Optimum Resolution and Minimum Time Operation in Gas Chromatography: Effect of Various Parameters on Resolution Under Normalized Time Conditions GEORGES GUIOCHON €cole Polytechnique, Iaborafoire du Professeur I.Jacqui, 7 7 rue Descarfes, Paris (5ime), France The use of normalized time conditions makes it possible to optimize column length, column temperature and capacity factor, carrier gas velocity, and particle or column diameter to obtain maximum resolution, by using the general well-known equations for elution time, resolution, and flow rate through porous materials (Darcy) or open tubes (Poiseuille). The optimum values obtained for these parameters agree fairly well with experimental data previously published b y several authors. Theory and experiments agree on very simple conclusions. For example, they show that in most cases the highest resolution in a given analysis time is obtained when the column is operated a t a value of the capacity factor k' of about 3 for a packed column and 1.5 for a capillary column, with a carrier gas velocity 1.5 to 2 times higher than the optimum velocity, and when the packed column is made with support particles of about 100 mesh. It is also possible to calculate more exact column design and operating parameters for minimum analysis time and to compare the performances obtained in this respect with columns of different types.
I
in any other analytical method, there is a price to be paid for each analysis. This price is composed of two parts: analysis time and apparatus expenditures. I n gas chromatography apparatus expenditures result from column length, pressure drop, and temperature. Unless analysis time is very short the auxiliary equipment is the same, regardless if this time is a few hours or a few minutes. So, generally, time will be the most expensive factor of analysis cost, which explains the great attention which has been already paid to the theory of minimum analysis time ( 1 , 7 , N GAS CHROhfATOGRAPHY, BS
19, 2 5 ) .
Previous papers were for theoretical computations of the minimum time required t o resolve the peaks given by a pair of compounds eluted through a packed or capillary column under the most favorable conditions. S o t many 1020
ANALYTICAL CHEMISTRY
experiments were made to verify experimentally the conclusions of these theoretically papers, and to test extensively the influence of various parameters. Karger and Cooke (16, 17) studied experimentally the effects of various parameters on resolution under normalized time condition. Although this condition may seem a t first to be somewhat arbitrary, it is a very useful and interesting step, both experimentally and theoretically, in the process of making fast columns. From a theoretical point of view, any procedure which is experimentally sound and is readily amenable to calculation is worth a detailed examination. From an experimental point of view, since the exact value of minimum analysis time is unknown a t first, the work of searching for an optimum column can be economically accomplished in two ways: by varying parameters and resolution while keeping the analysis time constant, or by varying parameters and analysis time while keeping the resolution constant. This last method leads to experiments which are more difficult to carry out and will give less precise data. It will not be considered here since it will lead substantially to the same results and because there are no experimental data to test the intermediary results of the theory. In this work we shall explain how the results obtained by the first method of Karger and Cooke can be accurately predicted by a theory based on very simple and general equations, and we shall show how the same theory can be used to optimize various parameters and obtain the minimum analysis time. For this purpose we shall use the equations for the gas flow velocity through a packed or capillary column, for the elution time of a peak through a column, and for the resolution of two peaks. From these equations and the condition that t~ remains constant, we can derive one general equation for the resolution. Using this equation and the relevant plate height equation, it will be shown that in most cases the resolution is maximum for a specific value of each parameter which has to be considered.
We shall then discuss the experimental results of Karger and Cooke and see to what extent they agree with the values derived from the equations. hlore general results can be obtained from this theory, dealing with comparison of performances of columns of various types and the search of conditions for minimum analysis time. Flow Rate, Column Length, ResoluRATE tion, and Analysis Time. FLOW EQUATIONS.It is well known t h a t the outlet carrier gas velocity is given by the equation:
where k is the column permeability (for the other parameters, see list of symbols a t the end of the paper). For a capillary column, k = r,'/8 (see, for example, Ref. 22 or 23). The average carrier gas velocity is:
3kp, (P' - 1)' 4qL P3 - 1
= juo = -
(2)
ANALYSIS TIME. By definition of the mean velocity, the retention time is: L tR=--=-
P3 - 1 -
4qL'
R F I ~ 3 R ~ k p 0(P'
+
- 1)'
-
4qLz(1 k ' ) P 3 -1 3 kp, (P' - 1)'
(3)
The analysis time is equal to the retention time of the last component. The time needed to elute the second half of this peak ( 2 t R / Z / N ) will be neglected. RESOLUTION.Among the possible equations we shall use the following one, proposed by Purnell(d4) .-
This equation is the most often used. IC' and N refer to the second peak of the pair. CoLuhn EFFICIENCY.The efficiency of various columns with different lengths may be compared by using Equation 5 : L = NH
(5)
which is merely the definition of the height equivalent to a theoretical plate. RASGEO F VALIDITY OF EQUATIOSS 1 TO 5 . Equations 1 and 2 are based on the two following assumptions alone: (1) The carrier gas is ideal. Martire and Locke (21) have shown that this assumption is valid in all experimental conditions of gas Chromatography. With P = 5.0 they found a deviation of the mean column pressure from the value obtained for an ideal gas of 0.1670 for carbon dioxide and less than 0.02% for hydrogen, helium, argon, and nitrogen. (2) Darcy’s law is valid. This is true for conventional packed columns of A-inni. i d . with flow rates up to about 600 61-11.~per ininut’e of argon, which means a carrier gas velocity of about 100 cm. per second (18). Poiseuille’s lan- is valid in a much wider velocity range. It is therefore hardly possible to find in gas chromatography any conditions where Equations 1 and 2 are not exact. To be allowed to use Equation 3, we need to add only one assumption to those previously dealt with: We are dealing with linear chromatography. This is always valid in analytical gas chromatography. The resolution between two Gaussian peaks is by definition:
R
=
2
tR 2
- I -
wz
- tR,1
+
w1
where the subscripts 1 and 2 relate to the first and second peaks to be resolved. This may lead to various well-known expressions depending on the peak which is considered the most important. These equations have been discussed in detail ( 2 , I S ) and it was shown that none is rigorous. They are known as Knox’s (19), Said’s (M), and Purnell’s (24)equations. The first one makes use of the parameters of the first peak and the last one of those of the second peak of the pair of compounds to be resolved. Although the second equationlooks more general than the other two, it makes use of average values of k’ and N , which are more complex to handle. Since we are interested in analysis time and hence in the retention time of the second peak, we shall use Purnell’s equation, although it is not exact for the resolution of peaks of compounds of large relative retention. I n this last case the following equation suggested by one of the reviewers may be used :
where k’ and N are measured for the second peak and n is w1/w2. I n the following discussions most of the results obtained will be the same if n is taken equal to unity, which explains the use of Equation 4.
PLATEHEIGHTEQUATIOS. We shall need in our optimization calculations the relation between the height equivalent to a theoretical plate and the various column parameters. Although there is no general agreement on that point, we shall use the empirical van Deemter equation: H
=
A
+ Bu, + CU,
(6)
The actual meaning of the three coefficients is still open for discussion (8, 10, 20). However, this equation will apply for most experimental data. We are not a t all concerned by the fact that A arises from a contribution of the apparatus or from any process of peakbroadening a t work in the columns, since we are dealing with a problem of the overall resolution which can be solved only by using an actual apparatus (always somewhat imperfect). In most cases, however, we can assume that A will be small. Giddings’ coupled theory can now be considered as valid, since it is supported by experimental data (18, 20). This means that good results could be obtained a t very high flow rates. HoFvever, this would need very complicated apparatus, so that practically gas chromatography will be limited for a long time to the flow rate range around the van Deemter optimum flow rate. Practically the following results will be valid as long as the use of very high inlet pressure will not be standard, I n Equation 6, B and C can be written as follows: B = 2yDg; C = Cld2 Cz; where d is the particle diameter (packed columns) or the tube diameter (capillary columns). For packed columns y is roughly equal to 0.7 and is independent of the kind of packing. Cld2 is the term of resistance to mass transfer in the gas phase. C1 is proportional to D,-1. C2 is the term of resistance to mass transfer in the liquid phase. Except with glass bead columns, CZ is independent of d. This is only a first approximation, If A , B , and C1,which refer to gas phase processes in the column, may be considered as independent of P , C2 should be written as C2j. For high values of P this takes the form 3C2/2P. This alteration will cause some change in the derivation of the optimum parameters, which will be discussed in the fourth part of the paper. The System of Equations Used, Equations 1, 3, 4, and 5 enable one t o compare the performances of various columns of different types ( 2 , 14) The experimental results will agree with those results which are predicted from these equations as long as it is not necessary to use a detailed equation for H , because Equations 1 to 5 have great generality (IS). They have
+
gained the widest acceptance, which is not yet the case of inoat equations derived from the kinetic theory of gas chromatography for the esl)reshion of I f . However, to simplify the calculations, we shall not use thew equations directly. I n most cases we shall suppose that CY is independent of temperature, so that optimization of the resolution may be replaced by that of the separation number:
S
S(RF - 1)’
(7)
The carrier gas outlet velocity is a very important’ parameter because its value determines the column efficiency. When a column is designed it is impossible to operate it a t any flow rate, but the carrier gas velocity will be chosen by the analyst in a range depending on the column design. The pressure drop appears only as the price to be paid to obtain this velocity. Therefore it is necessary to derive a relation between the retention time and u,. This is very complicated to do exactly and the resulting equation is difficult to use (1). It is possible to assuine that P 3 - 1 and P 2 - 1 are equivalent to P 3 and P2,respectively. The error is acceptable when P is greater than (less than 10yG) 3, which means that the inlet pressure is more than 2 bars ( 2 ) . This will not seriously restrict the validity of the following results because, as was shown by Knox (19), the shortest analysis time will be obtained by operating the relevant column a t the maximum pressure drop allowed by the apparatus. Difficult separation in which ’ire are mainly interested requires long columns (working at high pressure drop), and the optimum particle diameter (being, as we shall sholv, in the 100- to 120mesh range) will lead to high pressure drop columns. Equations 1 and 3 become :
When columns are made with the same stationary phase, they have the same permeability 12, and liquid-to-gasphase ratio, V,/T’G. If they are used a t the same temperature with the same carrier gas (same a ) and the same outlet pressure and velocity (pol u,) they will give the same value of peak-to-carrier gas relative velocity RF and of average plate height, H . Then the retention time increases as L3l2and the resolution as L1lZ. This means that the analysis time in which a given resolution will be obtained, using the same stationary phase and experimental conditions, increases as the cube of this resolution. Figure 1 s h o w that even at low values of P,the trend is the same but analysis VOL. 38, NO. 8, JULY 1966
1021
time is soniewliat longer than predicted by Equation 8 for a given column length. No~11.mzi:i) Tram COSDITION. In tinit) norin:tliznt ion tlic colinnn tempcrature is adjustrd so that the analysis tinir. ivnininq thr sanw for various columns 01' for a given colunin in the useful range of the psramcter, the inflwncc of wl~ic~hon t h e rrsolution is studir(l. C'liangr of resolution may be easily mcwiired (6,I S ) FOYc~sninl~lr,colrimn length and teililwrat tirtb ('~111 bt: simultaneously changed, in order to keep t R constant while pi is rhangrd in order to keep uo and hence H eonstant. Equation 8 indicates that L2l3/RFwill then remain constant, hence :
0
2
6
4
8
10
LILO
Figure 1.
Variations of q with trmperaturc can bc ncglrctcd since q increases somrwhat slomr than 2' (22), whereas k' increases exponentially with 1/T; they would have to he coniidercd only when k' is negligible coniparrd to one. This is nevrr the caw near the 01)tiniuni of k' as will be \homn latcr. The dimination of RI. and Ar betwern Equalioni 5 , G , 7, and 10 gives:
This very simple rquat'ion will be used to optimizr various column design and opcrnt'ing 1)aranieters. OPTISIIZATIOSPROCEDURE. Equation 12 is of general validity. H and X arc implicit functions of the column parnmctrrs. In a first simplified theory wc shall nssunie that I-I is indeprndcnt of k', T , and L , because the mathematics arc much simpler and because this is approsimatively the case where Iiarger and C'ooke (16) performed their espcrimciit,s. Then we shall show how thesc results arc niodificd in more comp1t:x cases. Using Equation 11 and the relevant plate hcight equation, we shall srarch the conditions whtw S is maximum. First we shall fix the analysis time. This is the normalizcd time condition, and we shall derive t'he optimum length of a colunin made with a given packing or a given tribe and stationary phase (capillsry column). This optimum length depends on various parameters &I.
. .I.
We shall compare the resolution given by column of opt'iinum length working at difftwnt carrirr gas velocities and made with different packings or tubes, but giving the same analysis time. This method gives the optimum condi1022
fR -
=
P3
fo
with
(uo,
Variation of analysis time with column length
ANALYTICAL CHEMISTRY
1 1 versus - = P2 IO
-1
--... ...
actual variations of = i3I2
fR
fR
tions, allowing us to have the maxinium resolution of a given pair of compounds in a given time. However, because of experimental difficulties, Karger and Cooke could not follow this procedure entirely. After having optimized the column length and capacity factor in a first paprr (16), they were obliged to work with columns of the same length to optimize carrier gas velocity and particle diameter. The temperature, and hence the capacity factor, were varied to satisfy the normalized time condition. This more practical experimental process does not guarantee that the column length is optimum at the resolutionoptimum velocity for example. However, the optimization process followed by Karger and Cooke is also amenable to theoretical calculation and we shall show it in the discussion of their experimental results. : Optimum Column Parameters. H and a Independent of t h e Temperature. Since the influence of temperature on the height equivalent t o a theoretical plate, H , and the relative retention, a , is very difficult t o account for, we shall discuss first the case where H and a do not vary with the temperature. Furthermore, this appears t o be the case where the experiments in normalized time conditions were performed by Karger and Cooke (16, 17). Comparison between their experimental data and the theoretical results will show us to what extent it is useful to refine the theory. OPTIMUMCOLUMNLENGTHAND CAPACITY FACTOR. In normalized time conditions, the estreme of the separation number and hence of the resolution is given by :
[4(;)"'
- 11 = 0
(13)
The first root ( L = A) gives a minimum of the resolution, ( R = 0). The second root gives a maximum of the resolution. It is obtained for a column length equal to the optimum value:
Equation 10 indicates when the column length is optimal:
RF
=
1 -;
4
IC'
=
3.
Equations 12 and 5 give the maxinium values of the separation number and the resolution :
(15) -The highest resolution which can be achieved in a given analysis time is obtained when the column length is optimum (Equation 14), and the column temperature is adjusted so that the capacity factor for the second peak is equal to 3. Inversely, when the second compound of a pair has a capacity factor equal to 3, then Equations 10, 13, and 14 indicate that the column length is optimum for that value of the analysis time; that is, it is impossible to get a better resolution without increasing the analysis time and lengthening the column. The improved resolution will then be obtained with the same k' value (that is, at the same
temperature), if the same packing is used to make the longer column. It must be pointed out, however, that with a column of given length L it is possible to obtain a better resolution than that which is achieved a t the temperature where k‘ = 3. But this resolution is obtained a t the expense of the analysis time. The two problems are quite different. I n time normalization we look for the highest resolution which can be obtained in a given time, whereas in length normalization the time needed for the analysis is not considered. We simply look for the highest possible resolution obtainable with a column of given length. It is easy to calculate dSldtR, using the same method as above, and t o show that it is always positive. But it is apparent from Equations 5 and 7 that the resolution ill be maximum for RF = 0, which means an infinite analysis time, and, since the analysis time is adjusted a t constant outlet flow rate, a temperature of 0” K. Then, from Equation 5 the resolution will be :
or capillary column diameter, or carrier gas velocity, can be discussed using the same equations. Although this process is not absolutely exact, for the sake of simplicity we shall use the following procedure. First, we shall optimize the column length for each set of values of d,, r,, and uo. The resolution obtained is given by Equation 15. Then we shall study the variation of the resolution given by these optimum columns when successively uoand d, or ro is varied. The optimum value of uois that which gives the maximum of the separation number (Equation 15) :
OPTIMUM PARTICLE A S D TUBEDIAMETER. The same method may be used
to calculate the optimum particle diameter for a packed column or the optimum tube diameter for a capillary column. R e shall study the variation with the particle or column diameter of the resolution given by the column of optimum length operated a t the resolution-optimum velocity. R e shall assume that A and B in Equations 17 are independent of the particle or tube diameter d; the actual dependence of A on d is still unclear in the present state of the kinetic theory (IO). For both types of columns C is given by : C
The results will of course depend on the variation of H with d, and u,. Using Equation 6, we can calculate the derivative
dSu ~
duo
which is given by:
Yk- - k’
VG
K
From what is apparent from a survey of the literature, this will lead to an increase of the working temperature or a decrease of the liquid phase ratio of the packed columns, and an inverse effect for the capillary columns. OPTIMUM CARRIERGAS VELOCITY. The influence of other parameters such as particle diameter in packed columns,
4c
=
(19)
.\lzcB -
The resolution-optimum carrier gas times higher than velocity is then the conventional optimum. When A is the most important contribution to the plate height, rendering 32BC negligible before AZ, uo becomes equivalent to A/2C. Of course, in the last case H is slightly dependent on uo and it would not be very profitable to work a t the optimum efficiency since with a low loss of efficiency it would be possible to use a much longer column a t a higher velocity (Equation 14). These results are much more complicated for the capillary columns since in most cases H depends on k’ and the two problems of optimizing u and L cannot be treated separately. This problem will be discussed later.
%‘s
=
0, this simplifies to:
H = 1.5 d 2 B C (23) Equation 15 allows us to calculate that dSM/dd has the same sign as:
2 dA2
The negative sign before the root will lead to a negative solution, which has no physical meaning. This resolutionoptimum carrier gas velocity is appreciably higher than the efficiencyoptimum. When A is negligible, for example in the capillary columns, then UKO
+ 3 A d A 2 + 32BC + 24BC 2 ( A + d A 2 + 32BC) (22)
This expression is positive for the lower values of uoand negative for high values of uo. It is null and RM is maximum for:
u.?&o =
(21)
H =
for A
+ d A Z + 32BC
+ Cz
U M ,0 :
3AZ
A
Cid2
when C1 and C2 are both independent of d . The column permeability k is proportional to dZ (12). Using Equation 19 we can calculate the value of the plate height a t the optimum velocity
&d = 0.232 [ t ? $ ~ ] ’ : ~ duo
For a column of given length, this is 33ojo better than the resolution obtained with the same column at k‘ = 3. This is a low gain compared to the high cost of an infinite analysis time. More practically, for a 10-fold increase in the analysis time (RFbecoming 0.025) the gain in resolution is 30%. Equations 14 and 15 show that if the normalized time is multiplied by 2, the optimum column length is multiplied by 1.6 and the resolution by 1.26. The theory of normalized time condition gives a much better solution to the problem of increasing the resolution. Another practical consequence, pointed out by one of the reviewers, is that not only the increase in resolution with an infinite analysis time is slight in comparison to k’ = 3, but also the peaks are so broad that we no longer can detect them. On the other hand, a short analysis time will allow a much higher sensitivity. The choice of the liquid phase loading depends on the value of the partition coefficient, K , a t the temperature which appears to be the most convenient :
=
+ 32BC (AT2 + 8BC2 8BCI2d4)+ 2ACz(A2 + 64BC2 + 64BCid’) (24)
This expression is positive for d = 0 and negative when d is very high. When A = 0 its root is:
Hence, C = 2C2. This means that at the optimum diameter both terms of mass transfer resistance are equal. If A is not zero, the greater A is, the greater the optimum diameter. It is easy to calculate this optimum diameter in two cases: in a capillary column, if the variations of H with k’ can be neglected and, in a packed column, if A is negligible. C2 is the term of resistance to mass transfer in the liquid phase; it may be anything besecond. With tween 10-4 and helium or hydrogen as carrier gas, C1(C,/d2) is generally a few units, say between 1 and 5. This makes the optimum value of d between 5 X and 5 X cm. The lower values are the most probable. This is well within the usual range, a t least for particle diameter. Capillary column diameters are generally somewhat greater than the optimum. Desty VOL. 30, NO. 8, JULY 1966
1023
Table I. DLl" q
x
108 (20" C.)
Do -
Comparison of Carrier Gases
H2
He
0,277 88 3150
0.248 196 1260
A 0.0587 222 264
N2 0.0726 175 415
9 a
n-octane, 30' C., 1 bar ( 3 ) [Doand q in C.G.S. units].
et al. used columns with different diameters and found that the narrowest columns always gave much shorter retention times with an improved efficiency at constant k ' ; hence, a higher resolution would have been obtained in normalized time conditions But the efficiency does not improve as predicted by theory, probably because of an increasing contribution of the apparatus to peak broadening ( 5 ) . It must be pointed out that if it is easy to make packed columns of different particle diameters with the same liquidto-gas-phase ratio (and hence the same k' a t the same temperature because, a t least in a wide diameter range, particle porosity does not vary with particle diameter) this is not the same with capillary columns. In order to keep the value of k' a t its optimum, it is necessary either to reduce df, which reduces C2accordingly and the optimum diameter d M (Equation 25)' or to keep the same df and to increase the temperature as the liquid to mobile phase ratio increases. I n glass bead columns the liquid mass transfer term is also proportional to d2. I n this case the optimum particle diameter will be derived from the expression (25) which is always negative. The smallest possible glass beads have to be used. This can be seen directly from Equation 17 if A is null. Then, H is proportional to d (Equation 23) and uo is proportional to d-l (Equation 20). Since k is proportional to d2, S M is proportional to d--213 and the resolution is proportional to d - l I 3 . The optimum column length is proportional to d-213. NATURE OF CARRIERGAS. There remains a last factor to optimize; the nature of the carrier gas. This calculation cannot be made in the same way as the previous ones because it is not standard practice to use mixtures of different carrier gases. Thus diffusion coefficients and viscosities cannot vary progressively. Therefore, we shall calculate which carrier gas gives the maximum resolution from Equation 17 and we shall assume that all the other parameters have been previously optimized. I n other words, we shall compare the resolution given in the normalized time by the different optimum columns working with the different carrier gases available. 1024
ANALYTICAL CHEMISTRY
Equation 25 shows that C is independent of D, when the optimized columns are compared. Column permeability k , which is proportional to d2, is proportional to l/Cl (Equation 25), hence to D,. Both uoand H (if we suppose that A is negligible) are proportional to B112and hence to Equation 17 indicates that S.hi will then be proportional to and independent of D,. The resolution is proportional to The best carrier gas in this condition (a = 0) will be the least viscous (Table I). So, hydrogen will be the best choice, and carbon dioxide the second. Helium appears to be a bad carrier gas; it is very viscous and expensive, and it gives plate heights almost as high as hydrogen. The difference however, is not very great. By turning from argon or helium to hydrogen, the resolution obtained in a normalized time is increased by 17% and 14% respectively. However, if approximately the same overall resolution can be obtained with columns using helium or argon as carrier gases, this cannot be done with the same column since the optimum particle diameter, the resolution optimum velocity, and the optimum column length depend on the nature of the carrier gas chosen. If A is so large that BC becomes negligible, uo and H are independent of D,, so that SM is proportional to (D,/~7)l/~and the resolution is proportional to ( D , , / T ) ~ ~ ~In. this case, hydrogen is also the best carrier gas, as can be seen in Table I, and helium is the second best. Hydrogen allows achievement of a resolution 1.17 and 1.51 times better than those obtained with helium and argon, respectively. Generally A and d/BCare of the same order of magnitude, and hydrogen will remain the best carrier gas, whereas the other gases will give roughly the same performances. OVERALLRESOLUTION.The overall resolution can be obtained by eliminating uo,k , H , C, and d, between Equations 17, 18, 19 (or 20), 21, 22 (or 23), and 25. If A is negligibly small the result is very simple :
where x = kdPp2. C2 is the resistance to mass transfer in the liquid phase.
BC1 is independent of the nature of the carrier gas, but represents the resistance to mass transfer in the gas phase. Together with x it depends on the quality of the packing. Comparison between Theoretical and Experimental Results. OPTIMUM LENGTH. Karger and Cooke (16) have performed most of the experiments of optimization suggested by the theory discussed above. First they have searched for the optimum column length in various normalized time conditions, using packed and capillary columns. Table I1 summarizes their experimental conditions, the optimum conditions obtainedfrom the experiments, and those derived from Equation 14. I n the two experiments made with packed columns and one of the two experiments made with capillary columns, the agreement between observed and calculated optimum column length is close. The calculated values are 10% lower than the observed ones. This discrepancy comes probably from the approximation made in estimating the values of colunin permeabilities. The capillary column permeability was calculated using Poiseuille's law, but it is known that this law gives permeabilities which are often well in excess of the observed values (16, 24). If k were 30y0 smaller, which would agree with some published results ( 1 4 , a perfect agreement between calculated and observed values could be obtained for the capillary column. The packed column permeability was taken from values measured in the author's laboratory, and it is our experience that packed column permeabilities vary greatly with the packing process used. The variations, however, rarely exceed 30%, which means a 10% difference on the optimum column length. Another source of discrepancy could come from the approximation made on the atmospheric pressure, the value of which was not stated. However, the dependence of the optimum column length on the cubic root of this pressure makes it improbable that this would be the major source of error. There is a much more important cause of discrepancy, which plays a major role in the last example given in Table 11. That is, the variation of H with T and k'. In the last experiment made with capillary columns, two isooctanes were analyzed in a normalized time of 405 seconds. The optimum column length would have to be the same as that obtained when dodecane and tetradecane are analyzed in the same normalized time, since Equation 14 is independent of the nature of the compounds studied. A comparison of Table I and Table I1 of reference (16) shows that in the first case H decreases steadily with k' when temperature and column length are increased.
It is impossible to account for variation of H with k' in Equation 15 without using a particular expression for H . This is not useful for a packed column where H does not vary greatly with k', although in some instances variations of H with the temperature would be important (16). This is necessary for the capillary columns. The optimization process must then be applied to Equation 15 and not separately to H and L. This will be discussed later. The optimum column length is proportional to the power 2/3 of the normalized analysis time, and tc the cubic root of the column permeability, the carrier gas mass flow rate (p,u,), and the inverse of the carrier gas viscosity. This means that for a given analysis time the optimum column length will be the longest for hydrogen, which has the lowest viscosity and gives the highest optimum velocity. Argon, which has a much higher viscosity and a lower optimum velocity, will allow for a shorter column length. The maximum resolution which will be obtained depends on the relative variation of the optimum column length and the HETP. It follows from this discussion and from Equation 14 that, when properly used, and if cy is independent of temperature, the capillary columns which have the lowest permeability and plate height always give the best resolution in a given time. However, Equation 17 is valid for both types of columns. A packed and a capillary column of the same permeability working a t the same outlet velocity will have the same optimum length. CARRIERGASVELOCITY.Karger and Cooke could not follow the optimization process we discussed above. They always used the same column at different carrier gas velocity. The temperature was adjusted so that the normalized time condition was fulfilled. The resolution is then plotted us. ii (reference 17, Figures 6 and 7). Equation 8 indicates that during such an experiment, in which t~ and L are /~ also conconstants, R F U , ~remains stant:
d
03
32
4-
O L O
I
The separation number becomes:
S
=
-[(-) L w H
uo
1/2
- 11'
/
.. : M
(29)
The derivative is: 01. 30, NO. 0, JULY 1966
0
1025
This expression is positive for u, =
d:-,
which gives the minimum value
(34)
of the plate height. It is negative when uois large. The value of uo which gives the maximum resolution is the root of Equation 31 :
I n the experimental conditions chosen by Karger and Cooke, and assuming a permeability of 10-7 cmS2for a column packed with Chromosorb of an average particle diameter of cm., which agrees with several published data (12-Q), this gives:
B
+ A ( W U , ) ~+’ ~
~ C U , ( W U , )” ~CuO2= 0 (31)
This equation cannot be solved, but it is easy to calculate its root when A , B , C, and ware known. Karger and Cooke gave two plots of H and R us. fi. It is possible to calculate u,if we assume that the column permeability is 2.0 X lo-’ cm.2 Equation 3 gives P and Equation 2 gives uo. Table I11 gives the experimental conditions and the optimum values of fi and uo derived from the theory and from experiments. Two theoretical estimations are given in this table: one assuming that A = 0 (then Equation 31 depends only on B / C , square of the efficiency-optimum velocity), and the second assuming values of A , B , and C which agree with the experimental values of H minimum and of the efficiency-optimum velocity (given in Table 111). The agreement between the observed and calculated values of the resolution optimum velocity is very good, in view of the approximations made in deriving the coefficients of the van Deemter equations. OPTIMUMPARTICLEDIAMETER.In this case again the theoretical optimization process differs from the experimental procedure followed by Karger and Cooke (17). They prepared for experimental convenience several columns of the same length, using support particles of different mesh size and selecting temperature and inlet pressure so that the average velocity and the analysis time would be the same for all the columns. I n these experiments the pressure drop is large, so we may simplify Equation 2 and replace P z - 1 and P3 - 1 by P z and P 3 , respectively. We obtain Equation 32:
p = -417La 3p0k
I n the same conditions Equation 1 simplifies to :
If the average velocity is constant, which results from the normalized time condition, tl is constant in Equation 32. The outlet flow velocities will vary accordingly to : 1026
ANALYTICAL CHEMISTRY
uo =
0.0072 d2
(34’)
~
Equation 34 indicates that in the experiments, ku, is constant. Since L , tR, and P oare also constant, Equation 10 shows that RFwill be constant. This was independently demonstrated by Karger and Cooke and is consistent with their statement that the difference in temperature between the various columns does not exceed 6” C. This variation of 6” C. results either from minor variations in the packing densities or from the fact that the variation of uo and fi with P is more complicated than we assumed. So S is inversely proportional to H , which is the only variable remaining in Equation 16. Hence, _1 -dS =
- -1 -dH
S dd
(35)
H 2 dd
Assuming as above that H is given by
a general van Deemter equation, in which uois a function of d by equation 34’: H=A+-
Bd2 0.0072 + 0.0072 (C1
+ $)
(36)
the sign of dS/dd will be that of:
- dH dd
= ~ ( 0 . 0 0 7 2c 2
d3
-
m2
The resolution is maximum when: d =
4
4
5.3
x
10-51 B
(37)
For a column of the type used by Karger and Cooke, B and C are estimated a t 0.2 cm.2 per second and 2.7 X 10-3 second, respectively (Table 111), when the average particle diameter is 3 x cm. Cz may be estimated at about 6 X second and C1 at 2.3 seconds per cm2. This gives a n optimum particle diameter of : d
=
19 X
cm.
15 X cm. Again the agreement between theory and experience is very close. In another series of experiments made on the same columns, Karger and Cooke (17‘) kept constant the elution time, t ~and , the inlet pressure, P,. Equation 1 indicates that the outlet velocity, uo,is proportional to the column permeability, k , or to d2. Since L , t ~and , p , are constant, Equation 10 shows that kRF, or d2RF must be constant for the various columns. I n the experimental conditions of Karger and Cooke the numeral values are: uo = 2.45 X 105d2, and ( 1 k’) = 2.730 d Z . Using the same method as above we can calculate that the optimum diamem. The experieter is 23 X mental data (reference 17, Table I) are compatible with an optimum value cm., which again of d around 16 X is in fair agreement with the theory, Optimization Problems in Complex Cases. The preceding results are mainly based on the two following assumptions: H is independent of T , k’, and L; and CY is independent of temperature. The first assumption applies mainly to packed columns, the H E T P of which depends but slightly on k’ alid T in a fairly large temperature range. It mas shown by Dal Nogare and Chiu (4,and results from a theoretical discussion by Giddings
