146
Anal. Chem. 1981, 53, 146-155
Optimization of Separations of Homologous Series in Reversed-Phase Liquid Chromatography Henrl Colin and Georges Gulochon” Ecole Polytechnique, Laboratoire de Chimie Analytlque Physlqoe, Route de Saclay, Q 1 128 Palaiseau Cedex, France
Jose Carlos Dlez-Masa Institute de Qulmica Organlca General, Calle Juan de la Cierva, 3, Madrid-6, Spain
This paper describes various approaches to the optimlzatlon of the separation of homologues in reversed-phase liquid chromatography (RPLC). The separations can be optimized on the basts of the minimum analysis time necessary to achleve a glven resolution between the peaks of the first two components of the mixture or on the basts of the maximum resolution obtained In a glven analysts time. The optlmlratlon requires the adjustment of the mobile phase composition so that the slope and the Intercept of the plots of log k’vs. carbon atom number have given values. It Is shown that in many cases isocratic elution is not more t h e consuming than gradlent elution.
In a recent paper (1)we derived an equation giving the resolution Rs between two successive members of a homologous series under isocratic elution conditions in reversed-phase liquid chromatography (RPLC). This equation, which is similar to that established by Said (2) but restricted to homologous series, shows that Rs increases with increasing carbon atom number in the alkyl chain (n) and tends toward a limit depending only on the efficiency of the column and on the selectivity of the system. The comparisonbetween data obtained for various packing materials for RPLC (nonpolar chemically bonded silica gel and silica modified by pyrocarbon) has shown that carbon adsorbents yield the highest limit resolution and that this limit is obtained for shorter alkyl chains than with bonded silica gels. However, very large degrees of resolution between two successivehomologues are not always necessary. The purpose of this work is to show how the separation of homologous compounds can easily be optimized for minimum analysis time or maximum resolution in a given time, in isocratic elution. Such an optimization is all the more helpful aa it appears that most often the isocratic analysis time is not longer than the time corresponding t o the best gradient elution analysis, when taking into account the time necessary for equilibration of the system before another analysis can be started. The isocratic elution mode is more convenient than the gradient one for many reasons, such as smaller base-line drift, better quantitative accuracy, lower detection limits, and lower equipment cost. In addition, with isocratic elution it is possible to work with a refractive index detector or to use a UV detector operating at low wavelength. It is particularly interesting to optimize the analysis conditions when the capacity ratio for the last peak is rather large. This corresponds to the separation of many homologues or to the separation of a small number of compounds with a large number of carbon atoms, that is when the analysis time can be expected to be long. Several optimization procedures are developed in this work. They are (i) the “absolute” optimization, which involves the 0003-2700/81/0353-0146$01.00/0
preparation of a column specially designed for a specific separation and the use of a particular mobile phase, (ii) the “relative” optimization, which is probably more realistic and allows the determination of the optimum mobile phase composition to be used with a given column for the achievement of a specific separation, and (iii) the maximum resolution optimization, which determines the conditions giving the maximum resolution between two peaks for a given time of analysis. In the following, determination of the optimum conditions of analysis means wentially derivation of the optimum values of a and 8, defined by eq 1,which expresses the observation log k,’ = n log a
+ log p
(1) that in homologous series the logarithm of the column capacity factor increases linearly with the number of carbon atoms of the alkyl chain. The parameters a and p depend on the composition of the mobile phase; a is a function of its water content, whereas P depends on the water content and also largely on the nature of the organic solvent used. A further paper will discuss the methods of calculation of the composition of mobile phases allowing the achievement of given values of Q and fl (3). For those readers who do not wish to follow the derivations in details we have summarized in table form the procedure to calculate the optimum values of a and ,d (Table 11). THEORY It has been shown in an earlier paper (I) that the resolution
+
RS(n,n 1)between two consecutive homologues containing respectively n and n + 1 carbon atoms in their alkyl chains is given by
MI2 Rs(n, n + 1) = -
a-1 2 a + 1 2/anp
+
(2)
where N is the column plate number, a the methylene group selectivity, and 8 a constant characterizing the retention of the functional group of the homologous series. Equation 2 is based on the assumption that the plate number is the same for the solutes containing n and n + 1 carbon atoms. This has been previously discussed (I)and it has been shown that for the systems studied here no significant change in the column efficiency occurs between k’ = 0.2 and k’ = 30, provided a well-designed system is used and the column is correctly packed. Moreover the effects of minor variations of N are dampened by the square root dependence of RS on N . Thus in the following, N will stand for the average plate number of the column, The homologue containing i carbon atoms will be referred to as compound i. Let us consider the separation of homologues with i between n and n + An. The capacity ratio of the last compound is
k ‘,,+an = aAnk,,’ 0 1981 American Chemical Society
(3)
ANALYTICAL CHEMISTRY, VOL. 53, NO. 2, FEBRUARY 1981
147
+ l ) ]- (1 - ro)a (@a) [(a- 1) - ro(a + l)I2
2r,JVaP"n-1H(u) A n [ ( a - 1) - r o b U
N
U2
(
%)u,a
Flgure 1. Plots of the absolute optimum selectivity (aoA) and reduced resolution (roA)VB. An (numerical calculations from eq 14 and 15).
-[ -[
u[dH(u)/du]-- H(u)
H(u) =
2r0aAn 1 + (a- 1 ) - ro(a
I
+ 1) (9b) + rOaAn[(a - 1 ) - 2ro(a + I ) ] [(a- 1) - ro(a+ 1)12
I
(94 Absolute Time Optimization. In this condition the required resolution between the two homologues with n and n + 1 carbon atoms will be obtained in the shortest posriible analysis time, that is, the minimum value of t R according to eq 8. Because the resolution between consecutive homolo,gues increases with carbon atoms, assumption of a given resolution between the first two peaks guarantees a minimum resolution between each pair of consecutive homologues. The values of u, N, and a corresponding to the minimum time of analysis are obtained by solving the system
If Ro is the resolution hetween the first two compounds ( n and n 1) combination of eq 2 and 3 yields
+
4&tP"
k',+A,, = -W 2 ( a- 1) - Z R o ( a
+ 1)
- (a- 12r0aAn ) - r o b +1) (4)
Equation 8 can be written tR fI(u)fZ(N,a) and then eq 10 is equivalent to
(11)
where ro is the "reduced" resolution
ro = 2Ro/N1/2
(5)
Because k' 1 0, eq 4 shows that for any value of ro there is a minimum selectivity required and that for any value of a, ro has an upper limit. These limits are given by the eq 5.
+ r o ) / ( l - ro)
(54
ro < (a- l ) / ( a+ 1)
(5b)
a ;> ( 1
Introducing the usual notations ( L = column length, u = mobile phase velocity, and t~ = retention time of the last compound = time of analysis), it follows that tR
= [ l + 2 r O u A n / [ (a 1 ) - r O ( a+ l ) ] ] L / u
(6)
The parameters L, N, and u are also related by
L = NH(u)
+
-
(7)
where H(u) is the plate height equation. An, n, and Ro are determined by the analytical problem. Between the five main parameters ( t ~L,,N, u,and a)there are only two relationships (eq 6 and 7). It follows that the combination of eq 6 and 7 can be written
t~ = f(u,N,a)
m.4 -dH(u) --= (13) au U Whatever is the plate height equation, the lhs of eq 13 is always negative which leads to the trivial conclusion that the analysis time decreases always with increasing mobile phase velocity. The minimum value of t R is determined by the pumping system. Anyway, optimization of the mobile phase velocity is not within the scope of this work which deals with thermodynamics of the system, not its dynamics. The optimum values of ro and a (roAand aoA,respectively) which depend only on An are given by the following equations (see Appendix 1): aoA(A,n- 1) An 2 anA(An- 1) - An
(8)
and u, N, and CY can be considered as independent variables in this equation. After combination of eq 6 and 7 and differentiation of the resulting equation with respect to u, N, and a,taking into account that ro is a function of N, we obtain the relationships given in eq 9. These equations can be used to solve different optimization problems.
aoA(An- 1) - An roA
=
(15)
+
An aoA(An-1) The plots of aoAand roAvs. An are given in Figure 1 in logarithmic scale. Figure 1shows that, when optimization becomes helipful (An > 4),the value of aoAis not very large ( 10 000) the optimization of the mobile phase is quite independent of the plate number for the separation of a given number of successive homologues. When N is larger than 5000, increasing n reduces very significantly p, a t nearly constant a. Maximum Resolution Optimization. The optimum P value is obtained from eq 22 and 25. The relationship between aORand PoR is very simple fioR(a0R)n+An kL’
(27)
Some plots of log aoHvs. log PoR are shown in Figure 9 for different pairs of n and kL’. PoR has a finite limit (0) at large An values. In practice, however, PoRnever becomes very small. The curves corresponding to a given value of n but to different values of kL’ are parallel but their shapes are unexpected: when An increases, aOR decreases while PoRincreases first and then decreases. The comparison between Figures 7-9 shows the different aspects of the optimization conditons: For the absolute time optimization, log a and log ,8 are linearly related. When An becomes infinite, a tends toward 1 and @ toward 1.2. For the relative time optimization, log a and log (3 are not linearly related; log a varies less than log when An increases. When An becomes infinite, a tends toward 1 and P toward infinity. For the maximum resolution optimization, there is a maximum value of 0, depending on n, kL’, and An. When An becomes very large, a and tend toward 1and 0, respectively. Performances in Optimized Isocratic Conditions. Comparison with Gradient Elution. The quality of the isocratic optimization can be evaluated by comparison of the analysis times in isocratic and gradient optimized conditions. It is also interesting to estimate the resolution between a given pair of homologues as well as the ratio of peak height in the two modes of elution. These aspects are discussed now. Comparison of Analysis Times. The isocratic analysis time can be calculated by use of eq 6. It depends so much on the characteristics of the specific separation in question that a detailed discussion is impossible. A good way to check the quality of the optimization, however, is to calculate the ratio T of the time of analysis in isocratic optimized conditions to the analysis time in the best possible gradient Conditions. In addition, this will provide us with a useful way to decide when it may be necessary to search for optimum gradient conditions. The best possible gradient conditions would provide for the elution of the first component of the series at K’ 0 and for the elution of the successive compounds with
’I-
I
An
5
0
x)
20
15
Flgure 10. Plot of T (ratio of isocratic analysis time to gradient analysis time) vs. An for different values of r,: (solid lines) relative timle optimization; (dashed line) absolute time optimization; (1) ro = 01.1, N = 800, and Ro = 1.4 or N = 20000 and R, = 7.1; (2) ro = 0.0618, N = 2000, and Ro = 1.4 or N = 20000 and Ro = 4.4; (3) ro = 0.028, N = 10000, and Ro = 1.4 or N = 20 000 and R , = 2.0; (4) r o = 0.0198, N = 20000, and Ro = 1.4. Rp
It can be
2roaAn+ (a- 1) - ro(a 1) [(a- 1) - ro(a 1)](1 2roAn)
(28)
constant peak width and constant resolution shown (see Appendix 3) that T is given by T =
+
+
+
Equation 28 assumes that the resolution between successive homologues is at least as good in isocratic as in gradient analysis. It should be underlined that such a gradient run is highly hypothetical (5) and even theoretically quite impossible (k’ = 0). Moreover after a gradient analysis is aompleted, it is necessary to come back to initial conditions which is time consuming. Often the reequilibration time olf the column is of the same order of magnitude than the anailysis time itself. Consequently, from a practical point of view, values of T must exceed a few units to make worthwhile the search for optimum gradient conditions. Some plots of T vs. An are shown in Figure 10 for different values of ro in the case of relative time optimization (given N value). Resolutions larger than 1.4 between two successive peaks are necessary when peak impurities or branched isomers have to be inserted between the peaks of linear homologues. The solid lines in Figure 10 correspond to the relative time optimization while the dashed line corresponds to the absdute time optimization. It is remarkable that the relative optimization seems always better (in terms of T ) than the absolute one. This is because for a given An value, the columns used are not the same for the relative or absolute time optimizations. The column used in absolute time optimization performs better in gradient elution at small An than the column used in relative time optimization. To clarify the situation further, we give in Table I the results corresponding to An = 10 for the absolute time optimization and the relative one with N = 500 and N = ij000. It is assumed that the mobile phase flow rate is the same in
152
ANALYTICAL CHEMISTRY, VOL. 53, NO. 2, FEBRUARY 1981
Table I. Example of Calculation of r for An = 10 Absolute Time Optimization a o A = 1.260 roA = 0.0628 NoA = 1990 L = LoA k,' = 1.06 kfn+A,,= 10.73 t R = 1.0 t G = 0.19 T = 5.20 Relative Time Optimization
For a1 1 chromatogram.
A I rborot i o
Rr4. B
~ A A , AA
I
.I
N = 500 aoN = 1.429 roN = 0.125 N = 500 k,' = 1.99 k',,+A,, = 70.71 t R = 1.53 t G = 0.07 r = 20.5
N = 5000 N = 5000
aoN = 1.204 roN = 0.040 kn' = 0.69 k',,+A, = 4.42 tl? = 1.16 t G = 0.38
-_
7
=
L = Lo%
N-20,000 An19
, A
A
I1
*I
hI
hI
.s
1. I
1. s
LI
.s
ZI
1.
s
LI
1. s
21
Gradi rnt
Rr4. B
L = 2.5LOA
t
.I
.C
3.02
Iroorat i o
Rrl. 4
the different cases and that the resolution required between the two first peaks is 1.4. t G is the analysis time corresponding to the best possible gradient. The length of the column ( L ) is calculated assuming that 2000 plates correspond to L = LoA. The time unit is chosen such that the analysis time ( t R ) is 1 for the absolute time optimization. It can be seen that, although the values of t~ corresponding to N = 500 and 5000 are larger than 1, the value of 7 corresponding to the absolute time optimization is intermediate between those corresponding to the relative time optimization with N = 500 and 5000. It also appears in Figure 10 that the smaller ro becomes, the smaller 7 becomes. Such a result could be expected as decreasing ro (decreasing Ro and/or increasing N) results in a smaller k' for the last peak and consequently a smaller difference between the isocratic and the gradient analysis times. The value of r increases very rapidly when r0 increases, especially at moderate or large values of An, Because of the quality of today columns, a large ro corresponds usually to a large value of Ro (and not to a small plate number). Consequently, when a large resolution is required between the successive peaks (Ro > 4) such as in the analysis of chain isomers, it is probably better to work in gradient elution conditions. In most practical cases, however, ro is between 0.03 and 0.02 and An between 5 and 15. It can be seen in Figure 10 that in such conditions, 7 is between 1.5 and 3. Consequently, in practice, isocratic elution is not more-and is even sometimes less-time consuming than gradient elution for the separation of homologous compounds, provided the composition of the mobile phase is carefully optimized. This may be true, even when 15 compounds have to be separated. This result is illustrated with some simulated chromatograms in Figure 11where a constant peak area is assumed between chromatograms A and B and C and D, respectively. With base-line resolution, the isocratic analysis time of 10 homologues is only 1.7 times larger than the best gradient analysis time (chromatograms C and D). Because of the reequilibration time needed after the gradient run is finished, and because we do not know yet how to derive conditions in which the performances of the best possible gradient run are approached (although we know how to calculate the optimum conditions for the isocratic run), it can be said that the two modes are time equivalent. When Ro increases, however, the isocratic elution becomes quickly more time consuming. In the conditions of chromatograms A and B in Figure 11, tI 4tG for Ro = 4. It must be emphasized,however, that the serious advantage of gradient elution over isocratic analysis is in the increased sensitivity. Larger volumes of sample can be injected while using a weak solvent and the eluted peaks have a narrow bandwidth, so they are higher, especially for the most retained ones. Comparison between the detection limits is discussed below.
k .I
D
Gradi rnt
Rrl. 4 1
I .I
.I
1. I
Flgure 11. Simulated separatlons of 10 homologues (An = 9): N = 20 000 (A) and (C) isocratic relative time optimizations, (B) and (D)best possible gradient; for (A) and (B) Ro = 4; for (C) and (D) Ro = 1.4.
Comparison of the Resolution between the Homologues. According to eq 2, the resolution between the successive members of a homologous series increases with increasing carbon number and tends toward a limit value. Consequently if the resolution is the same between the first two homologues in isocratic and gradient time optimized analysis (absolute or relative time optimizations), the other pairs of peaks will be better resolved in isocratic than in gradient analysis. The ratio of the resolution between homologues i and i + 1(n < i < n + An - 1)in isocratic analysis to that in gradient analysis (which is constant and equal to rG) is given in Appendix 4
For absolute and relative time optimizations, ro = rG. For maximum resolution optimization, ro and rG are different. rG is given by rG = kL'/2An (30) and rO/rG(=RJis smaller than 1. Ri has been calculated for the three optimizations in various cases, and the results are plotted in Figure 12. It is obvious that Ri is smaller than 1for maximum resolution optimization. Increasing the capacity ratio for the last peak decreases the resolution in isochratic condition, whereas it stays constant (and equal to ro) in gradient condition. Consequently, increasing kL' improves more rapidly the resolution in gradient than in isocratic analysis, and thus .Ri decreases. As far as minimum analysis time optimizations are concerned, Ri is larger than 1 and increases almost linearly with (i - n). In the case of relative optimization, R,decreases slightly with increasing ro;for typical conditions (ro between 0.02 and 0.08) the resolution between the last two homologues in isocratic elution is about twice that in gradient elution. This is no longer true for small A n values or large FO values, but optimization is not interesting in these conditions. Consequently, very often optimized isocratic analysis is no more time consuming than optimized gradient analysis and gives better
153
ANALYTICAL CHEMISTRY, VOL. 53, NO. 2, FEBRUARY 1981
Table 11. Organigram for the Search of Optimum Condit Optimum Conditions Problem: Separation of homologues containing n to n + An carbon atoms. The capacity ratio of the homlogue containing i carbon atoms is k j = a'P
Step 1 . Decide the characteristics of the analysis desired: n , A n , R,, or kL' (capacity ratio of the last solute) Step 2. Decide the optimization needed: absolute minimum analysis time (R, given); relative minimum analysis time (R, given); maximum resolution (kL)given). (A) Absolute Minimum Analysis Time Optimization
(the column is especially prepared for this analysis) An An t 0.688 Step 3 4 calculate: a o A = A(16b) n - 1 A n - 0.688 Step 4A, calculate: r, = I
0
2
I
4
I
I
I
6
8
0
,
I
1 2 1 4
(15)
4RO1
Isi-n 16
Figure 12. Variation of the ratio of the reduced resolution in isocratic elution to that in gradient elution between homologues land i + 1 for various optimizations: (upper curves) fuii lines, relative optlmizatlon, r o = 0.028; dashed lines, absolute optimization; (lower curves) maximum resolution optimization.
I
a o A ( A n - 1) - A n aOA(An- 1) t A n
I
Step SA, calculate: N = r,l Step 6A, calculate: PoA =
(5)
kOA(A.n - 1 ) - A n (25)
(a,A)n+'
The column is defined by its plate number, the composition of the mobile phase by the necessity to achieve the values of a o A and poA for 01 and p , respectively. (B) Relative Minimum Analysis Time Optimization
(the column is given) N is determined by the choice of the column and solvent velocity. Step 3B, calculate: r, = 2R,/N'/z (5) 1 + ro 1-r,
An
Step 4B, calculate: a o N = -An-1
Step 5B, calculate: PoN =
The composition of the mobile phase is determined by the necessity to achieve the values aoN and PoN for a and p , respectively (C) Maximum Resolution (the column and kL' are given) Figure 13. Plots of the ratio of the height of the last peak in lsocratic elution to that in gradient elution vs. An for different values of ro: (1) r o = 0.0198, (2) ro = 0.028, (3) ro = 0.0628.
resolution between the peaks. For the absolute time optimization the resolution between the last two homologues is independent of An (1.65). At high An values, the curves corresponding to the absolute and relative time optimizations are very close. This is because the ro value chosen for the relative optimization (ro = 0.028) becomes close to rg* for An larger than 50. The curve corresponding to the absolute optimization is above that corresponding to the relative optimization if the value of ro chosen is smaller than ioA. Comparison of the Peak Heights. The last aspect of the comparison between isocratic and gradient elution that must be discussed is related to the detection limits. Obviously because the solutes are eluted with practically constant bandwidth in gradient elution, the peak height is larger than in isocratic elution. The ratio 7ft of the height of peak i in isocratic elution and in gradient elution is given by (see Appendix 5) (a- 1) - ro(a 1) Hi = (31) (a- 1) rg% (+ - 1- a)
+
+
Step 3C, calculate: log oroR =
0.2433 t 0.011:) x An- 1 I
kL'(a - 1 ) t kL'(aoR
Step 4C, calculate: r, = 2(a,R)An
Step 5C, calculate: R = r, N1I2/2 Step 6C, calculate: poR = k'/a,R)n+An
+ 1) 1
(21)
(5) (27)
7 f i depends on rg and An as shown in Figure 13 where some results for the last homologue (i = n + An) in the case of relative time optimization are given. The values of ro selected correspond to efficiencies of 2000,lO000, and 20 000 plates with Ro = 1.4. As was predictable, the larger ro becomes, the smaller becomes. The curves show that 7f,+,, decreases very rapidly when An increases. If one considers curve 2 as typical ( N = lOOOO), 7f,+,, is about 0.3 when An = 10 and only 0.1 when An = 20. This is apparently a serious drawback to isocratic elution when a sensitive detection is required. However, it must be kept in mind that even with high-quality solvents, a gradient
154
ANALYTICAL CHEMISTRY, VOL. 53, NO. 2, FEBRUARY 1981
run cannot be performed with a very high detector sensitivity (except when working a t detection wavelengths larger than 300 nm with usual solvents in RPLC). Consequently, the theoretical loss in sensitivity suggested by Figure 13 is partly compensated for and sometimes overcompensated by the possibility of working a t high detector sensitivity in isocratic analysis and also to use low detection wavelengths (Le., 200 nm) which can drastically improve the signal to noise ratio.
CONCLUSION The separation of homologues in RPLC can be optimized in isocratic elution in various ways: absolute time, relative time, and maximum resolution. Among these various optimizations, the relative time optimization which assumes a given column efficiency is probably the most useful in practice. With a given column, the optimization of the mobile-phase composition requires the independent adjustment of the slope (log a) and the intercept (log 0) of the linear plots of log k' vs. carbon atom number in homologous series. The optimized values of a and P can be easily calculated. a depends on the number of consecutive homologues to be separated, on the resolution required between the first two peaks and on the column efficiency. The optimized intercept P depends on a and on the carbon atom number in the alkyl chain of the first homologue. The calculations of the various parameters are summarized in Table I1 for the different optimizations. Optimizing the mobile phase can save much time, especially when the high-efficiency columns presently available are used. Moreover, with such columns, the separation time in isocratic elution is not significantly longer than that with the best possible gradient elution and sometimes is probably shorter, even when a large number of homologues have to be separated. The loss in sensitivity associated to the decrease in peak height for the last homologues to be eluted can often be compensated for by the possibility to use a higher detector sensitivity (and sometimes a lower wavelength detection) in isocratic conditions. Moreover it has often been observed and reported that the stability of the columns is smaller when they are used in gradient than in isocratic elution (packing compaction). ACKNOWLEDGMENT We thank M. Martin and A. Jaulmes for interesting discussions and J. L. Escoffier for writing the program for chromatogram simulation. APPENDIX 1 Equation 12 is equivalent to An[(. - 1) - ro(a l)]- (1 - ro)a = 0
+
[(a - 1) - ro(a
where c is small. In such conditions, eq 26 gives
For evaluation of e A n at large An values, eq 14 can be rewritten 2
(1+ €)(An - 1) + An (1 + €)(An - 1) - An
Because of the relation
(1
+ e)An
ro =
-"-[ a-1
(1
+ 4%)"-
(38)
cAn - 1
-
(39)
The numerical solution of eq 39 is eAn = 2.2 and then An-
PoA-
1.2
APPENDIX 3 The optimum gradient described assumes that the first homologue is eluted with k' = 0 and the peak width is constant, as well as the resolution between each peak. If this resolution is Ro,the elution time of the last homologue (n An) is
+
t~ = (1
L w); +
+ 4RoAn
= (1
L
2roAn)-U
(40)
APPENDIX 4 The change in the resolution between consecutive homologues is accounted for by eq 2. Rewriting this equation for n = i and n in terms of reduced resolution gives
k,' =
2r0 (a - 1) - ro(a
k[
2ri (a- 1) - ri(a + 1)
+ 1)
the combination of eq 1, 41, and 42 yields
ro(ari =
+
ro(a
- 1) + (a- 1)
(43)
If rG is the resolution between each consecutive homologue in gradient elution, then Ri = ri/rG is
a(An - 1) - An a(An - 1) + An
combining eq 33a and 33b gives a(An - 1) + An 2 a(An - 1) - An
etAn
--4eAn - &,((I + 8e-'&")1/2 + 4e-e*" - 1)
(32b) The solutions of eq 32a and 32b are 2(a - 1) ro =
r?l
it follows
(32a)
+ l ) I 2 + roaAn[(a- 1)- 2ro(a + l)]= 0
-
(33b)
APPENDIX 5 The ratio of the height of peak i in isocratic conditions and in gradient conditions is given by
5-41
(1 - a + l
(34)
APPENDIX 2 For large An values, aoAis close to 1. We can thus write
(45) where W ~ ( G and ) Wi(1)are the peak widths in gradient and isocratic conditions, respectively. Assuming a constant plate number in isocratic conditions and a constant peak width in
ANALYTICAL CHEMISTRY, VOL. 53, NO. 2, FEBRUARY 1981
gradient conditions results in
and
k[ is given by (47) then 7fibecomes
gi= H(u)
Bi h
k,' kL'
L N NoA n
RO
Rs(n,n + 1) Ri
1 := 1 + k[
( a - 1) - ro(a + 1) (a- 1)
+ r0(2@
- 1- a)
(48)
GLOSSARY plate height equation ratio the height of the peak corresponding to the homologue containing i carbon atoms in optimized isocratic conditions to the height of the same peak in optimized gradient conditions reduced velocity capacity factor of the homologue containing n carbon atoms capacity factor of the last homologue column length column plate number absolute time optimized plate number number of carbon atoms of the first homologue resolution between the two first homologues resolution between the homologues containing n and n + 1 carbon atoms ratio of the resolution between homologues containing i and i 1 carbon atoms in isocratic optimized conditions to the resolution between
+
155
the same compounds in optimized gradient conditions reduced resolution between each peak in optimized gradient conditions reduced resolution between the two first homologues absolute time optimized ro retention time of the last homologue = analysis time values of t R in isocratic and gradient elution modes ratio of the analysis time in isocratic optimized conditions to the analysis time in optimized gradient conditions mobile phase velocity selectivity absolute time optimized a! relative time optimized a for a given column lenlgth relative time optimized a for a given column lenigth and mobile-phase velocity maximum resolution optimized a minimum value of aoN(high plate number) capacity ratio of the functional group of the homologous series absolute time optimized P relative time optimized 0 for a given column length and mobile-phase velocity maximum resolution optimized difference between the carbon atom number of the last and first homologues mobile-phase reduced velocity
LITERATURE CITED (1) Colln, H.; Guiochon, G. J. Chromatogr. Scl. 1080, 18, 54-63. (2) SaM, A. S. Sep. Scl. Technol. 1078, 13, 647-679. (3) Colin, H.; Zehou, Y.; Gonnord, M. F.; Gulochon, G. XIIIth Internatlonai Symposium on Chromatography, Cannes, 1980. (4) Halasz, I. J. Chrometogr. 1070, 773, 229-247. (5) Jandera, P., private communication, 1980.
RECEIVED for review June 9,1980. Accepted October 6,1!380.
