Kinematic Contraction and Expansion of Chromatographic Bands M. Jaffar and M. A. Khan, institute of Chemistry, University of Islamabad, Islamabad, Pakistan
A fairly generalized account of the effect of variation in the carrier gas velocity in the retentive zone on band kinematics is given. The analysis presented shows that the conventional definition of “ideal” column with zero band broadening becomes untenable unless the additional condition of neglecting the effect of retention on flow is augmented with the classical set of conditions. It is theoretically shown that some bands expand whereas others contract on separation.
The study of the size and shape of chromatographic bands is not only important from the view point of separation but also for obtaining accurate thermodynamic functions and other relevant data by chromatography. The width of a chromatographic band is ordinarily stated to depend upon the mode of injection, the carrier gas flow profile, equilibrium lag between the two phases dependent on the sorption and desorption kinetics, irreversible (diffusion etc.) effects, and the nature and mode of dispersion of the stationary phase. A clear exposition on molecular basis of these factors is given in detail elsewhere by Giddings ( I ) . Theoretically, rectangular bands of constant width are produced in an “ideal” column, usually defined as a “packed” medium which assumes instantanous equilibrium (zero transport free energy), absence of irreversible effects, plug flow, and uniformity of “packing.” These coupled with the requirement of plug injection are not only regarded as the necessary but also sufficient conditions to ensure resolution of the mixture into sharp rectangular bands, each having the same width as that of the initial unseparated “mixed-band.’’ Apart from practical realization of these conditions, there are conceptual difficulties, especially the very existence of equilibrium in a flow system, the departure from which has been quantitatively discussed previously by one of the authors ( 2 , 3 ) . Even admitting that the above conditions may be realized practically in the limit, the separated bands having absolutely the same width as that of the initial “mixed band” cannot be achieved in the so-called “ideal” column because of the unrealistic assumption that the mobile phase velocity remains constant across a band. The retentive zone can be pictured as a constriction in the chromatographic column where the gas must move faster to make up for the loss of vapor in the liquid phase if the column is to operate a t a constant flow rate. This was first visualized by Bosanquet and Morgan ( 4 ) , who studied the effect of finite concentration on the rate of solute migration. Using a dynamical diffusion model of the (1 J. C. Giddings in “Dynamics of Chromatography.” Marcel Dekker, New York, N.Y., 1965. (2) M. A. Khan, Nature (Londoni. 186, 800 (1960). (3) M . A. Khan in “Vapor Phase Chromatography,” International Symposium, Hamburg, 1962. (4) C. H. Bosanquet and G. 0. Morgan in “Vapor Phase Chromatography,” D. H. Desty. Ed., Academic Press, New York, N.Y., 1957, p 35.
1842
chromatographic process, they showed that finite concentrations could cause asymmetry in chromatographic peaks even if the concentrations were smaller than those required to produce linear isotherms. Mathematically, a more refined treatment of the problem was given in a subsequent publication by Bosanquet (5), showing a more exact agreement between the theory and experiments. However, in each case, the analysis was confined to a single-component system which has no direct application in practical chromatography except to provide a fundamental understanding of the unsymmetrical character of peaks. Golay (6) worked out a correction factor to allow for the effect of finite concentration on the retention volume, in which he showed that the actual retention volume is always smaller than that calculated on the basis of zero concentration. He provided a criterion on the basis of which maximum permissible sample size, consistent with the precision required in the measurement of the retention volume with negligible concentration effect, can be calculated. Peterson and Helfferich (7) gave an analysis of the sharpening and nonsharpening of chromatographic bands due to variations in the flow velocity arising from sorption and desorption of solute vapor in the band region. They showed that symmetrical peaks may be expected even with nonlinear isotherms. They derived a general expression for the retention volume for a multicomponent system, but the generalized formulation is regarded inadequate because of the lack of a unique relationship between the concentration differentials of the various components necessary to define a stable boundary. Bearing in mind the importance of the concentration factor on band width (the stress hitherto placed on the retention volume), the present analysis offers a different approach for investigating the effect of finite concentration on band width. It is the purpose of this paper to evaluate, in the first instance, the contribution of velocity variations, arising from retention of solutes, on band width and then to specify conditions under which this effect will be minimum. A fairly generalized formulation of the kinematics of partially overlapped and separated bands of a multicomponent system is presented, although the complete solution is worked out only for a binary system. The complexity of the situation does not permit incorporating in the generalized expression the effect of diffusion as has previously been done in the simplest possible case of a single-component system.
FUNDAMENTAL EQUATIONS FORTHE BAND KINEMATICS Let A be the effective cross-sectional area of the gas phase of the column, operating a t the temperature T and the pressure P. Further suppose that C, and x i represent (5) C. H. Bosanquet in “Gas Chromatography,” D. H. Desty, Ed.. Academic Press, New York, N.Y., 1958, p 107. (6) M. J. E. Golay, Nature /London). 202, 489 (1964). (7) D. L. Peterson and F. Helfferich, J. Phys. Chem.. 69, 1283 (19651.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
'2
:
I
I
!
I
I
I
li
I
I
I
I
I
'2
I
I
!
I
'1
1'
'3
'3
Figure 2. The stage of development just at the separation of
components 1 and 3
the concentration and mole fraction, respectively, of component i (mol wt = wi) in the gas phase of the composite band. The same letters bearing a prime will be used to denote the corresponding quantities for the liquid phase, so that Ki = C',/C, (partition coefficient of component i between the two phases) and C, = alxl,where ai = hU,/
&, = ( A + K , A ? a , { A ( r ; r p + l + 6 r P + J ( x L+ 6 ~ ~ ) "+' P'I 1-1
-
c n-1
Q, =
(AC,'P)+ A'C',(P))rprp+1+ ( A C L ' n+ )
P=i
A'C','"')r,fl
+ 2 (AC,'"''') + A'C',(n-9')f9fq+1(1) 9-1
6f9+l)(Xl
+
(IC)
6x,i:n-q'}
On comparing Equations l b and IC, under the condition that 6 x 0 for infinitesimal displacement, we have
RT. Let RLbe the migration factor, representing the velocity of the band of component i relative to that of the carrier gas. Further suppose that a sample containing n components (with Rl < Rn,where i = 1, 2, . . . , n - 1) is introduced into the column as a plug. As the sample travels down the column, the initial stage of development can be represented by the various zones situated between the rear of the most retentive and the front of the least retentive components, where r, and f, represent the rear and front of component i, respectively. After an infinitesimally short interval of time 6t, the rears and fronts of all the bands will be slightly shifted to new positions as indicated by the dotted lines in Figure 1, the band r,fl for the ith component occupying the position r',fll. Let 6r, and 6 f l denote the infinitesimal shifts of the rear and front of component i, respectively, with x L ( p ) representing the mole fraction of component i in the band region containing 1 to p components and x l ( n - q ) representing the mole fraction of the same component in the region containing all but 0 to q components. If Q 1 is the quantity of component i in the mixed band r l f l , with the rear ri and the front f l situated at given points of the column at the time t , then it is obvious from Figure 1that
+
C(fq'fq+l 9-0
The redundant terms 6r, + 1 and 6qo in Equations l b and IC have been introduced to get the mathematical expression in a compact form. Recalling that 6rp/6t and 6fq/6t (when 6t 0) represent the instantaneous velocities of the rear of component p and the front of component q, to be denoted by Upr and Upf,respectively, expression Id can be written in the form
-
TWp' P"'
r-1
+ ccuq' - U f q + l ) X , ( n --q0i -
urp+l)x!lpi
(2)
q=o
A partially resolved chromatogram of a three-component system will consist of five concentration zones, for which, on applying Equation 2, we have
On further development, when the rear of component 3 has overtaken the front of component 1, i.e., when r3 and f1 exchange their positions, the situation can be represented as shown in Figure 2. Under this condition, the set of equations in (3) takes the form (ulr - u2')x,0+ (Uzr - V l f ) X 1 ( 1 . 2=) 0
Or, on substitution of K , for C',/C,, expressing concentrations in terms of mole fractions, and further putting
+
r p r p + l = rprIp r n f l = rnrIn fqf,+l
=
+
fqf'q
r ' p r p + l= 6 r p
r ' , f l = 6rn
+
f'qfq+1
+
+
r',fl
= Sf, +
loa)
rtprp+l
f'qfq+1
the right-hand side of Equation 1 can also be written in the form
Q, = ( A
For a binary system, the set given in (4)is simplified to give the velocity concentration relationships for components 1and 2 as
+ K , A ) a ,{ c ( 6 r p + r;r,+Jx;" + P-r
and
F(6f9+ f 4 f 9 + l ) x t ' n - q i } Ob)
q-0
After the infinitesimal displacement down the column, the whole gamut of concentrations may, in the general case, change from x , to x c + 6 x L .As there are no sources or sinks, either of physical and/or of chemical nature, which may produce or consume any of the components, the quantity of component i shall remain the same. Hence, the quantity Q Lfor the band r'$, may be expressed as
Equations 5 and 6 show that velocities of the rear and front of component 1 relative to the velocity of the rear of component 2, or for that matter, the velocities of the front and rear of component 2 relative to the velocity of the front of component 1, are in the inverse ratio of their concentrations.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
1843
VELOCITIES IN THE MIXED (RETENTIVE) REGION T o maintain continuity of flow, it is necessary that the amount of matter flowing past any point of the column must be the same. It implies that the carrier gas must move faster in the retained regions containing solute molecules than in the solute-free zones. Thus, if Uco and Ucm were the respective velocities of the carrier gas in the free and mixed region containing n components, then according to the law of conservation of matter
Because of the physical symmetry of the binary system, Equations 13 and 14 can be obtained from each other by simply interchanging subscripts 1and 2. The values of the variables X I and x z can be obtained effectively by solving Equations 13 and 14 through the determinants. Thus, the simultaneous solution for XI yields
n
Ucopo=Uc"(p"
+ CRipi) 1
(7)
The symbols po and pm refer to the densities of the carrier gas in the free and mixed states, respectively, whereas p L and Ri are the density and the rate of migration of component i relative to the velocity of the carrier gas. The left-hand side, when multiplied by the area of cross section and time, gives the amount of fluid flowing a t any point in the solute-free region, whereas, in the same way, the second term on the right-hand side refers to the amount of the solute vapor and the first term that of the carrier gas passing in the retentive region. Or, expressing in terms of mole fractions of the species involved
U," =
UCO
1-
k(1- R J x ,
(8)
1
1-
whereby the velocity of component i in the mixed region can be written as -n I,
U," =
fi,uc
0
1 - g(1- R,)x,
(9)
By symmetry or otherwise
(16)
VELOCITIES IN TERMS OF THE SEPARATED CONCENTRATIONS I t is more convenient, as will be seen later, to express the velocity of the carrier gas in the mixed region in terms of the concentrations of the separated bands. For a binary system, the carrier gas can have three velocities, Uc(1s21, U c ( l ) ,U c ( 2 ) referring , to the mixed-band, the first-band, and the second-band regions, respectively. Combining Equations 8, 15, and 16, the value of Uc'lqzl in terms of the separated concentrations is given by
U,"."
=
U C o [l (1 - R J x ? - (1 - Rz)x; -
I'l
MIXED CONCENTRATIONS IN TERMS OF SEPARATED CONCENTRATIONS In order to calculate changes in the band width, it is required to express mixed concentrations in terms of the separated concentrations. Thus, combining Equation 5 with 9 for a binary system, we obtain for component 1 XI -xlo R1[1 - (1 - R z ) x ~-] Rz[1 - ( 1 - Ri)x1'] - R i ( l -
R1)~cl (10)
which on simplification can alternatively be expressed in the reduced form z ] Rz[1 - ( 1 - R1)xio](11) -xi- R1[1 - ( 1 - R z ) ~ x10 R1 - R2[1 - ( 1 - Rl)xiO] Similarly, either starting from the equation for x z / x z O given in (6), or by simply interchanging subscript 1 and 2, we have l -X Z -- Rz[1 - (1 - R J x J - Rl[1 - ( 1 - R 2 ) ~ z ~(12)
Rz
- R z + R2(1 - Rl)xlo]xlo (13) R ~ ) X ~ '+ X , [Rz - Rl + R l ( 1 - R 2 ) x z 0 ] x 2= [Rl
-
[R2 1844
n
The velocity of the carrier gas in the region where it is mixed only with component 1 is obtained by putting xz0 = 0 in the above expression. Hence
ucO 1 - ( 1 - R1)X1°
(17a)
Similarly, putting x10 = 0 in the same expression, the velocity of the carrier gas mixed with component 2 only is given by
Ucf2)=
ucO 1 - (1 - R,)x,O
(1%)
On multiplying Equation 17 in turn by the migration factor R1 and Rz, the velocity of the front of the first band (VI') and that of the rear of the second band (Uz') in the mixed region are calculated as
- Rl[l - (1 - RJ~z01
Rearranging Equations 11 and 12, we obtain the pair of simultaneous linear algebraic equations in x 1 and x p . [Rl - Rz + Rz(1 - R ~ ) x ~ O + ] X R~l ( 1 - R2)xl0x2 =
Rz(1
IT
U,"'
(Ri - R,)[1 - (1 - RJxiO]
x20
or
-
R,
+
Rl(l
-
On multiplying (17a) by R1 and (17b) by Rz, the migration velocities of the rear of the first band (VI') and the front of the second band (Uzf) in the separated regions can be written as
R z ) x z 0 ] x Z 0 (14)
ANALYTICAL CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
TI
U1' =
n
fi1uc-
1 - ( 1 - R1)x1°
(17e)
and
KINEMATIC EXPANSION AND CONTRACTION OF BANDS The above analysis shows that due to retention, affecting the flow velocity, concentrations in the separated zones in general differ from those in the mixed zone. I t follows in the light of the conservation of matter that separated bands must undergo changes in width, the mixed and separated widths being in the inverse ratio of the respective concentrations. If L were the initial width of the mixed band, and L1 and Lz widths of the first and the second bands in the separated states, then
On substituting the value of x L / x ~ Oand xz/xzO from Equations 15 and 16 in the above equation, we obtain
R,(1 - R J x,O)L - R, - R,
(18)
Translating the deductions from the analysis of the binary system consisting of one retained (R1 < 1) and one unretained ( R z = 1) components, the concentration x l 0 in the separated region a t any stage of development must be equal to the concentration x1 in the mixed region. Thus, the rear and front of the retained component should travel with the same speed throughout the column length. This is in harmony with our physical understanding that the unretained component plays the same role as the carrier gas and hence the retained band should behave as if it were effectively moving in the atmosphere of the carrier gas alone. Moreover, because of the invariant character of x l and the unretentive role of component 2, it is physically expected that the relative concentration of the latter in the mixed region must remain constant. Using the results of the present theory, where XI = x1O and L z / L = x z / x z 0 = 1 - xl0, we find that xz/(l - X I - xz) = xz0/(l - X Z O ) . It is evident that on separation (XI = 0 ) , component 2 must acquire a higher value, causing thereby contraction of the unretentive band to conserve matter. It will be noted, though only for a simple system, that the mathematical construction and analysis of the suggested model of chromatography fit in consistently with the predictions and consequences based on commonly recognized concepts. At least, the above example offers a check on the correct formal working out of the analysis presented. Taking the typical values of R1 l13, Rz l/~, and x1O = x20 10-2, we find from Expression 18 that the first band expands by about 1% whereas the second one contracts by about 2%. Although the expansion or contraction may not be substantial, it is not too insignificant to be ignored in very accurate chromatographic work.
-
-
-
As in each case the second term within brackets on the right-hand side represents a positive quantity, it is clear that for a binary system, in which both the components are retained by the stationary phase, the first band (more retentive) expands whereas the second band (less retentive) contracts on separation. Although band expansion in a nonideal column was recognized as a common phenomenon since the very birth of the technique, the contraction SINGLE BAND WITH DIFFERENT of bands can also be conceived. The quantities [(R1(1 CONCENTRATIONS R z ) ) / ( R z - R ~ ) I xand z ~ [(&(I - R d ) / ( R z - R d l x ~may ~ We can visualize a single-component band or simply a be termed as the coefficients of kinematic expansion and single band with two or more concentrations of the same contraction of chromatographic bands. As the retention of species. This may, in principle, be realized by placing tocomponents forms the basis of chromatographic separagether plugs of different concentrations in immediate contion, these coefficients will never vanish altogether, even tact so as to leave no gap between the various fronts and if infinitely dilute samples were used. I t is interesting to rears. note that each of the coefficients depends upon the retenConsider, for example, a simple system of a single band tive powers of both but the concentration of one compocharacterized by two regions B1 and Bz only, the common nent only, the expansion of the more retentive band deline of contact serving as a rear for Bz and front for B1. pending upon the concentration of the less retentive one We have two possibilities to consider, that is to say, that and vice versa for contraction. B1, as compared to Bz, is either a t lower or higher concenThe expansion in one band will be equal to the contraction in the other band when x1O/xZo = [Rz(l - R1)]/[R1(1 tration. In the former case, the rear of the composite band BI will tend to lag behind, as the lower concentrations - R z ) ] ,that is to say, when the partition coefficients are travel more slowly than the higher concentrations, wherein the ratio of the separated concentrations. The requireas the front of the composite band Bz will move faster to ment for no expansion in the first band demands that x10 stretch itself. The overall result is a single elongated ho= 0 or Rz = 1. The former (x1O = 0) refers to the initial mogeneous band. On the other hand, if B1 is a t higher mixed state of complete overlapping which is obviously of concentration, then , because of its faster movement, it no practical interest, whereas the latter (Rz = 1) signifies will tend to overtake BO,with the result that a superposed that to maintain L1 = L, the second component must shorter but more concentrated band is resulted. have no avidity for the stationary phase. Hence, the more One of the novel features of the present analysis is that retentive band will migrate without any expansion, proit demonstrates the existence of a single band a t different vided the second band were unretained in the column. concentration, then, because of its faster movement, it Further, putting RZ = 1 in Equation 18, we have LZ = (1 of such a band. To illustrate, we shall start with a binary - xlo)L, which shows that when expansion in the retensystem of the type in which the fronts of components 1 tive band is zero, the contraction of the unretentive band and 2 travel a t the same speed, so that the two bands increases with its mole fraction but is independent of the progress without ever getting mixed with each other. On retentivity of the first band. Furthermore, Rz > RI and RI or RZ < 1 , the inequality Rz(1 - Rl)xlo + R1(1 - Rz)xzO equating the expressions for r;lf and U z f , we obtain the following condition under which the front of the first will > 0 will always hold. Hence, no chromatographic separajust accompany the rear of the second without any intertion can be visualized in which the two bands, whatever val between them their retentivities and concentrations, will have the same length. R , = R,[1 - ( 1 - R,)x,O] (19) ANALYTICAL CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
1845
which on rearrangement may be written as
tions x10 and X I . On putting x10 and X I , in turn, equal to zero, Equation 22 reduces to the expressions
or (19b) Substituting Equation 19a in the expression for xl/x10 given in Equation 15, we get
(20) NOW1 - (1 - Rz)xz0 = Rz[l + i(1 - R z ) / R 2 l (1 which on combining with Equations 19b and 20 gives
1 - (1 - R,)x,O = ( R , / R , ) [ l - (1 - R , ) x , l
~ Z O ) ] ,
(21)
Combining Equations 17d and 21, we finally obtain LTxi'bxi
=
R1U;
11 - (1 - R l ) x l ] [ l- (1 - R 1 ) ~ l O (22) ]
This constitutes a band occurring in two different concentrations of the same component, where U X I ~ * X repre~ sents the velocity of the boundary between the concentra-
which are the velocities of component 1 a t concentrations and x l O , respectively. Thus, from theoretical considerations, a single band a t different concentrations can exist, although with time and the downstream passage, the concentrations produce a single homogeneous band of longer or shorter length depending on whether XI > or < x1O. On applying the condition RI = Rz[l - (1 - R I ) X I ~to ] the expression for x~ given in Equation 16, we arrive a t x z = 0. Thus, a single band in two concentrations may be produced in two ways, either by placing two plugs of different concentrations in contact or by letting a two-component band travel down the column and then a t an appropriate time withdrawing instantaneously from the mixed region all the molecules of one of the species which may be regarded to act as a kinematic catalyst. x1
Received for review April 6, 1972. Accepted February 22, 1973. One of us (M. J.) wishes to express thanks to the University of Islamabad for the award of a research fellowship.
Gas Chromatographic Degradation of Several Drugs and Their Metabolites Albert0 Frigerio, Kenneth
M. Baker,
and Giorgio Belvedere
lstituto di Ricerche Farmacologiche "Mario Negri," Via Eritrea, 62 20157 Milano, ltaly
The gas chromatographic behavior of some drugs and their metabolites has been examined, particularly with regard to decompositions and, hence, misidentification. The structures of decomposition products were elucidated using gas chromatography-mass spectrometry, or in some cases collection and identification. Drugs which have been studied are oxazepam, N-methyloxazepam, nitrooxazepam. o-chlorooxazepam. carbamazepine, carbamazepine-10.11-epoxide, 1 0 , l l -dihydroxycarbamazepine, and floropipamide. Each of the drugs oxazepam, ochlorooxazepam, and nitrooxazepam underwent a decomposition to give 4-phenylquinazoline-2-carboxaldehyde derivatives while N-methyloxazepam is stable. This reaction also occurred on heating the compounds to 200 "C. Carbamazepine partly decomposes when injected as a methanol solution to give iminostilbene and 9-methylacridine. Carbamazepine-10.11 -epoxide decomposes completely to 9-acridinecarboxaldehyde. while the same degradiation product is obtained from 10.11-dihydroxycarbamazepine. Floropipamide undergoes a thermal dehydration of the amide position of the molecule to give a nitrile. Mechansims for most of the reactions are discussed.
Gas chromatography (GLC) is widely used for separating complex mixtures arising from biological sources ( I , 2 ) ; however, the technique suffers from the fact that it 1846
permits determination of only one property of a compound uiz. its retention time. Since many compounds may have the same retention time for a given set of conditions, the exact identity of a material cannot be established. Hence GLC is more valuable when it does not give a peak, rather than when it does; however, when this concept is used, it is possible to have a misidentification of drugs ( 3 ) . The rearrangement or degradation on the gas chromatographic column of the material under investigation can also give false information concerning the presence of a molecular species. The combination of GLC with mass spectrometry overcomes the problem of positively identifying drugs and their metabolites present in a biological system ( I , 4-9). Hammar, E. Holmstedt, J.-E. Lindgren, and R . Tham. Advan. Pharmacoi. Chemother.. 7. 53 (1969). (2) S. P. Cram and R. S. Juvet. Jr., Anal. Chem.. 44. 213R (1972). (3) L. R. Goldbaum, E. H. Johnston, and J. M . Blumberg, J. Forensic Sci.. 8 . 286 (1963). (4) C. Merritt, Jr.. "Applied Spectroscopy Reviews," Vol. 3, Marcel Dekker, New York, N.Y.. 1970, p 263. (5) E. C. Horning and M . G. Horning, J. Chromatogr. Sci.. 9. 129 (1971). (6) C. J. W. erooks. "Mass Spectrometry," Vol. 1. 0 . H. Williams Ed.. The Chemical Society, London, 1971, p 288. (7) G. A . Junk, Int. J , Mass Spectrom. / o n Phys., 8 . 1 (1972). (8) A. L. Burlingame and G. A. Johanson, Anal. Chem.. 44. 337R (1972). (9) C. J. W. Brooks, A. R. Thawley, P. Rocher, E. S. Middleditch., G. M. Anthony. and W. G. Stilwell. Advan. Chromatogr. Proc. Int. Symp., 6th. 262 (1970). (1)
ANALYTICAL CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
