Efficiency of Chromatographic Columsns Connected in Series Nonadditivity of Theoretical Plates in Chromatography Joel Kwok and L. R. Snyder Union Research Center, Union Oil Co. of California, Brea, Calif. 92621
J. C . Sternberg Research Department, Beckman Instruments, Inc., Fullerton, Calif. 92632 The total number of theoretical plates for two or more chromatographic columns connected in series is often less than the sum of theoretical plate numbers for the individual columns. As a result, it i s generally unwise to couple together columns which differ markedly in efficiency(HETP values), column diameter, or composition. Corresponding variations within a single column or bed should also be avoided. These and other aspects of coupled-column operation are examined.
IT IS COMMON practice in various chromatographic procedures to couple two or more columns in series so as to increase separation efficiency (total' theoretical plates) or to change selectivity (relative retention volumes of different compounds in a sample). While at first it might seem that the total number of theoretical plates in such a combined system should equal the sum of the theoretical plates for each column (at least where such complications as junction effects and compression of the moving phase can be ignored), this is not necessarily true. In some cases the separation efficiency of the combined system can be less than that of one of the columns making up the system. To our knowledge, this fact and its practical consequences have not yet been pointed out explicitly, although several workers have discussed the theory of coupled columns in gas chromatography (1, 2), and the basic concept is implicit in the discussion of Giddings (3). The question of separation efficiency in coupled column systems is important for practical reasons, and a more detailed examination of this subject therefore seems desirable. Recently there has been increasing emphasis on high efficiency separations by liquid-solid chromatography (usually using coupled column units), both in adsorption and gel permeation applications. The maximization of separation in a given system requires that we consider every possible contribution to separation efficiency. THEORY
Assume that we have n columns (1, 2, 3, . . . i, . . . n ) connected together in series. Further assume that for some compound of interest, under the conditions of separation, the retention volumes in each column are V I ,V2, V3, . . . V I , . . . V n and the band widths (standard deviations) are ul, u2, aa, . , , ut, . . .un. V I and ut are expressed in the same volume units (e.g., milliliters) and are corrected for any expansion which the mobile phase might undergo in proceeding from column average pressure, p i
soLi
(=
p t d z / L L ; h),to
atmospheric pressure, pa. No such correction is required in the case of isothermal separations by liquid chromatography, (1) G. P. Hildebrand and C . N. Reilley, ANAL.CHEM.,36, 47 (1964). (2) R. A. Keller and G. H. Stewart, Zbid.,p. 1186. (3) J. C . Giddings, Zbid.,35, 353 (1963).
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ANALYTICAL CHEMISTRY
because of the incompressibility of the liquid phase. In the case of a compressible mobile phase (gas chromatography) the quantities V t and ut for column i in a series of n columns will in general not be the values measured for column i in the absence of the other columns, unless column i is operated at the same average pressure it has in the combined column system, (see Appendix I). The efficiency of the connected set of n columns is defined by the bed plate number N , equal to V 2 / u 2where , V and cr refer to the overall observed retention volume and bandwidth for the combined columns. For n
a noncompressible moving phase, V is given by
Viand a* i=l
n
by
oi2 for the individual columns.
It is important to
i=l
note here that it is u2, the variance or second moment, and not the standard deviation, cr, that is an additive property (see Appendix 11). N for the combined columns is therefore related to the retention volumes Vi and the plate numbers N t (equal to V t 2 / a a of z ) the individual columns by
N =
-
i=l ~
i=l
i=l
i=l
Thus, in the general case, N is not given as the simple sum of the theoretical plates in the individual columns (N # n
N a ) . In the special case where the height equivalent of a i l l
theoretical plate H i (= L t / N t )is the same for each individual column, and where Vr/Ltis also the same for every column (Lt equals the length of column i), it may be shown that n
N t . The quantity V t / L rmay be related to the distri-
N = i=l
bution coefficient Ki (ml/gram) of the eluted compound (from column i), the cross sectional area of column i (At), the fraction of the total column volume (column i ) occupied and the density p t of the stationary by the moving phase phase in column i
ut)
Vt/L=
4 [Kt P t ( 1 - f t )
+
ftl.
(2)
That is, additivity of column plate numbers generally requires columns of equal efficiency per unit length (Ht), equal column diameters or cross sectional areas ( A ( ) , equal ratios of stationary and moving phase volumes in each column (constant fi), the same stationary phase (equal values of ,&), and (in gas chromatography) negligible pressure drop across the series of connected columns. The form of Equai
tion 1 is such that N 5 Nr: the total number of theoretical plates in a multicolumn system is generally less than the sum of plates in the individual columns. This fact has a number of important consequences in design of high efficiency separations. We will examine some practical examples below.
For a compressible moving phase (see Appendix I), the additive relationships apply only to the retention volumes and standard deviations (or, more properly, the variances) evaluated at the proper average pressures, p i , for the individual columns in the combined column system. Thus the observed retention volumes and variances will be additive only if the values for the individual columns are determined under, or corrected to, conditions where they each have the same average pressures as they have in the combined system. Equation 1 is not directly useful for a compressible moving phase because the plate number N i is itself affected by decompression of the moving phase. It is more desirable, therefore, to re-express Equation 1 in terms of local plate height rather than plate number, so that pressure effects can properly be taken into account. Equation 1 may be redefined in terms of the plate height Hi of column i, using the relationship H i = Li/Ni, which gives
[2
i=l
N = -
4’
(3)
i=l
(Vi/Li)
(4)
C ( Vi/Li) Hi i-1 In the limit, for a column (or columns) of total length L , with properties which may change continuously from one end to the other, we have
soL
[
N =
similarly replaced by the individual columns.
V o i , the sum of void volumes for
Columns of Unequal H1. In high efficiency separations, it is frequently necessary to combine several individual columns, each of which are characterized by low values of Hi. In liquid-solid chromatography, it is sometimes difficult to pack columns of uniformly low H i , and we then have the problem of deciding which columns to include in a group of combined columns. Equation 1 permits us to calculate N as a function of the N , values of the individual columns. For the case of nominally equivalent columns (Le., V , and Li values constant) of varying efficiency, Equation 1 reduces to
where n refers to the number of separate columns in the system. This may be expressed in terms of Equation 5a:
(Viz/Liz)dz]’
1
(5)
(VirlLir)zHi,dz
with Viz/Liz defined for the point z in column segment i by Equation 2. Hi, is here the local plate height [which may be obtained, for example, from theoretical expressions, such as those of van Deemter (4, Golay (J), or Giddings (6)]. The case of variable H (along the column or bed) is normally handled [e.g. (6, 7)]in terms of the relationship
iL s,
HidL
H =
n
KI and Kz are replaced by the weighted quantities KtWt for each compound, where K , and W , refer to corresponding values of Kl or K2 and W for each separate column i. V o is
SOhlE PRACTICAL CONCLUSIONS
I n the case of a compressible moving phase, the plate height H t for column i is the apparent plate height for that column under the pressure conditions which apply to it in the combined column system. Equation 3 may be extended to the case of a large number of very small column elements of length Az as
[5
Here Kl and Kz refer to equilibrium distribution coefficients (milliliter/gram) for two compounds 1 and 2 which give adjacent bands on the chromatographic column, V” is the void volume (ml) of the column, and W is the weight (grams) of stationary phase in the column. For a multicolumn system
n
(Vi/Li)
C (VtiLi)’ HiLi i=l
N =
N as calculated from Equations 1 or 5 may be regarded as equivalent to the corresponding quantities for a single-column system. Thus it can readily be shown that resolution as normally defined for a single column (R,)can be calculated for a multicolumn system using the value of N given by Equation l [e.g. (S)]
(5a)
dL
This distance-averaged value of H is equal to the effective value of H (Le., the value of H corresponding to L / N ) only when Ai, piKi, and f i are constant, and the moving phase is incompressible. Equation 5a represents a special case of 5. (4) J. J. van Deemter, F. J. Zuidenveg, and A. Klinkenberg, Chem.
Eng. Sci., 5, 271 (1956). (5) M. J. E.Golay, in “Gas Chromatography-Amsterdam, 1958,” D. M.Desty, Ed., Butterworths, London, 1958,p. 36. ( 6 ) J. C. Giddings, “Dynamics of Chromatography. Part I. Principles and Theory,” Marcel Dekker, New York, 1965.
i.e., the effective value of H for a combined-column system (R) is equal to the average value of H i for all columns in the system. One case is of particular interest. When will the elimination of a particular column i from the system be desirable? Le., under what conditions will N for all columns but one be equal to or greater than N for all n columns? Let us define the average value of H for the n - 1 columns of lowest H i as fi;, - (from Equation 4a) and the value of Hffor the column of highest Hi as H,. Then N for the n - 1 columns is equal to (n - 1) L,/R, - 1. Similarly N for all n columns is equal to n L i / { [ ( n- l)lf, - 1 Hn]/n). The condition that N be equal to or larger for n columns than for n - 1 columns is then
+
That is, whenever Hifor one column in a group is greater than (2n - l)/(n - 1) times the average H i for the remaining columns in the group, that column should be rejected. Its addition to the remaining columns in a coupled-column unit will result in a decrease in N for the total unit, relative to a (7) G.H.Stewart, Separation Sei., 1, 135 (1966). (8) L. R. Snyder, ANAL.CHEM., 39, 705 (1967). VOL. 40, NO. 1, JANUARY 1968
119
unit without that column. The ratio (2 n - l ) / ( n - 1) varies from 3 for n equal 2 to 2 for n equal a. That is, a column whose Hi value is three or more times the average of a group ,of columns should never be included in a coupled-column unit, while a column for which Hiis less than twice the average of the group will always increase N for the coupledcolumn group if it is included. Columns of Unequal Diameter. It has been considered advantageous in preparative separations to use a column of wide diameter (to which the sample is charged), followed by a longer column of narrow diameter. Presumably the larger capacity of the initial, wide column will tivoid column flooding (because of its larger capacity), while the narrow second column will provide efficient separation. The reason for avoiding a wide column throughout is that wide columns generally give substantially higher values of H i(and lower plate numbers), as well as inconveniently large retention volumes. At the same time the greater number of plates in the second column (because of lower H i and greater length) should provide a sufficiently large value of N for the total separation, The same idea is probably present in the proposal that conically shaped columns be used in preparative separations [e.g. (9)]. Actually, the preceding reasoning is fallacious, as an analysis of this situation in terms of Equation 1 shows, Because V i is larger for the first column, and Ni is smaller, this column will largely determine the effective plate number N for the two combined columns. That is, only a small fraction of the theoretical plates contained in the second column will be added to the value of N ifor the first column. For example, assume the following characteristics for two connected columns of this type:
individual columns are packed with different stationary phases, the corresponding values of K i and V , will vary, and N n
to Equation 7 gives Vizequal V i Vi), which in turn requires that V I = Vz = V i = V,. A maximum value of N requires that all column volumes V ibe equal. When the
will be less than Ni. The column in which an eluting compound spends most of its time largely determines the value of N for that compound. This means that the value of N for a coupled-column series depends upon the compound used to measure N , particularly the relative Ki values for that compound in the various columns which are connected in series. For elution from a single column, N tends to be independent of K i[e.g. (IO)]. However, closely eluting compounds (Le,, those difficult to separate) will exhibit similar values of N , unless their relative K i values in two or more columns are widely different (a fairly rare case) and the H ivalues of those columns are also different. The form of Equation 7 is interesting with respect to a question that has been raised previously. In the case of separations which require two different stationary phases, is it better to use two separate columns (one for each phase) or one column filled with the mixed phases? It is assumed that the total column length and the values of Hi for all columns are the same for each situation. Keller and Stewart (2) have argued that if solute retention volumes in the two cases are the same (Le. “additivity assumption”), separation should be equivalent. This is not true, however, whenever (Vi/Li)is not the same for the two single-phase columns. For single-phase columns of equal length, N for the two columns (in series) will approach the value of N for a single column as V i for one column becomes large with respect to V i for the other column. Consequently, it appears that in the general case it is preferable to use mixed-phase columns, rather than two or more columns packed with separate phases (so far as separation efficiency is concerned). This advantage of mixed-phase columns is probably marginal in many cases, and the greater convenience and flexibility of single-phase columns will then be a more important consideration. It should also be emphasized, for separations on liquid stationary phases, that the “additivity assumption” is sometimes inapplicable [e.g. (]I)]. In such cases overall sample resolution may be poorer for a single mixed-phase column, even when separation efficiency (the value of N ) is greater for a single mixed-phase column. That is, separation selectivity can be more important than separation efficiency in controlling sample resolution. Another situation which can be profitably examined in terms of Equation 1 (or Equation 5 ) is that of a partition column with varying amounts of stationary phase distributed along the column. This is equivalent to a variation in (1 - fi)along a given column. As discussed by Stewart and Keller (12), this situation arises in gas-liquid chromatography as a result of column “weathering.” The latter authors have devoted a detailed study to the effects of column weathering on separation efficiency, but they appear to have overlooked an important effect: namely that predicted by Equation 1 (or Equation 5). For the hypothetical case examined by them, that of a column with (1 - f i ) varying from c/2 in the first half of the column to 3c/2 in the second column half (we assume Kipi (1 - f i ) >> fi), we can calculate from Equation l that N for the latter column will be only 80% of the N value where fi is constant (Le. where the concentration of stationary phase is uniform) throughout the column.
(9) H. M. Stahr, R. M. Ikeda, E. T. Oakley, and B. M. Carter! ANAL. CHEM., 38, 1974 (1966).
(10) L. R. Snyder, ANAL.CHEM., 39, 698 (1967). (11) A. B. Littlewood and F. W. Willmott, Zbid., 38, 1031 (1966). (12) G. H. Stewart and R. A. Keller, J. Chromafog.,12, 150 (1963).
Column 1 Column 2
diameter length 50 cm 3 cm 200 cm 1 cm
Vi
Hi
Ni
1000 ml
0.4 cm 0.1 cm
125 2000
444ml
From Equation 1 we calculate N for the combined-column system as 257 plates; i.e., much less than the number of plates contained in column 2. It is also interesting to note that if column 2 is replaced by a less efficient 3-cm column of the same length (200 x 3 cm; N i= SOO), N for the combined column system is increased to 625 plates. It can be argued that a single long column is always better in elution chromatography (from the standpoint of separation efficiency) than a wider column of the same total volume but smaller length, or a wide plus a narrow column of equal length. Columns of Different Composition. A series of columns of equal efficiency and equal length, but varying stationary phases (and varying Ki’s), will have an overall value of N which is less than the sum of N i values for the individual columns. Thus Equation 1 for the case of equal N ivalues takes the form
(7) The condition for maximum N is that aN/aVi equal zero for each column volume Vi. Application of this operation
5
120
0
ANALYTICAL CHEMISTRY
(5
Equation 7 also applies to the case of bed gradients; e.g., a column packed with stationary phase of varying composition along the bed. Bed gradients have found occasional application in thin-layer (adsorption) chromatography [e.g. (1311, but it must be emphasized that the separation efficiency of such beds (Le., the value of N ) should be less than for comparable beds of uniform composition or constant adsorbent activity. Accidental bed gradients can also rise in adsorption chromatography as a result of transfer of water between a water deactivated adsorbent and a dry solvent during separation [e.g. (141. The adsorbent in the inlet side of the column then becomes more active than that in the remainder of the column, K becomes larger, and N smaller. It might appear that Eq. 1 also applies to the case of stepwise or gradient elution from a single column-i.e. that N for such separations will be less than N in normal elution from the same column. Actually the apparent value of N in such separations, as measured by V Z / a 2has , no real meaning. A proper selection of experimental conditions-Le. the form of the solvent gradient-can yield any value of N desired (from 0 to m ), for the same column. The true value of N for a column in gradient or stepwise elution is better defined in terms of resolution, i.e. R,,and it can be shown that N and resolution are the same in gradient or stepwise elution as in normal elution (see discussion of 15). That is, gradient or stepwise elution does not reduce separation efficiency per se. Fischer and Kabara (16) have recently shown that the separation of lipids by stepwise elution is substantially better in coupled columns of varying diameter than in constant-diameter columns of equal length. This result apparently contradicts our preceding conclusions on the use of coupled columns of varying diameter. We believe the data of Fischer and Kabara can be explained as follows. In stepwise elution as carried out by the latter workers, solvent composition is changed between the elution of each sample component, and each sample band is eluted from the column as rapidly as possible. Under these conditions it seems likely that each band is pushed from the column by the advancing solvent front which forms the boundary between the previous and present solvents in the column. The apparent widths of the elution bands (and the resolution of adjacent bands) will then be determined largely by how sharp the solvent fronts are. The width of the solvent front depends upon factors other than those which determine the width of elution bands, and can be better compared with the separation of adjacent bands by displacement chromatography (see discussion of 17). Here it is well known that band sharpening (or narrowing of a solvent front) occurs at the junction between columns of differing diameter (the advantage of so-called “coupled columns” in displacement chromatography follows from this effect). Thus it appears that the improved separations shown by Fischer and Kabara result not from an increase in the elution efficiency (value of N ) of their coupled-column units, but from an increase in the sharpness of the solvent fronts associated with stepwise elution. This aspect of separation efficiency in stepwise elution will be examined in detail in a later paper. APPENDIX I
In general, measurements on an individual column i in gas chromatography will be made at some average pressure pi’ (13) A. Niederwieser and C. G. Honneger, Adcarices in Chromatography, 1, 123 (1965). ( 1 4 ) G. Hesse and G. Roscher, 2. Anal. Chem., 200, p. 3 (1964). (15) L. R. SNYDER, Chromatog. Recs., 7 , 1 (1965). (16) 0. A . Fischer and J. J. Kabara, Anal. Biochem., 9,303 (1964). (17) L. Hagdahl, “Chromatography,” 1st Ed. Heftrnann, Ed., Reinhold, New York, 1961, Chap. 5.
which is not equal to p i . This then gives a retention volume Vi’and a standard deviation ut’. The idealized quantities VtO and ~ $ for 0 elution at atmospheric pressure (column inlet and outlet pressures equal to Pa) can be calculated through the relationships Ft’
V,’ = - Vi0 Pa
In gas chromatography V i ois often called the “corrected retention volume.” When column i is in the n-column system, it will in general operate at another average pressure, pi, such that
where Vio and aio are calculated as in Equations 8a, 8b for the run on the isolated column. Then, of course,
Pi
ut = -
Ui’
Pi‘
In general, an element of retention volume, V i t , contributed at a position z in column i, where the pressure is pir,will make a contribution Vt,O to the “corrected retention volume,” where
If the substitution of Equation 11 is made for Vi,in Equation 5, and a single column i of uniform packing with constant Vo/Lis considered, the expression becomes
or L N
=
Happ= -
so” (E)2
Hiz dz
=
This is the proper expression for the apparent plate height, Happ, of the column, as discussed earlier by Stewart and Keller (12) and by Sternberg and Poulson (18). If the column is a series of individually uniform segments, differing from one another in length and in capacity (thus in Vo/L),the more complex expression below is applicable.
(18) J. C. Sternberg and R. E. Poulson, ANAL.CHEM., 36, 58 (1964). VOL 40, NO. 1, JANUARY 1968
121
According to the above relationships, the value of N for a given column is reduced as the ratio of inlet to outlet pressures increases. At very high pressure drops, this places an upper limit on the maximum value of N which can be attained, regardless of column length. It is interesting to note that this adverse effect of column pressure drop can in principle be cancelled by the use of coupled-columns of increasing diameter. Thus V J L , can be held constant by proper design of the overall system. A theoretical analysis of some practical situations of this type, however, suggests that this expedient would be only marginally effective. APPENDIX I1
It is not obvious that the relationship u 2 =
n
cut2holds
i=l
when values of ui are defined in terms of eluate volume, and K , is permitted to vary among the various columns. Part of the difficulty in accepting this relationship arises from the fact that u Z values measured in terms of column length (the usual convention) change across column junctions when Kt changes, For example, assume that K I increases by a factor of 2 in going from column 1 to column 2, and u iis equal to 2
cm at the end of column 1 and 5 rnl in the eluate from column 1. As the sample band enters column 2 it initially occupies 1 cm (std dev or value of u ) at the front of the column. That is, the band has been compressed from 2 cm to 1 cm in crossing the junction between columns 1 and 2. But this phenomenon has no effect on the final contribution of ut to u, as long as values of ui are computed in terms of volume. To see this more clearly, assume that column 2 is completely equivalent to column 1 except for the difference in Kt values, An initially narrow sample band will then have u2 equal to 2 cm (or 10 ml) at the end of separation on column 2. Combination of columns 1 and 2 is seen to give a value of u2 equal to (12 22)cmz, or 5 cm2. This is equivalent to 125 m12 in the eluate from column 2. But the same result is obtained by simply adding u i 2 and u t 2 expressed in volume units: u 2 = 5 2 lo2. The additivity of ai2 in terms of volume units (or equivalent time units) in this situation has also been discussed by Giddings (3).
+
+
RECEIVED for review July 24, 1967. Accepted November 14,1967.
Rapid Procedure for the Chloroacetylation of Microgram Quantities of Phenols and Detection by Electron-Capture Gas Chromatography Robert J. Argauer Entomology Research Diaision, Agricultural Research Seroice, U.S . Department of Agriculture, Beltsuille, Md. Aqueous sodium hydroxide solutions containing as little as 0.01 ppm phenol were treated with choracetic anhydride in benzene to prepare a relatively stable phenol derivative sensitive to electron-capture detection and analysis by gas chromatography. The benzene and aqueous phases were mixed for 2 minutes, and a 5-pl aliquot of the benzene phase was injected onto the column of the chromatograph. Retention times were tabulated for the chloracetate derivatives of 32 phenols.
PHENOLS and their derivatives are used as antiseptics, disinfectants, and pesticides, as antioxidants in food:products, and in many other chemical applications. The use of the carbamate derivatives as pesticides has therefore generated renewed interest in a rapid and sensitive means of detecting phenolic compounds. Gutenmann and Lisk ( I ) hydrolyzed microgram quantities of the carbamate insecticide carbaryl, to 1-naphthol, which then was brominated and acetylated to prepare an electron-capture sensitized derivative. However, a procedure that would allow the preparation of a sensitive derivative with a minimum number of physical-chemical manipulations was still desired. When Landowne and Lipsky ( 2 ) prepared various mono-, di-, and tri-haloacetates of cholesterol in tetrahydrofuranpyridine solution and measured the response of the recrystallized derivatives to an electron-capture detector, the chloro(1) W. H. Gutenmann and D. J. Lisk, J. Agr. Food Chem., 13, 48 (1965). (2) R. A. Landowne and S. R. Lipsky, ANAL.CHEM., 35, 532 (1963).
122
ANALYTICAL CHEMISTRY
acetate derivative was the most sensitive. In the following procedure, a reaction medium of aqueous sodium hydroxide and chloroacetic anhydride in benzene was used. The prepared chloroacetate derivatives can be injected into the gas chromatograph without further physical-chemical workup. EXPERIMENTAL
Apparatus. A conventional gas chromatograph (Varian Aerograph Model 200) was used. It was equipped with a 5-foot X lfrinch i.d. stainless steel column containing 2 x (w/w) XE60 @-cyanoethyl methyl silicone polymer) coated on 50- to 60-mesh Anakrom ABS (Analabs, Hamden, Conn.) and maintained at 170" C. The tritium detector was held at 205' C, the injection heater at 210" C, and the nitrogen flow at 30 ml per minute at the column outlet. Screening of Phenols. To 10 p1 of each phenol in benzene (1 pg/pl) in a 125-ml Erlenmeyer flask were added 15 ml of 0.25N NaOH and 10 ml reagent [l gram of chloroacetic'anhydride (Eastman White Label) per 200 ml benzene]. The flask was shaken for 2 minutes on a mechanical shaker. A 5-kl aliquot of the benzene layer was injected into the gas chromatograph. Retention Times. A mixture of five chlorophenol chloroacetates were used as references to obtain over 2 days the retention times (expressed in minutes from the benzene front) of the chloroacetate derivative of 32 phenols. In these measurements, 1 pg of each of the respective phenols was treated as in the screening tests. The retention times obtained for benzene dilutions of the chloroacetate derivatives of 10 phenols prepared in 50-gram quantities in our laboratory by S. I. Gertler 15 years ago were identical with those obtained by this procedure.
