The Determination of Group Contributions to Activity Coefficients by Gas-Liquid Chromatography through Homologous Trends William A. Scheller,* J.
L. Petricek,'
and G. C. Young2
Depart?iient of Chemical Engineering, The 7'niversity o j Sebraska, Lincoln, Kebr. 68508
A rapid and accurate method for determining group contributions to liquid phase activity coefficients for a binary system of groups from measurements of net retention volumes b y gas-liquid chromatography has been developed. Group contributions determined b y this method for four different pairs of groups at a number of different temperatures were used to calculate 1 10 equilibrium compositions for 1 1 different binary systems of nonelectrolytes. These were compared with experimental equilibrium data and indicated an overall absolute average deviation of 5.9%. Of the 1 10 deviations, 19 were greater than 10% and of these, three were greater than 30%. It also appears from these studies that the partial residual energy of mixing at infinite dilution i s a constant for a given solute in a homologous series of solvents.
T l i c generalization of \-npor-liquic1 and liquid-liquid equilibi~iunidata. to permit tlie cstinintioii of equililxiuni conitenis for n-hich 110 reliahlc data e s k t is of considerable importance to tlie clieniicnl engineer. In 1959, Pierotti, Ileal, and n e r r (Pierotti, et al., 1959) puhlislied a correlation for estimating binary liquid pliase activity coefficients a t infinite dilutioii from a iiunibcr of acniien~pi~ical interaction coiistaiits a n d the number of hydrocarbon carbon atoms in each of tlie tIvo componeiits, thus quantifying tlie effects of lioniologous trends on activity coefficient.: a t infinite dilution. l'heir correlation implies tliat a gloul) coiitribution approacli to these calculations should be feasible. -4 niethod for estimating liquid phase activity coeficiciits from grouij coiitril~utionshas been discussed and illustrated by Kilson and Ileal (1962) and Scheller (1965). Ileal and Derr (1968, 1969) have continued t o publish inforinatioi~on the correlation of zictivity coefficients ill t e r m of p ~ r a m e t e r s associated with pairs of structural groups. One of tlie liinitations to a broader application of this grouli contrihutio~i nietiiod u p to now lias been tlie lack of sufficient data to establish complete sets of groulj contribution curaes;. This paper deicrihes a niethod 113- which a gas-liquid chroniatograph inaj- be used to establish these curves rapidly and accurately.
(3) (4) i;
Z&7p"
=
(5)
0
E'ioni tliese equations we shall no^ draw some conclusions c~oiiceniinglRiRo aiic~ BrG. If n-e consider the isotlitrinnl residual energy a t infinite dilution! AR~",of $1 solute i witli tIic f o r m u ~ aA4m~3n in a liquid solvent, L, with the forniula A l l tlieii ZA = 1.0 and as c:iii be seen from Figure I , RAG= 0 mid RBG= R B . Substituting these values into eq 3 gives
+
wliere f;TAG*and ABG* arc evaluated a t Z A = ? i i / ( m n ) (see Figure 1). However, this meaiis that iu this system RiRo/RT is independent of I , tlie niiniber of -1groups in the liquid A I , and depends only on m and 12. Tlius the effect of the liquid - 1 l on the infinite dilution activity coefficient of L4nc137L in .4i must be completely accouiit,ed for b y the Flory-Huggins entropy a t infinite dilution, i.e.
Theory
Five equations discussed by Sclieller (1965) are used in the determination of group contributions from activity coefficient, data or activity coefficients from grouli contributions. These are
ABiR -
111 y z. -- ~-
RT
1
~
R
(1)
Alt'liough this coiicliisioii points out the rather crude basis of this group contribution t,lieory, we shall see below that this does hold moderately well) a t least for a variety of compounds in n-paraffins. Let us again consider the solute X,B, a t infiiiit,e dilution in a liquid solvent A l . If we differentiate eq 3 n-ith respect t o m (m = T L A , ~at' ) constant' T ! P, I , and n ( n = n B , i ) and then apply eq 5 , we obtain the relationship
Preient addresis, Dow Chemical Co., Torrence, Calif. Present addre-, Continental Oil Co., Poiica City, Okla. Ind. Eng. Chem. Fundam., Vol. 11, No. 1 , 1972
53
Substituting eq 7 and 9 into cq 1 and rearranging then gives ARiRn ~
RT
=
[-VI. Vi
In (V,P,O) - In
Vi] +
+
[hi (wzRT)
111
VI, - 11 (10)
If the liquid coating, L, on the chromatographic column is a straight chain coinpouiid -IL, if tlie elution gas is a very short chain of A groups, aiid if tlie solute (sample) has tlie formula A,13,, t,heii eq 10 may be differeidatcd wit,li respect to rn at constant^ T , P, I , and n aiid substituted into eq 8 to give the relationship RAG*
RT -
H :* RT
0
2, =rn/(rn
GROUP F R A C T I O N
'K'I N
+n 1
1.0
SOLUTION
Figure 1. Typical group contribution-group tionship for A and 6
fraction rela-
-
a 111 (P,oVN) btn
Thus the group coiitribution-group fractioii relationship for A may be calculated with the aid of glc net retention volume (V,) measurements for a homologous series of ?nof compounds A,,& and from infornlation on t,he molar volumes of the same series a t the glc temperat'ure. Experimental evidence indicates that for inaiiy compounds the quantity I n (PioVX)is quite linear wit'li respect t'o m and n (see Figure 2). W i e n this is true, we may write
In ( P t V N ) = m P A
+ nPB
(12)
Similarly, the liquid molar volume also can often be written as a linear function of m and n
Vi
-3I 1
t
1
1 -I
9L
7 1
0
I I
1
I
I
4 5 NO O F C H 2 GROUPS
6
7
I
2
I
I
3
Figure 2. Relationship between In (PtoVn) and the number of CH, groups in the homologous series of: 0,n-primary alcohols; 0, n-alkylbenzenes a t 150°F; and 0, n-alkyl bromides at 150°F.
+
again wliere RAG*is evaluated a t Z A = m / ( m n ) . But for the conditions described, the value of is a part of the curve of RA' us. ZA (see Figure 1). Thus the group contribution-group fraction relationship for A in a biliary system of groups, A aiid 13, can be established with the aid of experimental data on t'he variation of A n i R o with rn and eq 8. The relationship for the group contribution of 13 may then be established t'hrough the integration of eq 5. We shall next discuss holv gas-liquid chromatography (glc) may be used to evaluate eq 8. It has been shown by many authors including Porter, et al. (1956), t,hat from the principles of glc the infinite dilut'ion act,ivity coefficient for the gas solute, i, in the liquid solvent (column coating), L, is giveii by t,he relationship ln 54
yio =
ln (wzRT) - In (PioVN)
Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 1, 1972
(9)
=
7nV.4 f nVB
(13
If eq 12 and 13 are differentiated with respect to m a t coiistant T , P, 1, and n and subst'ituted into eq 11 upon rearrangement we have RAG*
___ RT
-
[PA
-
vA/vL]
+ VA/Vi
(14)
This equation, however, is not consistent with the required behavior of the group contribution relationship; Le., RG*/RT must be zero when rn = m (ZA= 1.0). Rather, eq 14 shows t,hat when rn = , RG*/'RThas a constant value of ( P A VA/VL). This is most likely due to the failure of eq 12 and 13 over a very large range of m. I n order to retain the advantages of the linear eq 12 and 13 but also t o assure the proper value of R A ~ * / R at T m = m, eq 14 was multiplied b y the empirical factor Bn/(m n) t o give the working equation
+
where C = P A - VA/VL.Since the value of ~ A ~ * / depends RT only on T , P, ZA,A , and B and not on the chromatographic systeiii used, C must be a constant if T , P , A , and B are constant. Differentiation of eq 15 and substitution into eq 5 then gives the second relationship
Experimental Equipment
Net retention volumes, VN,were determined experimentally in the usual manlier with the aid of a Perkin-Elmer Model 154-C chromatograph equipped with a thermal conductivity cell and a Vestronirs ;\lode1 S5.i strip chart recorder. The
Table 1. Comparison of Activity Coefficients and Residual Energies of Mixing at Infinite Dilution in Two Different Solvents at 50°C In
-(LO
_ _ _ _ _A>,R'/RT
._
Dev,
Solute H-(CHz),-B
CL8H38
Ethanol Propanol Methyl formate l l e t h p l acetate .Ic et one Propanol
2 2 1 0 1 0
68 53 18 99 46 98
C24H 10
2 2 0 0 1 0
52 36 97 78 25 80 Av
~
Dev,
%
C18H38
6 0 6 7 178 212 144 183 14 1
357 323 202 163 216 169
CZ4H50
%
363 1 7 327 1 2 203 0 5 163 0 0 216 0 0 172 1 8 Av 09
the average was never more than l.8yc. This, nlong with the precision of the carrier gas rate inemireinelit, indicates that the precision of the net retention volume is probably between k2.Oy0 and k2.5y0, F o r methanol a t 50°C there might be a n additional error in t h e activity coefficient' in the order of 5% to 10% resulting from unmeasured surface adsorption effects, This error, if present, would be considerably less for the components other than methanol and for all components a t temperatures above 50°C. Complete det'ails of the equipment, experiment,al procedure, and all data and calculation results were reported b y I'etricek (1966) and Young (1967). Discussion of Results
rate of the carrier gas (which was 99.99 mol yc CHI) was measured to kO.2 cni3,/min (about 0.5y0)with a calibrated r o h n e t e r . Oven temperature was controlled to =kO.l"C and nitasured to *0.2"C wit,h a calibrated copper-constaiitaii thermocouple and a Leeds and Northrup semiprecisiori potentionieter. Pressure a t the column outlet was atmospheric preswre as meawred on a mercury barometer to *0.1 nixn pressure and column inlet pressure was measured by n gauge to k 0 . 2 psi. Two columns were prepared for use wit,h this equipment. The first, which was used at 5OoC, contained n-octadecane on 20 mesh tin, while the second, which was used between 75 and 15OoC, contained n-tetraeosane on 20 mesh tin. Tin was chosen as a packing material in order to minimize surface adsorption effects which would tend to increase the net retention volume. Five measurements of net retention time were made for each sample a t each temperat'ure and the average value was used. The maximum deviation of any one measurement from
To test the conclusion drawn from eq 6; i.e. ARtR0/RT is independent of the number of A groups in the liquid r l l , net retention volumes at 50" C were determined individually in n-octadecane, H-(CH2)18-H, for six different compounds having the formula H-(CHz),-13. VN was determined for the same six compounds in n-tetracosane, H-(CH&-H, at 75, 100, 125, and 15OOC and these values were extrapolated to 50°C (log V N us. l / T is linear). The log yio and ARiRo/RT were t'hen calculated and are compared in Table I. AnfR0/RT in bhe bwo solvents is found t o be reasonably constant with a n absolute average deviation between solvents of 0.9% whereas the average deviation of In yzObetween solvents is 14.1%. T o further test this conclusion, values of yio for ten different compounds in five normal paraffins a t 40, 60, and 100°C were calculated from the values of AniR'/RT which were obtained using the chromatograph and t,he n-tetracosane column. The 4OoC values of V X were obtained b y extrapolation. These values of yio mere then compared with experimental values published in the literat'ure as shown in Table 11. Again the results are moderately good with a n absolute
Table II. Comparison of Experimental y$ from the literature with Values Calculated from Glc AHiRo Y%O
Solute
Solvent
Acetone
n-Cd34
Butanol
n-C&6
2-13utanone
n-C6H14 n-Cl& n-C~H34
Ethanol
n-C~H34 n-C20H42
Ethyl formate Methanol Methyl acetate 3-Pentanone
n-Cl& n-C& n-C16H34 n-C16&1 n-CdL n-C16H34
Propanal Propanol a
n-C1& n-C&31
Temp, 'C
40 60 100 60 100 100 60 100 60 100 60 100 60 100 40 60 100 40 60 100 60 100 40 40
Lit.
Calcd
5.15 4.2 3.1 13.6 6.2 2.8 3.5 2.8 2.9 2.3 13.5 6.0 12.0 5.3 3.2. 29.6 9.3 3. l a 2.9 2.3 2.6 2.1 3.45 19.5.
5.4 4.3 2.8 12.5 7.0 3.2 4.0 2.7 3.1 2.2 14.4 6.9 12.2 6.2 3.2 28.4 9.1 3.1 3.4 2.5 2.5 1.8 3.4 17.5
These data are from Kwantes and Rijnders (1958). All other data are from Pierot,ti, et al. (1959).
A
0.3 0.1 -0.3 -1.1 0.8 0.4 0.5 -0.1 0.2 -0.1 0.9 0.9 0.2 0.9 0.0 -1.2 -0.2 0.0 0.5 0.2 -0.1 -0.3 0.0 -2.0 __Absav A 0.47
Ind. Eng. Chem. Fundam., Vol. 11, No. 1 , 1972
55
m
Table 111. Constants for Eq 17 A
B
-CHz-
-OH
-CHz-
-c 1
-CHz-
-B r
-CH1-
-C6Hs
T, OC
VB/VA
50 75 100 125 150 50 75 100 125 150 50 75 100 125 150 150
1.377 1.618 1.494 1.616 1,731 2.502 2.956 3.509 4.558 4.573 2.265 2.386 2.542 2.713 2.863 5.948
For all of the compounds studied, n = 1. In this case = ZA/(1 ZA)and eq 15 can be written as
-
C
0.3318 0.2600 0.1564 0.1605 0.1319 0.1070 -0,1900 -0.1018 -0.1131 -0.0737 - 0.0502 -0.1221 -0.1019 -0.0999 -0,0809 -0.1032
Equation 16 tnay also be modified to give
average difference between experimental and calculated 72's of 0.5. Let us now turn our attention to the evaluation of eq 15. Using methane as a carrier gas, values of V Nwere obtained at 75, 100, 125, and 15OOC in n-tetracosane for homologous series of normal primary alcohols, chlorides, bromides, and alkylbenzenes and a t 50°C in n-octadecane. The values of PA were then determined from eq 12 by linear regression using literature values of P2. The values of VB/VA for the above five temperatures were obtained by least-squares fits of literature values of Vi to eq 13. Values of VL a t these temperatures are also available in the literature. The values of C as well as VB/VAare tabulated in Table 111. The value of p was found by evaluating eq 15 for methanol (m = n = 1, C = 0.3318) at 50°C using the value of gAG/RT at ZA = 0.5 in the table published by Scheller (1965). The calculated value of p was very close to 2.0 and it was decided to use the value = 2.
Comparison of experimental and calculated vapor compositions a t the same liquid composition for eleven systems are contained in Table I V . I n the point-by-point comparison of the 110 experimental and calculated values of vapor compositions, 19 values deviated more than 10% and of these 3 deviated inore than 30%. It is especially interesting t o note that the alcohol-water systems were predicted with fair accuracy even though no water data were used in evaluating the constants in eq 17. The two methanol-water systems in Table IV contain the three deviations greater than 30%. If these points are ignored the deviations are considerably less (about 6% and 4%). As discussed by Scheller (1965), water is counted as 1.6 OH groups for purposes of determining the group fraction and calculating ARR/RT for water. If one recognized that B In y?/Brn = B ln (PtOV,)/bm,then some of the yo's published by Harris and Prausnitz (1969) will be useful for establishing group contributions for other systems than those reported here. Conclusions
In view of the results reported above, it appears that for the systems studied, the partial residual energy of mixing a t infinite dilution can be considered to be constant for a given solute in a homologous series of solvents. It is further concluded that it is possible t o obtain a complete group contribution-group fraction curve from glc net retention volume data on a homologous series of solutes. An important con-
Table IV. Summary of Comparisons of Calculated Vapor Phase Compositions with literature Data System
Temp, "C
Methanol-water Methanol-water Et hanol-water 1-Propanol-water Ethanol-n-heptane 1-Propanol-n-heptane 1-Butanol%-heptane 1-Butanol-n-decane n-Butyl bromide-n-heptane n-Butyl chloride-n-heptane Benzene-t oluene
100.0 150.0 50.0 49.9 50.0 75.0 50.0 100.0 50.0 50.0 150.0
No. of data points
10 9 9 8 12 15 10 12 11 9 5
Liq comp range, mol
%
Abr av dev,
3.5-94.6 4.4-95.3 4.6-90.8 9.0-82.0 3.1-97.7 3.0-98.0 6.9-78.3 13.0-99.0 11.7-95.2 15,8-95.7 12.7-76.0
9.24 11.48 3.76 1.96 5.88 8.47 2.41 5.45 5.17 3.73 5.45 5.91
%
Footnotea
la la 2 lb 3 4 IC
4 Id le 5
~
Av
a Chu, J. C., Shu, L. W., Sherman, L. L., Rajendra, P., "Vapor-Liquid Equilibrium Data," J. W. Edwards, Publisher, Inc., Ann Arbor, Rlich., 1956: (a) p 420; ( b ) p 482; (e) p 323; (d) p 324; (e) p 127. Jones, C. A , , Schoenborn, E . 11, Colburn, A. P., Ind. Eng. Chem. 35, 666 (1943). 3 Smyth, C. P., Engel, E. W., J . Amer. Chem. SOC.5 1 , ?,660 (1929). Lee, L. L., Scheller, W. A., J . Chem. Eng. Data 12, 497 (1967). 5 Chi, J. C., Getty, R. AI., Brennecke, L. F., Paul, R., Distillation Equilibrium Data," Reinhold, New York, N. Y., 1950.
56 Ind.
Eng. Chem. Fundam., Vol. 1 1 ,
No. 1, 1972
sequence of this is that it, in effect, permits one to use infinite dilution activity coefficients obtained from the chromatographic data to predict activity coefficients at finite concentrations through the medium of the group contributions.
GREEKLETTERS
fl
=
an empirical constant in eq 15
y z = liquid phase activity coefficient of component i $ %= x z V z / % c i V z= volume fraction of component i
Nomenclature
A I3 C&HiR HkG
a structural group in the molecules A,B, and hl a structural group in the molecule A,B, a t e r m i n e q 15 = PA - VA/V, = partial residual energy of mixing of cpmponent i = group contribution of group k to AHiR 1 = number of groups of type -1in the molecule A l L = refers to the liquid solvent A i 711. = number of groups of type A in the molecule A,B, n = number of groups of type 13 in the molecule X,B, nk,i = number of groups of type k in molecule i PA = contribution of group A to ln (PIOVY) = contribution of group 13 to ln (Pi0VN) PB Pto = vapor pressure of component i R = gas constant ASiFH = Flory-Huggins partial excess entropy of mixing for component i T = temperature V A = contribution of group A to V i = coiitribution of group 13 t o V i VB V i = liquid molar volume of component i = liquid molar volume of component L VL = chromatographic net retention volume VN = nioles of L in the chromatographic column Tt’ = mole fraction of component. i in t’heliquid phase = gas compressibility factor = group fraction of group in a solution of d and 13 ZA Z k = group fraction of group k in a solution of groups = = =
2
SUPERSCRIPTS
0
*
= =
at infinite dilution in the standard state
literature Cited
Deal, C. H., Ilerr, E. L., Ind. Eng. Chem. 60, 28 (1968). Deal, C. H., Ilerr, E. L., Inst. Chem. Eng. Syrnp. Ser. N o . 3.2 3, 40 (1969). Harris, H. G., Prausnitz, J. PI., J . Chrornatogr. Sci. 7, 685 (1969). Kwantes, A., Rijnders, G., in “Gas Chromatography 1958,” p 125, 11. H. Desty, Ed., Butterworths, London, 1958. Petricek, J. L., M.S.Thesis, The University of Nebraska, Lincoln, Kebr., 1966. Pierotti, C. J., Deal, C. H., Derr, E. L., Ind. Eng. Chem. 51, 95 i l 9 . 3 \ I-
Porter, P. E., Deal, C. H., Stross, F. H., J . Amer. Chem. SOC. 78, 2999 (1956) Scheller, W. A., IND. EXG.CHI~M., FUNDAM. 4 , 459 (1965). Wilson, G. W., Deal, C. H., IND.ENG.CHEM.,FUNDAM. 1, 20 (1962). Y&ngJ’G. C., M.S.Thesis, The University of Nebraska, Lincolii, Yebr., 1967. for review October 26, 1970 ACCEPTED September 14, 1971
ItEcicIveD
Dispersion of Chbrged Particles in a Turbulent Air Stream under Transverse Flow Conditions Shan K. Sunejal and Darsh T. Wasan” Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Ill. 60616
Equations have been developed to predict the diffusion of electrically charged particles in a turbulent air stream within a circular straight porous pipe with uniform transverse flow at the walls. The calculated concentration profiles agree well with those determined experimentally in the absence of electric charge on the particles. In the experiments, an aerosol of 5-11 particles of potassium chloride was introduced from a point source located at the axis of a fully developed turbulent flow of air in a 6-in. diameter pipe, and timeaveraged concentration distributions were measured a t four distances downstream. Average velocities covered a range from 9.55 to 23.0 ft/sec (corresponding to entrance Reynolds numbers of 26,800-66,000) with transverse injection velocities ranging from 0 to 0.1 70 ft/sec. The highest injection volume corresponded to 29.0% of that of the main stream. The plume width of the dispersing aerosol i s affected b y the transverse flow. The effective dispersion coefficient and the radial mass flux for charged and uncharged particles increase with air injection and decrease with suction through the pipe walls. Also, for increasing values of aspect ratio (length/diameter), they increase with injection and decrease with suction.
A n understanding of the way gases and particulate matter diffuse from a point source in turbulent streams is important for the aerodynamic design of air pollution control equipment - Present address, Research and Development Department Amoco Chemicals Corporation, Stal~dardoil Research Center: Naperville, Ill. 60540.
and for numerous other applications in industry, in agriculture, and in public health. The movenient of particles in a turbulent field has been the subject of many studies, such as those by Tchen (1947), Hughes and Gilliland (1952), Corsin and Lumley (1956) , Lie (1956), So0 (1956), Lumley (1957), Friedlander (1957), Hinze Ind. Eng. Chem. Fundom., Vol. 11, No. 1, 1972
57
