ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
LITERATURE CITED
1693
(11) W. Dunaes. Anal. Chem.. 49. 442 (1977). (12) w. &,.A. eyer,. IK Male;, M. M~~IIw, k. Pietschmann, a.plachetta, R. Sehr, and H. Tuss, Fresenius' 2. Anal. Chem., 288, 361 (1977). (13) W. Dunges and N. Seiler, J . Chromatogr., 145, 483 (1978). (14) W. Wendelin and F. Knotz, Monatsh. Chem., 103, 1632 (1972). (15) F. Knotz, Sci. Pharm., 38, 227 (1970). (16) G. Gubitz, in preparation. (17) H. Bethke, W. Santi, and R. W. Frei, J. Chromtcgr. Sci., 12, 392 (1974).
A. Wnert and K. H. Wssler, Fresenbs' 2.Anal. Chem., 267, 342 (1973). W. Dunges, Chromatographia, 6, 196 (1973)., U. Hintze, H. Roper, and G. Gercken, J . Chromatogr., 67, 481 (1973). H. Ehrsson, Acta Pharm. Suecica, 8, 113 (1971). E. 0.Umeh, J . Chromatogr., 56, 29 (1971). 1. R. Politzer, G. W. Griffin, E. J. Dowly, and J. L. Laseter, Anal. Left:, 6, 539 (1973). Regis Chemical Company, Morton Grove, Ill. D. R. Knapp and Sh. Krueger, Anal. Lett., 8, 603 (1975). M. J. Cooper and M. W. Anders, Anal. Chem., 46, 1849 (1974). H. D. Durst, M. Milano, E. J. Kikta, Jr., S. A. Conneliy, and E. Grushka, Anal. Chem., 47, 1797 (1975).
RECEIVED for review February 14, 1979. Accepted June 4, 1979.
Estimation of High Pressure Liquid Chromatographic Retention Indices John K. Baker Department of Medicinal Chemistry, School of Pharmacy, University of Mississippi, University, Mississippi 38677
time may vary several orders of magnitude because of changes A method for the prediction of the retention times on CIB in the composition of the mobile phase ( 3 ) . It has also been reverse phase columns is developed and applied to a series observed that the substitution of a cyano reverse phase column of propranolol, anthranilic acid, and barbiturate analogues. The for a reverse phase column has little effect on the retention retention properties of the drugs are measured uslng a retention index of most compounds even though the act,ual retention index scale that is based on the relative retention of a series times on the two columns were quite different. Because of of 2-keto alkane standards. The retention index ( I ) of the test these properties, the retention index scale is very useful in compounds is estimated using the equation I = 2007~i- Ire, providing a uniform basis for reporting retention data and for where Ire, is the observed retention index of one reference correlating chemical structure with retention properties. compound that is structurally related to the other compounds The retention index scale was also constructed in such a and 7~ is the sum of the Hansch substituent constants for the manner that most probably it would be linearly related to the test compound. lipophilicity of the compound. Since the lipophilicity of drugs and other compounds can also easily be estimated in a linear manner ( 4 ) , it was anticipated that the two concepts could be combined to form a very simple method for the prediction of the HPLC retention properties of a compound based on an analysis of its chemical structure.
One of the major difficulties in any chromatographic method is the prediction of the retention time or retention volume of a new compound or a derivative of an old one. In the area of gas chromatography, the Kovats retention index scale has become widely used to standardize the reporting of GLC data and as a tool for the correlation of GLC properties and chemical structure ( I , 2). In the area of high pressure liquid chromatographic (HPLC) analysis, qualitative estimations of the retention times of compounds can be readily made for both adsorption and partition columns. However, little progress has been made in the development of methods for the precise prediction of the retention times based simply on the chemical structure of the compound. Recently a retention index scale suitable for use with reverse phase (fi-Bondapak c18 and fi-Bondapak CN) has been reported (3). The retention index scale is based on the relative retention times of a series of 2-keto alkanes. By definition, acetone is given a value of 300 and 2-butanone, 400; etc. A given column-solvent combination is calibrated by chromatographing the 2-keto alkane standards (C3-CZ3) and correlating the logarithm of the observed capacity factors in a linear manner with the defined retention indices. The retention index of a given drug or other test compound is then obtained by a mathematical interpolation between the values of the 2-keto alkane standards. I t has been found that the retention index of a given compound remains nearly the same even though its retention 0003-2700/79/035 1-1693$01.OO/O
EXPERIMENTAL Chromatographic Conditions. A 3.9 mm i.d. X 30 cm CIS reverse phase column (fi-Bondapak CIB,Waters Associates Inc.) with a 10-pm particle size was used for the study. The mobile phase flow rate was 2.0 mL/min and was prepared using 6.6 g K2HP04,8.4 g KH2P04,1.6 L CH30H, and 2.4 L HzO. The pH of the mobile phase was 7.0 before the addition of CH30H. A Waters Associates Inc. M-6000 pump, U6K injector, and Model 440 dual wavelength ultraviolet detector (254 nm and 280 nm) were used. Though the dual wavelength detector was not essential, measurements of the 254 nm/280 nm absorbance ratio greatly facilitated the identification of the drugs in the mixture with the 2-keto alkane standards (5). Materials. The 2-keto alkane standards (C3-C& were obtained from Analabs and the barbiturates from the Theta Corporation. The anthranilic acid derivatives were synthesized (6) by R. F. Borne, Department of Medicinal Chemistry, School of Pharmacy, University of Mississippi. The propranolol analogues were synthesized in these laboratories (7). The methanol used in the mobile phase was freshly distilled while all other chemicals were of reagent grade and were used as obtained. Measurement of Retention Indices. The capacity factor (12') of the drugs and standards were determined from the observed retention time ( t R ) using Equation 1. k'= (tR-tO)/tO
(1)
The retention index of a given 2-keto alkane standard was by
C
1979 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979 I300
1200
I100
IOoo 900
5
800
1 1 1
obtained earlier in these laboratories where it was found that log k’of the barbiturates was linearly related to the log of the partition coefficient (5). Since the value of k’ was directly related to the stationary phase/mobile phase partition coefficient, it was expected that both log k’ and I would linearly related to log P if the distribution mechanism of the barbiturates in the chromatographic column followed the same free energy relationship as the distribution between the octanol and water phase in the partitioning experiments. That is, log P would be linearly related to the partition coefficients through Equation 4 and the retention index would be related to the partition coefficients in the form shown in Equation 5.
b
0
I -
I
E
700
b-
w 600 (r
500
k’ = K (Vs/Vm)
-
log k’= a’ log P
-3
+ log (VJV,)
I = a log P + p A
/O
IO0
(3)
-2
-1
0
LOG
I
2
3
P
Flgure 1. Correlation between observed retention indices and octanollbuffer partition coefficients. (0)barbiturates, (0)propranolol
analogues, (A) anthranilic acid analogues. The retention index measurements were all made using a mobile phase of a pH 7.0 buffer and 40% methanol. The log Pvalues for the propranolol series were made at pH 7.0; the anthranilic acid series, pH 8.0; and barbiturates measurements were made for the free acid form definition equal to 100 times the number of carbons in the compound. Thus, 2-butanone was assigned a value of 400. The retention index (0 of a given drug or other test compound was calculated from the observed capacity factor for the drug ( k b ) , the capacity factor for a 2-keto alkane standard eluting just before the test compound (kh),and the capacity factor of the next higher homolog ( k k+J using Equation 2.
Measurement of Octanol/Water Partition Coefficients. The details of the measurement of the octanol/water partition coefficients of the propranolol series is described elsewhere (7). Basically, the method consisted of the measurement of the distribution of the drug between an octanol phase and a pH 7.0 phosphate buffer as measured by ultraviolet spectroscopy. The partition coefficients of the anthranilic acid series (6) was determined in a similar manner; however, a pH of 8.0 was used. The majority of the partition coefficients of the barbiturates have been previously published (8)and the values were determined for the nonionized form of the drug. The remainder of the partition coefficients were calculated using Hansch additivity constants ( 4 ) .
RESULTS AND DISCUSSION In earlier studies with the barbiturates, it had been found that the logarithm of the retention time was linearly related t o the octanol/water partition coefficient ( 5 ) . In the case of the barbiturates, it was observed that the retention index of the drug was linearly related to the octanol/water partition coefficient (log P ) and that the results were very close to that of the 2-keto alkane standards (solid line in Figure 1). These results were very similar to those
The anthranilic acid derivatives were also found to follow the same type of relationship (Figure 1). The retention indices of these compounds were also linearly related to the partition coefficients and the slope of the curve was essentially the same as that for the 2-keto alkane standards. The HPLC measurements were made with a p H 7.0 aqueous buffer-methanol mobile phase while the octanol-buffer partition measurements were made at pH 8.0. While the compounds are largely anionic under both of these conditions, the slightly lower p H would cause a larger fraction of the compound to be in its free acid form and therefore would have a higher retention index. This could then account for the slight displacement of the anthranilic acid series curve from the 2-keto alkane standards. The propranolol series also was found to be linearly correlated with the experimentally measured partition coefficients with the exception of compounds P-10 and P-12 (Table IV). All the compounds in this series contained at least one cationic group, while compound P-10 and P-12 each Contained two cationic groups. The most likely explanation for the slight upward displacement for the monocations (Figure 1) and the marked displacement of the dications was that anionic silanol sites (9) contribute to the binding of these compounds in addition to the partitioning mechanism. I t is possible to minimize this type of interaction either by silanization of the residual sites or by the addition of tetrabutylammonium chloride to the mobile phase. In some of the earlier literature, i t has been erroneously stated that log k’is linearly related to log P with a slope of unity (9, I O ) . This would be equivalent to a’ = 1.0 (Equation 4) which would be true only if the two phases in the HPLC column were precisely identical to the two phases in the classical skake-flask experiment. From previous studies in these laboratories (3) and by was near to unity Tanaka and Thornton ( I I ) , the value of CY’ only when the mobile phase was pure water and the value of a’ approached zero in an exponential manner as the percent of organic solvent modifier in the mobile phase was increased. Though experimentally determined log P values could be used to predict retention indices following a regression analysis of the data in Figure 1with Equation 5, this was not the major objective of the study. The objective of the project was t o develop methods of predicting the HPLC retention properties using a minimum of experimentally derived parameters. The basic approach that was taken to accomplish this link was to relate the experimental log P values of the 2-keto alkane standards t o their defined retention index values. The addition of each CH, unit to the 2-keto alkane standard increases the log P value by 0.50 unit and the log P value of 2-butanone is 0.29 (4). Using these data, the log P of any member of the series could be calculated using Equation 6 where N is the number of carbons in the series. In the 2-keto alkane standard series, the retention index (IN) of a specific
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
1695
Table I. Estimation of the Retention Indices of Barbiturates Using Equation 10
P
barbiturate barbituric acid barbital allobarbital phenobarbital
R, H H H H
R, H CH,CH, CH,CH=CH, CH,CH,
R, H CH,CH, CH,CH=CH,
metharbital apro barbital butabarbital
CH, H H
CH,CH, CH,CH=CH, CH,CH,
CKCH, CH(CH,), -CHCH,CH,
cyclobarbital
H
butethal butalbital mephobarbital
H H CH,
CH,CH3 CH,CH, CH,CH,
X
-Q
CH,CH,CH,CH, CH2CH(CH312
hexobarbital amobarbital pentobarbital
H H
retention index obsd calcd
log Pa
.b
0 0 0 0
53 423 511 523
23 423 503 571
-1.35 0.65 1.05 1.42
-2.00 0 0.40 0.77
0 0
532 552 600
535 523 583
1.21 1.15 1.45
0.56 0.50 0.80
0
604
665
1.86
1.21
0 0 0
610 616 624
623 583 689
1.65 1.45 1.98
1.00 0.80 1.33
0
648
677
1.92
1.27
CH,CH, CH,CH,
CH,CH,CH(CH,), -CHCH,CH,CH,
0 0
681 686
683 683
1.95 1.95
1.30 1.30
0
728
723
2.15
1.50
I
secobarbital
H
CH,CH=CH,
CH3 -CHCH,CH,CH,
thiopental
H
CH,CH,
CH3 -CHCH,CH,CH,
S
732
893
3.00
2.35
thiamylal
H
CH,CH=CH,
CH3 -CHCH,CH,CH,
S
760
933
3.20
2.55
methohexital
CH,
CH,CH=CH,
CH, -CHC=CCH,
0
776
731
2.19
1.54
I
I
I
I
CH, a
Log of the octanol/water partition coefficient of the nonionic form. The majority of the values are taken from ref. 8.
While a few were calculated using the standard substituent constants. member was defined as being equal to 100 times the number of carbons. These two relationships were then algebraicaly combined to give Equation 8 which could be used to predict the retention index of some new compound if it was assumed that it interacted with the HPLC column in the same manner as the standards. log P = 0.5 N
-
1.71
= 100 (N) (7) IN = 200 log P 342 (8) Though Equation 8 could be used to estimate the HPLC retention index of some new compound, the accuracy of the estimate would be limited by the accuracy of the log P prediction. Though log P values could be estimated ( 4 ,it was easier to utilize relative partition coefficients obtained through Hansch substituent constants (T values). The estimated retention indices in Tables I, 11, and IV were calculated using Equation 10 where rX is the Hansch substituent constant, many of which were available in the literature ( 4 ) and Iref was an experimentally obtained retention index of a parent or structurally similar compound. P, = log P, - log Pr,f (9)
+
+ Iref
Table 11. Estimation of the Retention Indices of Anthranilic Acid Derivatives Using Equation 10
(&',.;a -
(6)
IN
I, = 200 P,
Log P x - log Pb&,iM.
(10)
In the barbiturate series (Table I), barbital was used as the reference compound. The calculated values of the retention index of the remaining barbiturates were in good agreement
compound A-1 A-2 A-3 A-4 A-5 A-6
X
H F NO,
CH, C1
Br
retention index obsd calcd 530 565 586 606 656 678
530' 560 518 634 670 734
logPa
vb
-0.46 0.03 -0.03 0.05 0.54 0.83
0.15 0.24 0.52 0.70 1.02
0
a Log octanol/pH 8.0 buffer partition coefficient. Data from ref. 6. Standard aromatic substituent constants, ref 4. Reference compound for the series.
with the experimentally observed values even though their retention times varied by over two order of magnitude in the series. The average error in the predicted retention index was 43 units while the median error was 29 units. The largest errors were observed for the two thiobarbiturates. These two compounds eluted more quickly than estimated, which may be due to their slightly higher acidity. (thiopental pK, = 7.3, secobarbital pK, = 7.78 (12).) Since the mobile phase was buffered at pH 7.0, a slightly larger fraction of the thiopental
1696
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
shake-flask method (Table IV) and was similar to the results obtained by Tanaka and Thornton (11). The two values obtained for the protonated amine group were self-consistent; however the values were more positive than would be expected for an ionized species. The quaternary ammonium group was found to have a R value (-0.02) that was considerably more positive than the R value obtained by conventional partitioning measurements (-5.70 ( 4 ) ) . These results were consistent with those that had been observed for the two propranolol analogues that were doubly charged (Figure l). With both the two propranolol analogues and the cations in Table 111, the unusually high affinity for the stationary phase was most probably due to binding with free silanol sites. The retention indices of the propranolol analogues (Tabie IV) were estimated using the standard T values for most of the drugs, but the x values from Table I11 were used for the ionized groups. There was a good agreement between the predicted and the observed retention indices, though the errors were greater than seen in the two previous series. The two dicationic drugs, P-10 and P-12, were retained by the column more strongly than predicted. I t was also noted that the experimentally measured octanol/water partition coefficient for P-12 was 1.8 log units higher than calculated. Thus it would appear that the poor accuracy of the retention index predictions for these two compounds may not be due to deficiencies in the method per se, but were the result of deficiencies in the partition coefficient estimations. The fundamental reasons for the anomalous behavior of the dicationic compounds as compared t o their monocationic counterparts are not known. In summary, the prediction of the HPLC retention properties based on the use of the retention index scale and Hansch substituent constants proved to be very reliable in most cases. Protonated amines were found to be more strongly bound to the HPLC column than would be expected from experimentally measured partition coefficients. App:, 3nt R values for the amines and other ionic groups were e; v i mentally determined by HPLC and these values were used to predict the retention index of other compounds. Caution should be used in the application of the x value for protonated amines when columns from different sources are used. From experiences in these laboratories, the remainder of the R values should be applicable to other reverse phase columns, regardless of source, but variabilities in residual silanol sites on the column may produce variations with the cations.
Table 111. Apparent R Values of Ionized Functional Groups COmQQUnd
TTa
R e f . Compound
-0.75
+
@CH2CH2"
3
(o>
-0.80
CH2CH3
+ DCHzN (o>(CH3)3
.o. 02
CH3
Values were calculated using experimentally obtained retention indices and Equation 10. would have been ionized, thus it would have eluted more quickly. The agreement between the calculated and the observed retention indices for the anthranilic acid series (Table 11)was found to be even better than with the barbiturate series. The average error in the retention index was found to be 22 units and the median error was 14 units. In the anthranilic acid series, the carboxyl group of the compounds would have been nearly completely ionized; however this did not interfere with the estimation of the retei,+inn index because this was a constant factor for each of the compounds. The introduction of a new ionized group would have created more difficulties because x values for imized groups were not readily available. In the propranolol series that was studied, a number of the drugs contained carboxylic acid groups, quaternary ammonium groups, and amine groups in addition to the secondary amine group that was present in each of the drugs. Since these values were not available, the retention indexes of a number of reference compounds were measured and these were used to calculate apparent x values using Equation 10 in a reverse manner (Table 111). The R value obtained for the carboxylate group (-2.81) was close to the value that had been obtained by the
Table IV. Estimation of the Retention Indices of Propranolol Analogues Using Equation 10 c
compound P-1
P-2 P-3 P-4 P-5 P-6 P-7 P-8 P-9 P-lo P-11 P-12
R,
CLI / / \l
RZ
R3
(CH,),COOH CWCH,),
H SO,NH,
CH(CH3)Z
? "
H H H H H H H H H H Br H
CH(CH,), (CH*),CO", (CH,),OCH, CH(CH3)?.
(CH?),CH3 CH(CH3)Z
(CH?),"H, CH(CH3)2
(CH?.
)3"(CH3)3
OH H H H H C1 H Br H
- R,
retention index obsd calcd 546 561 590 667 720 901 912 950 1,047 1,104
1,215 1,316
438 548 666 778 610 858 912c 1,052 1,054 798 1,256 948
log P a 0.21
-0.51 -0.27 0.39 0.31 1.28 1.08 2.08 2.00 -0.19
Rb
-2.61d -1.82 -1.23
-0.67 -1.52 -0.27 OC
0.70 0.71 -0.57e
2.71
1.72
-1.89
O.lBf
Refa Log octanol/pH 7.4 buffer partition coefficient. Data from ref 7. Standard substituent constants, ref. 4. erence compound for the series. Calcd using n ( C 0 0 - ) = -2.81 and R (branching) = 40.20. e Calcd using n(-N'H,) = -0.77 and n (branching) = 4 0.20. f Calcd using n (-N+(CH3)3) = -0.02 and R (branching) = +0.20.
ANALYTICAL CHEMISTRY, VOL. 51, NO. 11, SEPTEMBER 1979
LITERATURE CITED Kovats, E . Helv. Chem. Acta 1958, 4 7 , 1915. Kovats, E. Adv. Chromatogr. 1965, 1 , 229. Baker, J. K.; Ma, C. Y . J . Chromatogr. 1979, 169. 107. Tute, M. S . Adv. Drug Res. 1971, 6 , 1. Baker. J. K.: Skelton. R. E.: Ma. C. Y.J . Chromatoor. 1979. 768. 417 Borne; R. F.; Peden, R. L.; Waters, I. W.; Weiner, M ;: Jor-din, R:; Coats, E. A. J . Pharm. Sci. 1974, 63, 615. (7) Rauls, D. 0.; Baker, J. K. J . Med. Chem. 1979, 22, 81. (8) Hansch. C.; Steward, A. R.; Anderson, S. M.; Bently, D. J , Med. Chem. 1967. 7 1 . 1. (9) McCall, J ' M . J . Med. Chem. 1975, 18, 549.
(1) (2) (3) (4) (5) . . (6)
1897
(IO) M i n k s , M. S.; Moulton, S.J.; Murphy, C. T.; Taylor, P. J. J. Med. Chem. 1976, 19, 615. (11) Tanaka, N.; Thornton, E. R. J . Am. Chem. Soc. 1977, 99, 7300. (12) Sunshine, I. "Handbook of Analytical Toxicology"; The Chemical Rubber Co.: Cleveland, Ohio, 1969; Table I.
RECEIVED for review January 29,1979. Accepted June 1,1979. This work was supported in part by the Research Institute of Pharmaceutical Sciences, School of Pharmacy, University of Mississippi, University, Miss.
Pulsating Column Separations with a Polyurethane Foam Syringe Tibor Braun" and Stefan PalCgyi' Institute of Inorganic and Analytical Chemistry, L. Eotvos University, 1443 Budapest, P.O.6. 123, Hungary
A pulsating column technique based on the resilience of open-cell polyurethane foam is described. After loading with a suitable hydrophobic organic reagent, a foam cylinder is packed in a conventional medical syrlnge (pulsating column) and made to pulsate in the solution containlng the species to be separated. Based on theoretical consideratlons, two experimental arrangements for a new separation and preconcentration method have been developed. The value of the method is demonstrated by the separation of "'I and *03Hg from large volumes of aqueous solutions.
Recently the use of open-cell polyurethane foams for the separation and preconcentration of various metal ions and organic compounds has attracted considerable attention ( I , 2). Because of their membrane-like structure, foamed plastics exhibit excellent sorption, mass-transfer and hydrodynamic properties, which permit their utilization in rapid separation procedures. Open-cell polyurethane foams used as resilient column filling offer a further possibility of application in analytical practice. A polyurethane foam cylinder impregnated with a suitable hydrophobic organic reagent and placed into a conventional medical syringe can be easily compressed and released by moving the plunger of the syringe. Forced pulsation of a foam syringe, with its tip in a solution, brings the liquid, containing a substance to be separated, into repeated contact with the immobilized reagent in the foam. The resilience of open-cell plastic foams (e.g., polyurethane) is a unique property, which is not found with any other kind of sorbent or column filling. Some time ago a similar technique was employed in the determination of Ce(IV), V(V) and FetIII) via a redox reaction with a pulsating foam filling (3). A recent study was dedicated to a special version of the same technique ( 4 ) . This paper describes a study on the pulsating column technique for the development of a general separation and preconcentration method applicable in trace analysis. The relevant theoretical principles have been worked out and confirmed for the separation of I 3 l I and *03Hgfrom aqueous solutions. '00leave from the Radiobiological Laboratory, Faculty of Sciences, P. J. S a f i r i k University, 04167 K o i i c e , Czechoslovakia. 0003-2700/79/0351-1697$01.00/0
THEORY Assuming that equal fractions of solution are drawn into and discharged from the foam-packed syringe, the process can be performed in open or in closed experimental arrangement. These are schematically outlined in Figure 1. In the open arrangement, after reaching the equilibrium, each fraction of the sample is rejected; in the closed setup, it is pressed back into the sample reservoir. Separation is performed by repeating the draw/discharge cycle. This cycle represents one pulsation. The equations for the calculation of separation efficiency are derived by assuming ( 5 , 6) that the ith fraction, W , = W o / n ,successively taken from the total sample volume W , with a solute concentration of mw,o(or a radionuclide with a radioactivity concentration of and pulsing through the syringe, comes into equilibrium with the reagent-loaded foam cylinder at the ith stage of separation. Further, it is assumed that in each stage the solute to be separated is distributed between the foam filling (0 and the aqueous phase (w) so that mf,+/mw,, = KD (or af,,/uw,, = FD)( 4 , 51, where KD is the distribution ratio and i = 1, 2, 3, . . ., n the sequence of separation stages. The concentration of solute mw,oin each volume W, drawn into the syringe is constant throughout the process. During the separation, it decreases to m,,,. The concentration of solute in the foam after the ith stage is mf,+ and for i = 0, meo = 0. It is also assumed that the volume of the relaxed (Vex)and compressed foam cylinder (V,) is also constant. Evidently, W , = Vex - V,. The ratio W o / V p= p represents the maximum volume concentration. Using the open arrangement, the volume of liquid in the reservoir decreases to 0 after the i = n stage. In the closed arrangement the volume of liquid remains unchanged and the separation process can be terminated after the nth stage. Open Arrangement. For the ith stage, the following balance equation is valid: (1) Vphf,+ - mf,,-l)= WL(%V,O - mw,,) Solving this equation for mf,,, after the last (nth) stage, an expression for mf,,can be obtained: n
mfsn
= P'mw,o
5p
i- 1
If the separation efficiency ( E n , o p ) is defined as En,op = mf,n/p.mw+o, the following equation can be derived for the 0 1979 American Chemical Society
