on the yo values generally gives a better over-all approximation than does an extrapolation to dilute solutions based on the same number of points in the concentrated regions of the binary. An example of a fairly complicated system, 2-propanol-water at constant temperature, is shown in Figure 1. Here, deviations from Raoult’s law are fairly large and the net association effects give rise to a fairly unsymmetrical system. A simple two-constant Margules prediction of the concentration dependency of the y’s produces an obviously unreasonable result. The smooth curve shown is obtained by the use of a three-constant Margules relation based on the yo’s and, in addition, the composition and boiling point of the azeOtrope in the system. This results in quite satisfactory prediction of volatilities throughout the concentration range. I n connection with this example, it may be explicitly noted that the yo’s alone in principle give the basis for unequivocally predicting the occurrence of an azeotrope-i.e., if yI/yz = p2*/pl* a t some point between the limits 7 1 / 7 2 = yIo/1and y1/y2 = l/yZo. Prediction of composition of an azeotrope from yo’s alone is in general accurate only if the system is reasonably well represented by a two-constant relation. I n systems of more than two components, the yo’s can likewise be used as base points for estimating concentration dependencies of the activity coefficients and volatilities by means of the accepted methods. Frequently these predictions give only a first estimate but are useful as approximate values. For some purposes, the yo’s may be used directly to obtain an idea of volatilities and separation possibilities in a system. In the limiting case of an extractive distillation at high solvent flow, for instance, components to be separated are practically at infinite dilution in the liquid phase. Here, the yo’s apply directly and can be used to calculate limiting volatilities. The volatilities of several series of components at infinite dilution in water are shown in Figure 2 to illustrate this use. Such data provides an immediate idea of the components which can be separated in aqueous extractive distillation. A comparison with similar data for a second solvent gives good basis for choosing between the solvents in so far as selectivity is concerned. The analogous use of yo’sin predicting
1.00
a *
g$
0.80
; 2o zm o
0.60
,y
K O I
d
p
p
0.40
g t
0.20
*g 1. 0
0.8
-$
3
0. 6 0.4
0.2 0.0 -0.2 -0.4
- 0. 6 XL,
MOLE FRACTION 2-PROPYL ALCOHOL IN L l Q U l D
Figure 1. Vapor-liquid equilibria for isopropyl alcohol-water are predicted
efflux times in gas-liquid chromatography and the converse measurement of y o ’ s by gas-liquid chromatographic methods has been discussed (74). Liquid-Liquid. I n binary systems where the y o ’ s are greater thanQ5 or 30 so that mutual solubilities are small, good estimates of mutual solubilities can be made on the basis of the y o ’ s alone, O n the other hand, in systems with more moderate deviation from ideality, concentration dependencies cannot generally be estimated from the y o ’ s alone with sufficient accuracy to predict the occurrence or concentrations of two equilibrium liquid phases. I n multicomponent systems, mutual solubilities are in general much too sensitive to concentration to predict the compositions of two or more equilibrium phases with useful accuracy from yo’s alone. O n the other hand, estimates of the distribution of a given component at low concentration between two substantially immiscible phases can usually be made with fairly good accuracy. Such predictions are frequently quite useful in screening extraction processes or solvents.
107
c
CARBON NUMBER
Figure 2. Volatilities of various homologous series at infinite dilution in water are predicted VOL. 51, NO. 1
JANUARY 1959
101
CXRBOV \ U U B E H
Figure 3. Limiting distribution coefficients, ko, of paraffins and aromatics between heptane and phenol at 7OoC. This pattern can apply to any set phases
A particularly interesting direct application of yo's arises in the extraction of a broad molecular weight range mixture of hydrocarbon types (here defined as homologous series), such as a mixture of paraffins, alkylbenzenes, and alkyl naphthalenes. For the limiting case of a duosol extraction at high solvent and countersolvent flows, all feed components are practically a t infinite dilution. Here, the limiting distribution ratio for any component is simply the ratio of its yo's in the respective phases. The logarithms of such limiting distribution ratios, k o , for paraffins and several homologous series of aromatics between assumed immiscible heptane and phenol phases are plotted against solute carbon number in Figure 3. These plots form a series of straight parallel lines which are described by the solvent dependent constants K1 for paraffins, K E ,B,, and B , for each of the two solvent phases. T h e analytical relations are, explicitly. for paraffins log k "
=
(K1 - Kl')
+ ( B , - B,')??,
and for alkyl aromatics or alkyl naphtheno-aromatics log k 0
=
+
( K , - K c ' ) ( B p - B,')n, i( B . - BoOna f ( B , - &)nn
Tvhere the solute carbon numbers n,, n,, and n, are counted and where the primed and unprimed coefficients refer to those of the respective phases. As is apparent from these y o correlations the simple pattern shown requires in principle but four experimental measurements of yo's for each of the solvent phases (five if the system includes naphtheno-aromatic solutes). Although here applied to pure solvent phases, the same sort of pattern can be expected to apply to any set phases. I n this connection the same kind of pattern has been shown ( 7 ) for partially miscible phases by directly measuring distribution coefficients for solutes between heptane and a number of polar solvents wherein the tu'o solvents exhibit intermiscibilit).. Although the average distribution ratio in the present example, say for total saturates or total aromatics, is not defined until the relative concentrations of
102
the various carbon numbered solute components are given, the pattern does lead to useful generalizations. First, since log k0's form a family of parallel lines (slopes = B , - B p ' ) ,the relative distribution ratio = k i o / k , ' ) for two materials of a given type and given carbon number difference are independent of the type and carbon numberLe., the molecular weight selectivity of the solvent pair does not depend on solute molecular weight or type. Second, for the same reason, the relative distribution ratio between representatives of different types which have the same carbon number does not depend on solute carbon number. Finally, the difference in carbon number between representatives of two types for which the relative distribution ratio is unity also is independent of carbon number. These generalizations lead to a useful means of characterizing a solvent pair. Thus, the limiting relative distribution ratio, a', for two type representatives of the same molecular weight----i.e., the antilog of the vertical separation of the k o lines in Figure 3-can be taken as a measure of the inherent type selectivity of the solvent pair in question. This is distinguished from an average selectivity for a given feed. Likewise, the carbon number difference at, which the relative distribution ratio for two types is unityi.e., horizontal displacement of the t'ico k o lines in Figure 3-can be taken as an inverse measure of the sensitivity of average type selectivity to the molecular iveight spread of the feed material. These two quantities then characterize a solvent pair in so far as selectivity is concerned and give an immediate idea of the value of that pair in separating broad range mixtures. The characterizing quantities for a paraffin-alkyl benzene separation for several polar solvents working in conjunction with normal heptane are: ao
Furfural
Phenol Ethanol Diethylene glycol Ethylene glycol
8.1 5.0 2.3
Anc 11
as in the recovery of aromatics from a concentrate of benzene, toluene. and xylenes. Also, they are suitable for simultaneously extracting aromatics and the lower molecular weight saturates from someivhat broader fractions as in the recover); of a stream of high octane number from a naphtha or gasoline. Acknowledgment
The authors wish to acknowledge the guidance and general support of lLIott SoudeIs, Jr.. and C. L. Dunn, and contributions of D. B. Douslin, A. K. Dunlop, R. J. Evans, R. L. hfaycock, H. -4. Meyer, R. A . Wilson, and P. S. zucco. References (1) Alders, L.: A@$. Sci. Research A4, 171 (1934). (2) Allen, C., ISD. ENG. CHEM.22, 608 (1930). (3) Barker, J. -4., J . Chem. Phys. 20, 1526 (1952). (41 Black, C., TND. ENG.CHEW50, 403-12 (1958).
( 5 ) Broensted, J. N., Koefoed, J., Kgi. Danske Videnskab. Selskab. Mat. jys. LVltdd. 22, No. i7, 1 (1946). (6) Butler, .J. A4.V., Ramchandi, C. K . , Thomson, D. W., J . Chem. Soc. 280, 952 (1935). (-) Carlson, H. C., Colburn, A . P., I N D . EX. CIIEM. 34, 581 (1942). (8)Colburn, A . P., Schoenborn, E. M., Trans. Am. Inst. Chert. Engrs. 41, 421 (1945). (91 Guqgenheim: E. A , , €'roc. ('London) A, 183 (1944).
Roy. Sor.
1~101Hildebrand, J. H., Scott, R . L., "Solubility of Nonelectrolytes," 3rd ed.: Reinhold, New York, 1950. (11) Slackor, E; I,., Rec. trav. chim. 75, 836, 871 (1956j. 112) Palit. S.. J . Phvs.€3 Colloid Chem. 51, 837 (1947): (13) Pierotti, G. J.: Deal, C. H., Derr, E. L., American Documentation l n stitute (1958). (14) Pierotti, G. J . , Deal, C. H., Derr, E. L.. Porter. P. E., J . Am. Chem. Soc. 78, 2989 (1956). \
-
'15) Prigogine, I., Bellemans, A., Disrurrions Faraday Soc. 15, 80-93 (1953). 0..Kister. A. T.. IND.ENG.
1.2 5
11.4
7
25.0
6
33, 160 (1937). ' (19) Tompa, H., Zbid., 45, (101) (1949). (20) Waals, J. H., vander, Hermans, J. J., Rec. trao. chim. 6 8 , 181 (1949); Jbid., 69, 949 (1950). (21) Wohl, Kurt, Trans. Am. Inst. Chem. Engrs. 42, 215 (1946). ~
As a rough rule, An0 decreases as a o increases. To obtain a reasonably high average selectivity between two types in a broad range oil, a solvent is chosen which represents a compromise between high a' and An'. From these magnitudes, it is clear that furfural and phenol represent solvents with fairly high An' but yet reasonably high a'. Likewise, in spite of high inherent type selectivity, diethylene or ethylene glycols have relatively low AnO's and are not useful for extracting broad range fractions. O n the other hand, in so far as selectivities are concerned, these two glycols are good solvents for extracting aromatics from narrow molecular weight range fractions
INDUSTRIAL AND ENGINEERING CHEMISTRY
RECEIVED for review October 21, 1957 ACCEPTEDJune 23, 1958 Material supplementary to this article has been deposited as Document No. 5782 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington 26, D. C. 4 copy may be secured by citing the document number and by submitting $21.25 for photoprints or $6.23 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.
