J. F. Parcher University of Mississippi University, M~ssissippi38677
Retention Volume Theories for Gas Chromatography
T o the casual observer, the chemical literature contains an apparently unlimited collection of chromatographic theories. There are several reasons for this surfeit, the most important being the basic classification of chromatographic theories into retention and dispersion theories. A truly complete theoretical treatment will ovcrcome this basic distinction; however, this "perfect" theory is not presently available; and the practicing chromatographer must deal with a number of t,heoriesof limited applicability. The dispersion theories attempt to relate the second moment of an elution peak, or boundary, to physical parameters such as flow rate, particle size, liquid phase properties, etc. The dispersion of a plug sample can be described in terms of the plate height or a mathematical description of the locus of the point C = f(V) or f(t). This work has been reviewed and evaluated by numerous authors (1-4) and consequently will not be discussed in this paper. The retention theories attempt to relate the first moment of a chromatographic peak or boundary to the same physical parameters. The first momcnt is usually described in terms of the retention volume. The retention time is often a more convenient parameter, but becomes meaningless undcr any conditions in which the flow rate is uncertain or inconstant. Another reason for the multitude of retention theories is that there are several differenttypes of chromatography. These different typcs are listed in Table 1. Because of these four types of chromatography, there Table 1.
Types of Chromatography
Type
Physical Basis of Classification Dilate/ The term "dilute" chromatography is used to indiNondilute cate 8. system in which the sorption effect i.,negligible, i.e.! 11%- 0. The term"dilute"hasoften been assocmted with the region of z3in which the isotherm is linear however, this is ambiguous. In this text, we Gill use the sorption effect criteria t o define theC'dilute" region. Perfect/ We will define "perfect" or "perfect gas" chrome Imperfect tography as chromatography for which the Ideal Gas Law provides an adequate description of the gaseous phase. The term "ideal" would he preferable, however this name has traditiondly been applied to chromatography involving elution peaks with a. negligible sccond moment. Linear/ If the region of the partition or adsorption, up to z. Nonlinear is linear, then the pesks will generally he symmetrical, neglecting sorption and imperfection and the type of chromatography is "linear." At suficiently low 2 2 all isotherms will be linear, however, due to detector limitations this linear region is often exceeded and the system is then nonlinear. Two/Three This classification is based on the number of phases Phase involved in the retention mechanism applicable t,o a eiven solnt,e.
472 / Journal o f Chemical Education
are sixteen possible combinations ranging from dilute, perfect, linear, two phase chromatography to the other extreme of nondilute, imperfect, nonlinear, three phase chromatography. Fortunately, not all of these combinations are observed and, in other cascs, the distinction between types is negligible. Another possible classification not appearing in Table 1is the equilibrium state of the chromatographic column. In most cases, two phase gas-liquid chromatography can be adequately described as an equilibrium process. However, this assumption is often dubious in gas-solid, three phase, or nondilute types of chromatography. This is an assumption which must be verified in all cases, since all of the theories and equations discussed in this treatment are equilibrium theories, based on the assumption of instantaneous equilibrium between the mobile and stationary phases. The purpose of this paper will be to provide the student of chromatography and the practicing chemist with a critical guide as to the applicability of the different theories to the different combinations of types of chromatography, and to minimize the common practice of applying the simple V, = Vm KV, equation to all types of chromatography.
+
Deflnition of Retention Volume
The retention volume and partition coefficient are the connecting links between chromatography and thermodynamics and there has been some confusion in the literature concerning the definition of the retention volume. For all types of dilute chromatography, the retention volume is defined as the volume of gas, a t column temperature and mean column pressure, which passes out of, or into, the column in the interval 0 5 t 5 t,. Where t, is the time of emergence of the peak maximum. The retention volume is related to the velocity of the peak maximum, v, by eqns. (1) and (2).
where F , = volume flow rate of the carrier gas corrected to column temperature and mean column pressure (ml/min) ; x = distance from the column inlet (cm) ; and A , = cross-sectional area of the mobile phase (em" ).
This work was supported by a Frederick Gardner Cottrell grant from the Research Corporation and Grant No. GP-27999 from the National Science Foundation.
The retention volume can be measured at either the column inlet or outlet. I n the case of dilute chromatography, the two measured volumes will be equal when corrected to column temperature and pressure. In practice, i t i. s rrlatively simple matter to measure the retention volume for ,111 dilute types of chromntography, since the column flow rate is usually constant. The retention volume is then simply the product of the flow rate and the retention time. This is the usual method of calculating the retention volume, however, it is based on the assumption of constant flow rate. If the flow rate changes in the interval 0 I t 5 t,, then the simple calculation is not accurate and eqn. (1) or (2) must he employed. I n the case of nondilute forms of chromatography, the sorption effect will operate and the flow rate within the sample plug will not equal that in the region of the column containing pure carrier gas. Frontal chromatography is an extreme case in which the feed gas is a binary mixture of solute and carrier gas and the volume of gas flowing into the column will not equal thevolume flowing out in the interval 0 I t I t,. Thus, the retention volume measured a t the inlet will not equal that measured a t the column outlet. This is a real distinction, and we will designate the two retention volumes as V,( and V,a, respectively, when discussing frontal or other types of nondilute chromatography. Still another factor is important in nonlinear types of chromatography because the retention volume is a function of the concentration of solute in the carrier gas in this case. We will designate a retention volume which is dependent on concentration as V,(C). It is more convenient to use concentration (mole/l), rather than mole fraction, since all of the solute a t a given concentration travels a t the same rate a t any point in the column. The concentration can be obtained directly from the mole fraction by use of the virial equation of state for a binary gas mixture. PV = nRT(1
C
=
+ BPIRT)
YP/(RT
+ BP)
(3)
(4)
where B = YaB,,
+ 2Y(l - Y)Bzsf (1 - Y)2Bns
(5)
For ideal types of chromatography, eqn. (3) would reduce to the usual C = Y P / R T . Complicating Factors i n Gas Chromatography lmperfed Gas Phose
The common chromatographic carrier gases, such as helium, hydrogen, and nitrogen, are not ideal gases and, especially when mixed with a hydrocarbon solute, show significant deviations from the Ideal Gas Laws. The Virial Equation of State can be used to describe such gases and Table 2 is a compilation of the second virial coefficients of several common gases and vapors along with the factor BP/RT a t 27°C and one atmosphere pressure. The last column in the table gives an indication of the magnitude of the error introduced by neglecting gas phase imperfections. This error is not large under normal conditions, however, it will increase with concentration, pressure, and decreasing temperature. Since the correction procedures are relatively simple and the virial coefficients of many common
solutes are readily available (S), it is advisable to use the Virial Equation for accurate work, especially for high pressure or low temperature work. Compressible Mobile Phose
If the mobile phase of a chromatographic system is a gas, the local volume flow rate will be a function of the position of the sample peak in the column and the true or mean retention volume will depend upon the pressure drop across the column. Cruicksha&, Windsor, and Young ( 6 , 7 )have shown that the retention volume of a small eluted sample in a nonideal compressible mobile phase is given by eqn. ( 6 ) . In VPo= InKoV, + BPaJa4 ~ ( P O J J ~.). ~. (6) where
+
+
and 0 = (2B1z - vZ0)/RT;PO= outlet pressure; Pi = inlet pressure; and 6 = (3C822 - 4B23B2~)/2(RT)2. The usual textbook approximation assumes an ideal gas, and eqn. (6) reduces to In V, = In KOVI
(7)
where V , is the retention volume corrected to column temperature and pressure and adjusted for the mobile phaHe volume. Equation (7) is an approximation and several authors have shown the second and third terms on the RHS of eqn. (6) to be significant, especially at high pressures. This equation has been used quite successfully to measure virial coefficients of gas mixtures by means of a plot of In V , versus PJa4 ( 7 , E ) . Sorption Effect
The sorption effect is the change in the mobile phase velocity in a region of significant solute concentration when a boundary between this region and a region of different solute concentration is within the column. Since the solute moves only in the mobile phase, the flux of solute must be different in the regions adjacent to a concentration boundary passing through the column, because a fraction of the solute is removed into or out of the stationary phase. This net absorption is limited to the transition region and nowhere else. Thus the sorption effect is only operative while a boundary is within the column and always results in a decrease in the outlet flow rate during saturation with an increase during the desorption process. The inlet flow rate increases during saturation and decreases during desorption. Table 2.
Magnitude of Gas Phose Imperfections
Gas Phase
B (ml/mole)
Hydrogen 14.8 ( 5 ) Helium 11.7 (5) Nitrogen - 4 . 2 (6) n-Hexane -1910 (5) Hydrogen n-Hexane (Y = 0 . 1 ) 60 Helium 2. n-Hexane (Y = 0.1) n-Heusne Nitrogen -4ZS (Y = 0 . 1 )
+
+ +
BPIRT
% Error
0.0006 0.0005 -0.0002 -0.0777
0.06 0.05 -0.02 -7.76
0.0002 0.0001
0.02 0.01
-0.0017
-0.17
B, estimated by the Hudson and McCoubrey Combining rule ( 6 ) . a
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473
Bosanquet and Morgan (9)first proposed this effect in an early attempt to account for asymmetric peaks in systems with linear partition isotherms. On the sorption boundary, the velocity of the leading, solute free, region of the boundary, u., is related to the velocity of the gas in the solute rich region, ui, by eqn. (8). uc/ui
=
1 - (kYl(1
+ kj)
(8)
The velocity a t any point on the boundary is given by ub
=
w/(l
+ k) = uJ(1 + k ( l - Y ) )
(91
The retention volumes corresponding to these velocities are V,,
=
V,
V,a(Cj = V ,
+ KV,
+ (1 - Y ) K V I
+
>
+ +
Where g+ = q j ( l Ic) - 7.g; and g = 1 k (kt kl)X?. One of the difficulties encountered in a mathematical treatment of these systems is the choice of a reference velocity, because both the inlet and outlet flow rates will change with time during the interval in which the peak or concentration boundary is in the column. ICrigc and Pretorius used the outlet flow rate after all boundaries were out of the column as the base velocity, uo. Experimentally, this is often hard to measure because thc effluent often contains one or more condensable components. An alternative base velocity is the outlet velocity prior to the entry of a boundary into the column, m*. I n this case the effluent is usually pure carrier gas and the flow rate can be readily measured, however, the inlet pressures must be accurately controlled. The two base velocities are easily related using Darcy's Law. uo*/uo = 8*nmix/Bne
All of thc previous retention volume equations have been based on thc assumption of a constant partition cocfficicnt, K. This term represents the ratio of the conccntrations of thc solute in the stationary and mobile phases, and is the usual link between chromatography and thermodynamics. However, all of the mass balance equations used to formulatc the basic theory of chromatography result in a term involving (bQ/bC),, rather than Q/C or K. Helffcrich and Iilein (14) have discussed the distinction bctween these two terms and \ye will mcrcly point out that K is a chord on the isotherm, while the differential term is the slope at a given point c on the isotherm. If the isotherm is linear, the two terms are equal and this is often the case for very low concentrations. Unfortunately, few systems have perfectly linear isotherms at moderate concentrations, and even nearly ideal solutions, such as n-hexane in n-hexadecane (15) show a significant deviation between the slope and the chord on the isotherm a t Xz 0.3. Most chromatographic experiments involve decidedly nonideal solutions due to polarity, size effects, and/or surface adsorption, so that low concentrations are not always an effective means of avoiding nonlinear isotherms, especially in the case of surface adsorption. The retention volume equations for a system with a nonlincar isotherm are given by eqns. (12) and (13) (15).
Journol of Chemicol Education
Dyson and Littlewood (17, 18) have observed that the viscosity of an eluted sample can have a significant influence on the mobile phase flow rate during an elution experiment. These workers did not propose a retention volumc equation, however, Krige and Pretorius (19-33) proposed the follolving equations for frontal chromatography with gases of differing viscosities.
(11)
Nonlinear Portition Isotherms
/
Gas Phose Viscosity Effects
(10)
Thus the sorption effect mill produce outlet retention volumes which depend upon the concentration and consequently produce asymmetric elution peaks. This effect has been recognized and studied by several authors (10-15). Obviously, this effect is not significant a t low conccntrations and cqn. (11) rcduccs to cqn. (10). The sorption cffcct will be significant in the case of large elution samples, frontal, or preparative chromatography. In cqns. (10) and ( l l ) , the term V , represents the volumc of the mobile phase i11 thc column. If this phase is a gas, then V , is the volumc of gas contained in thc columu at column temperature and mean column pressure. This term is usually measured from the retention volume of a sample which does not partition into thestationary phase, i.e., K = 0.
474
An equation similar to eqn. (13) has been derived by Conder and Purnell (16) except that V,, appears in the equation rather than VTOV,,.
(15)
The permeability of the column, P, may depend upon the composition of the feed gas if a significant amount of solute is absorbed into the stationary phase. However, it is usually assumed that @ = P*. This is equivalent to assuming that Vmis independent of the amount absorbed, Q, and this is a common assumption in chromatography. If this assumption is valid then the outlet flow rates prior to, and after, saturation are determined solely by the viscosities of the gases involved. For a sharp boundary or "stationary" front, the situation is relatively simple, however, for a diffuse boundary, the viscosity of the mobile phase will vary with the composition of the gas. The outlet flow rate will vary with composition due to the sorption effect and also due to the change in viscosity of the gas phase with concentration. Solid or Liquid Surfoce Adsorption
Any added retention mechanism other than partition of the solute into the stationary phase must also enter into the retention volume equations. I n the table of types of chromatography, this additional retention was given the title of three phase chromatography, even though the liquid surface does not represent a true third phase. This terminology is acceptable, since the retention volume equations have the same form for both liquid and solid surface adsorption, i.e., the liquid surface appears as if it were a separate third phase in the equations. Martin (83) and several other investigators (84-86)
have proposed an equation of the form of eqn. (16) to describe the retention volume of a chromatographic system involving liquid surface adsorption. V,
=
V,
+ KV, + K,A,
(161
This equation assumes a linear surface adsorption isotherm and neglects the sorption effect. A similar equation has been proposed for solid support adsorption in systems with nonlinear isotherms (25, 87). V,a(C)
=
Vm
+ @ Q / W 0 V ~+ @ Q , / ~ C l A
(17)
There is little data available for isotherms of coated supports and eqn. (17) has never been rigorously tested against a static isotherm. The equation has been used to predict elution retention volumes from isotherms obtained from frontal chromatography, however, this is not an accurate or conclusive test. Complete Retention Volume Equation
Almost all of the retention volume equations given previously were derived from a basic continuity e q u a tion and these derivations are given in many references (13, 14, 16). If the same derivation technique is applied to a chromatographic system with nonlinear surface adsorption isotherms and a significant concentration of solute, the following general retention volume equation will result V,JC) =
v,
+ (1 - Y ) ~ ( a Q , / a c ) , v+~ (aQ~,/bc),Au
+ (aQ,./bC).A,,l
(18)
A more convenient form of the equation can be obtained by defining the various Q values as the amount sorbed per unit length, then the total amount of solute sorbed per unit length, Q t , would be Qt = Q I ( V J L )
+ Q~a(Ata/Ll+ Q&L/L)
(19)
=
v,
(20)
This equation avoids the distinction between the various retention mechanisms and can be readily tested since most common isotherm measurements will give a value of Q,, rather than QI, &I,, or Q,,. Table 3 is a summary of the different retention volume equations and the applicable domain for each equation. I n this table the equations are presented in the form which assumes an ideal carrier gas and an ordinary pressure range, so that the p and 6 terms of eqn. (6) can be neglected. All of the equations in Table 3 can be corrected to allow for high pressures and nonideal carrier gases through the use of the treatment given by Cruickshank, Windsor, andYoung (6,7). Table 3.
Types of Chromatography
Nondilate, linear, three phase Nondilute, nonlinear, two phase Nondilute, nonlinear, three phase
-
A
Literature Cited (1) G I D D I N ~ S J.. C.. "Dynamic8 of Ch~om&topmph~." Vol. I , Part 1. Mar ce1 Dekker, 1nc.. New York. 1%5. (2) Kwox, J. H.,ANDPARCIIER, J. F., Anol. Chem., 41,1599 (19691. (3) DAI. N o a ~ n e S., . A N D JOVET.R. 8.. "GBs-Liquid Chromatography.'' Interaoienee Publishers. NervYork, 1962. (4) PORN EL^^, J. H., "G&s Chromatography." J . T i l e y & Sons. Nerv York. 1061
(5) D r ~ o n r o J. , H.. A N D SMITE. E. B.. "The Virial Coefficients of Gases," Cisrendon Press. Oxford, 1969. x, AND YOU NO.^. L.,PWC.Royal (6) C ~ u ~ C r s n n NA.J.B..WINDPOR.M.L., Soc.. 295A, 259 (1966). J. .B.. wmoson, M.L..ANDYouNB.C.L . , P m ~ . R o s a (7) C ~ n r ~ a m n n n . A Soe., 295A. 271 (1966). B. W.. HICHS,C. P., LBTCXER, T. M., (8) C R U ~ O R B H AAN. JB. .B.. GA~NEY, MOODY. a. W.. m o YOONO,C. L., T i o m Fawdoy Sm., 65, 1014 19f'.l. c. H h ~ r n a ~C. s . 0.. ,n"vs,..r ~ ~ a , ~ c ~ , r o r n n r n ~ '11 r.plh ' lE'dllnr r,r,rr D. I 1 ) I l u t t e n . ~ n : . 'l .~n. n l .n 1957,p. ?. (I01 Gol..oao, \.""",.
C O N D . ~J. . R.. .I. Chromoloa.. 39. 273 iI969I . . CADODAN.D. 5.. CONYER.J . R.,LOCIE. D. C., A N D P"RNEL'. J . H.. J.Phys. Chem., 73,708(19691. URONE. P.. PAROIER,J. F.. A N D BAILOR.E. N . , in "Separation Techniques in Chemistry and Biochemistry;' (Bdilor: KELLER,R. A,. MarcelDekker, New York. 1967, p. 193.
Retention8 Volume Equations
Equation No. (10) (16) (12) (17) (10) (11) (16) (21) (12) (13) (221 (18)
(131 P n ~ ~ n s o rD. r . L., AND HELPTERICH, F., J . Phys. Cham.. 69, 1283 (19651. F..AND KGEIN.G., " M n l t i c ~ m p ~ n e nChromatography." t (14) HGLFFERICX. Marcel Dekker, Ino.. New York, 1970. M.. L., A N D W ~ r ~ m w s A. o ~G.. . Tvonr. Foroday Soc.. 57, (15) M c G ~ ~ s n m
Retention Volume Equation V,',. = V, V,. = V ,
+ KrV,
+ K,V, + K,A,
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475
