Frontal Analysis Chromatography Calculation of Flow Rate Changes Due to Solute Sorption, and Retention Characteristics of Sharp Fronts G. J. KRIGE’ and VICTOR PRETORIUS Department of Physical and Theoretical Chemistry, University of Pretoria, Pretoria, South Africa If the concentration of a sample continuously injected into a chromatographic column is sufficiently high, considerable amounts of the solute are removed from the fluid stream by As a result the flow the sorbent. velocity of the mobile phase behind each front differs from that ahead of it. Expressions are derived for the mobile phase flow velocity in each region of the column in which the fluid stream has a different composition, assuming that the fronts are sharp and that the distribution isotherms are nonlinear. They are used to calculate the retention volumes and times of the constituent fronts of a binary mixture of solutes, and to obtain expressions for the fluctuation, due to solute sorption, of the flow velocity of the mobile phase.
T
use of frontal analysis as a chromatographic method when highly diluted solutes are continuously injected into a column has received considerable attention in the past ( 1 , 2 ,14-1 7 , 2 1 , 2 2 ) . These studies were based on the assumption that the distribution isotherms of the solutes between the stationary and mobile phases are linear, and that changes in the flow rate due to solute sorption (3, 4, 18, 22) are negligible. An increase in the solute inlet concentration affects the retention times and widths and heights of solute fronts, as well as the concentration profile which is observed when the column is flushed with an inert fluid (1-6, 8, 12, 22, 27). These concentration effects are normally ascribed to nonlinearity of distribution isotherms ( 1 , 2, 6, 6, 8, 12). Schay and his associates (9--11,26,23)and others (3, 4, 7, 13, 18, 24-26) have, however, pointed out that additional mechanisms, which may be designated as sorption effects, are also of importance. When considerable amounts of the sorbate are removed from the fluid stream by the sorbent, the flow velocity of the mobile phase behind a solute front exceeds that ahead of the front (18, 22). Changes in the temperature (24) and pressure (25) have also been observed. If sorption HE
1 Temporary address, University of British Columbia, Vancouver 8, Canada.
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ANALYTICAL CHEMISTRY
effects are taken into account, chromatographic processes may be classified according to whether or not they are ideal, the distribution isotherm is linear, and sorption effects are negligible. As a rule, only the first two of these three criteria have been employed in the literature, and even when the role played by sorption effects has been incorporated, results which are valid only under certain limiting conditions have been obtained. Boeke and Parke (1, 2 ) considered linear, nonideal chromatography a t low concentrations when sorption effects are negligible. Claesson (6, 6) and Glueckauf and Coates (12) investigated the situation arising when the distribution isotherms are nonlinear, but also neglected sorption effects, whereas Bosanquet and Morgan (3, 4) studied sorption effects, but assumed the isotherms to be linear. Only Schay and his associates (9-11, 22, 23) have attempted to treat nonlinear, nonideal chromatography by frontal analysis when sorption effects are significant, but they too have made a number of simplifying assumptions which considerably limit the application of their results to situagons arising in practice. An extensive study of frontal analysis is undertaken in the present series of papers. No restriction is placed on the inlet concentrations of the solutes, and the role played by additional effects which arise when highly concentrated samples are injected is investigated in detail. For simplicity, mixtures of more than two solutes are not considered. The extension of the results to take multicomponent mixtures into account involves only mathematical manipulation, and new concepts need not be introduced. Furthermore, the mobile phase is treated as incompressible. Unless the contrary is explicitly stated, theoretical sections of the study are applicable to liquid as well as gas chromatography, provided that volume changes due to changes in the composition of solute-diluent liquid mixtures can be neglected. The symbol X is used to denote mole fractions in gas chromatography and volume fractions in liquid chromatography-i.e., the fraction of the volume of the mixture
which is associated with a givei ponent. For convenience, thc “mole fraction” is, however, used. It is also assumed tk sorption process is isothermalheat liberated on sorption mediately dissipated. The may, however, easily be made a to the adiabatic sorption of since the wass distribution ( k X , will be uniquely determin value of the mole fraction, X is dissipated on sorption. T‘ need ther2fore only be re1 plot of k X against X for ad tion. Thc r y l t s obtainer will probably lie somew: those predicted for isothe on the one hand, and adia on the other hand. In this paper a theor made of the rate of flow analysis, and its effect ( times and volumes 0, The treatment is kept as general as possible by considering solute mixtures diluted with an inert fluid, which for convenience is referred to as a carrier. The term is used in the usual chromatographic sense (8). MASS BALANCE EQUATIONS
The mass balance equations a t each front boundary under the conditions given above have been derived by Schay (22). His expressions are, however, valid only when the distribution isotherms of the solutes are linear. This assumption is not made in the present treatment. Single, Undiluted Solute. Consider first the simple example of a single, undiluted solute, continuously injected into a packed chromatographic column which a t that stage is filled with an inert carrier. If it is assumed that it forms a completely sharp front, a volume A,6x cc. of the carrier will be displaced by the moving front in a time 6t. Let k f denote the ratio of the amount of solute sorbed by the stationary phase, a t equilibrium, to the amount of solute in the mobile phase when the concentration of the solute in the mobile phase is equal to the inlet concentration. A,6x CC. of carrier must then be displaced by A , ~ x
cc. of solute in the mobile phase, in equilibrium with kt times as much solute in the stationary phase. Since solute is transported only by the mobile phase, this is possible only if the linear flow velocity behind the front is (1 k') times that ahead of the front. This clearly shows that the flow velocity of the mobile phase behind a solute front exceeds that ahead of the front when sorption effects are important. Binary, Diluted Solute Mixture. The previous example is now extended to a binary solute mixture diluted with a carrier. The situation arising when two sharp fronts are formed is depicted schematically below.
+
column inlet Front progression velocity Mobile phase flow velocity ub Mole fractions
La
-W b
To derive the mass balance equations for the components contained within the first front, the section between x , - 6x and x. 6x is considered. Let the front move forward a distance ax, < 6x in time 6t.. The gain in the volume of carrier in the section is given by
+
=
- Xce)
A,GX,(X,a
(1)
This volume of carrier must be equal to the net volume which has streamed into the element, given by
-
6V = A,Gt,(u.X,a
U,X,C)
(2)
By equating the above two expressions, and setting 6 2 , = u& and X,' = I, the mass balance equation for the carrier in the first front is found as
(u,- V,)X,"
= uc -
vo
(3)
Similarly, for solute 1 in the first front v,(l
+ klU)
=
u.
(4)
The mass balance equations for the components contained within the second front are obtained in an analogous manner. They are, for the carrier, (ub
- ub)Xc'
for solute 1,
ub(1
f k2')
=
ub
(7)
Equations 3 to 7 are the five mass balance equations applicable to the system under consideration. They may be used to determine the value of X I " = 1 - X,", which is one unknown, and also to express four of the flow velocities in terms of the fifth. An expression for Xla is obtained by eliminating uo, ub, and v b between Equations 5 , 6, and 7-viz., k2'Xc%X1" =
[ki"Xi"
+ (kz' - k i 2 ) X ~ ' ] (-1 Xia)
(8)
Furthermore, from Equations 5, 7, and 8,
Equations 4 and 7 reveal that ratios v./ua and vb/Ub do not depend on and X z i when the distribution isotherms are linear.
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RETENTION VOLUMES OF FRONTS
Single, Undiluted Solute. The retention volume of a front is defined as the volume of fluid which leaves the column in the period 0 t 6 t, where fdenotes the retention time of the front. The retention volumes may be calculated from simple mass balance considerations. Let V' denote the total volume of fluid which has been introduced into the column from t = 0 to t = f, and let p denote the retention volume of the front. If ub and uc,respectively, denote
