Anal. Chem. 1995,67, 613-619
Theoretical Relationships between the Void Volume, Mobile Phase Volume, Retention Volume, Adsorption, and Gibbs Free Energy in Chromatographic Processes K. S. Yun, C. Zhu, and J. F. Parchec* Chemistty Department, University of Mississippi, University, Mississippi 38677
The critical roles of dead time and void volume in chromatographic measurements are discussed. Void volume measurements are particularly important for chromatographic systems in which one or more components of the mobile phase (eluent) form an integral part of the stationary phase. Such systems include HPK, SIT (especiallywith polar modifiers), and GC with adsorbable components in the carrier gas. In each of these cases, the measured column volumes determine the exact type of capacity factor obtained, the type of adsorption measured, and the experimentallydetermined volume of the adsorbed phase. Concise definitions are presented for the different types of void volumes and adsorbed phase volume of chromatographicsystems along with a detailed discussion of “excess” and “total” adsorption. Experimental methods are also discussed for the accurate determination of the entire amount or volume of eluent in a column, the void volume, the total amount of material adsorbed, and the excess amount of material adsorbed. The effects of these parameters upon the type of capacity factor, equilibrium constant, and free energy are examined in depth. Much has been written concerning the experimental measurement as well as the theoretical significance of the column “dead time” required to determine the “void volume”of chromatographic columns and the retention volumes or capacity factors of probe solutes. At least six different methods have been proposed for the experimental determination of the void volume of a chromatcgraphic None of these, however, seems to be entirely satisfactory, and the problem appears to be especially obdurate for RPLC systems. In addition to the experimental difficulties, the exact definitionsof commonly used terms such as void volume, mobile phase volume, capacity factor, and surface adsorption are not universally agreed upon, even though each term has been discussed extensively in the literature of chromatography and surface science. A secondary problem is the common use of the terms “void volume”, “dead volume”, “and holdup volume” to describe the same parameter. The three expressions appear with roughly (1) Zhu, P. L. Chromatographin 1985,20, 425-433. (2) Djerki, R A; h u b , R J. J. Liq. Chromatogr. 1987,10, 1749-1767. (3)Kazakevich, Y. L.; McNair, H.M. J. Chromatogr. Sci. 1993.31,317-322. (4)Sadek, P. C.; Cam,P.W.; Bowers, L. D.LC-GC 1984,3, 590-592. 0003-2700/95/0367-0613$9.00/0 Q 1995 American Chemical Society
equal frequency in the literature, and there is no consensus regarding the propriety of each term. Ettre5has discouraged the use of the term “dead volume” but made no recommendations regarding the other two terms. Most authors use holdup volume to mean a measured retention volume, whereas void volume is a k e d , physical parameter which represents the interstitial (both interparticle and intraparticle [pore]) volume of a given column. In this sense, the holdup volume for an unretained probe solute would provide an experimental measurement of the void volume of a column with nothing adsorbed on the stationary phase. The term “mobile phase volume” is ambiguous because of the implication of a moving phase volume and the fact that some components of the mobile phase may be resident in the stationary phase. Martire and ScottG have suggested that the mobile phase in HPLC systems may consist of as many as four distinct domains, with only one of the four actually representing a moving segment of the so-called “mobile phase”. Exact measurement and dehition of these terms are sometimes unnecessary. However, in many instances,precise defhition and accurate measurement of the void volume and holdup volume of an unretained probe solute are crucial for the correct thermodynamic interpretation of chromatographicdata. Examples where such parameters are critical include any chromatographic system in which signifcant adsorption or absorption occurs. Typical situations would include RPLC with multicomponent mobile phases or bonded stationary phases, SFC at high pressures, or GSC with adsorbable components in the mobile phase. The latter systems become most problematic at low temperatures or high pressures. In each case, signifcant adsorption or absorption of mobile phase components on or in the stationary phase can influence the holdup volume of unretained solutes and the capacity factors of analytical solutes. Moreover, when the concentration of an adsorbable component of the mobile phase becomes significant, a new problem is encountered, viz.,specifcation of the exact type of adsorption being measured, Le., whether the total or excess amount of material adsorbed is measured in a given chromatographic experiment. All of these parameters are interrelated, and the experimentally measured retention volumes must be interpreted with rigorous regard to the precise definition of the theoretical parameters. For example, a particular definition of the holdup volume of an unretained solute will determine the type of adsorption measured as well as the value of the retention (5) Ettre, L. S. J. High Resolut. Chromatogr. 1993,16, 258-261. (6) Alhedai, A; Martire, D.E.; Scott, R P.W.Analyst 1989,114,869-875.
Analytical Chemistty, Vol. 67, No. 3, Februaty 1, 1995 613
parameters and the exact free energy relationship obtained from those parameters. Chromatographic measurements can yield negative values for both the amount adsorbed and the capacity factor under certain conditions. Such results can be disconcerting without a clear understanding of the exact definitions which dictate the resulting chromatographic data. There are precise definitions and relationships given in the literature; however, these discussions are sometimes fragmented and often unrelated to chromatographic investigations. The objective of the present work is to idenbfy and illustrate exact definitions for the parameters discussed above and to illustrate the relationship between the chromatographic parameters of retention volume and capacity factor and the derivative results of the amount adsorbed and the free energy change associated with a well-defined chromatographic process. The initial discussion must necessarily center upon the exact definitions, symbols, terminology, and mathematical relationships among the measured and derived chromatographic parameters.
0
2
Distance Figure 1. Illustration of the concept of the Gibbs dividing surface.
system and Vis the volume of the bulk liquid, fluid, or gas out from the Gibbs dividing surface. That is,
V = A&
dz
TOTAL (ABSOLUTE) AND EXCESS ADSORPTION
The IUPAC definiti~n~-~ of np, the surface excess amount of adsorbed substance (Gibbs adsorption), is as follows: “(np)is the excess of the amount of component i actually present in the interfacial layer over that which would be present at the same equilibrium gas pressure in the reference system in which the gas phase concentration is constant up to the Gibbs surface, and the reference concentration of component i is zero beyond the Gibbs surface in the surface layer of the solid.” This definition involves the notion of a “Gibbs surface”, more commonly called the Gibbs dividing surface (GDS). The concept of a GDS is illustrated in Figure 1, where ci represents the concentration (molar density) of component i and z is the distance from the adsorbent surface. In particular, c:ds and cPLkare the concentrations of component i in the adsorbed and bulk (gas, liquid, or fluid) phases, respectively; zd represents the location of the GDS and zba the point closest to the surface at which ci = C ? ~ . In real systems, the interfacialboundary is probably less well-defined than in the example shown in Figure 1;however, the basic concept is still useful. Mathematically, the IUPAC definition of np is given by eq 1, where A, represents the surface area of a solid adsorbent:
An equivalent equation which is easier to visualize and leads to
simplified expressions is the following:
From any of these definitions or expressions, it can be seen that the amount as well as the exact type of adsorption is critically dependent upon the position of the GDS. Riedo and Kovats’O discussed several conventions for tiXing the position of the Gibbs dividing surface in liquid-solid systems. Such conventions include two that are commonly used for establishing the position of this theoretical boundary of the bulk phase, viz.,(i) the GDS is defined to coincide with the physical surface of the solid adsorbent and (i) the GDS is established at the interface between an adsorbed film with concentration cpds and the bulk fluid with concentration tiba, as illustrated in Figure 1. In addition, a thiid convention is applicable for chromatographic systems, viz., (ii) the GDS is defined to be the boundary between the stationary and moving phases in a chromatographic column. The latter convention is unique to chromatographic systems and provides an extra “dimension”which is not available with static volumetric or gravimetric methods for the investigation of phase distribution equilibria. Equations 1and 2 are general and applicable to any adsorption system; however, the GDS conventions form the theoretical basis for a subdivision of adsorption into two distinguishable, but complementary, types of adsorption commonly designated as “excess” and “total” (or “absolute”) adsorption. Excess Adsorption. Convention (i) is used quite commonly, and the resultant adsorption data are designated as excess adsorption, npxcess.The mathematical definition of this particular type of adsorption is given by the relation
Equation 2, along with the assumption that cpa is constant, leads to a working definition of np, viz.,
np =
Again, a working definition of npxess can be developed from eq 5, - ‘?Cv
(3)
where n? represents the entire amount of component i in the (7) Everett, D.H. Pure Appl. Chem. 1972,31,579-638. (8) Sing, K S. W. Pure Appl. Chem. 1982,54, 2201-2218.
where
(9)Everett, D.H.Pure Appl. Chem. 1986,58,967-984.
(10) Riedo, F.;Kovats, E. Sz. 1.Chromatogr. 1982,239,1-28.
614 Analytical Chemistry, Vol. 67,No. 3,February 1, 1995
lp
is the volume of the bulk liquid, fluid, or gas up to the
physical suvface of the solid. This volume is defined by the relation
v = A s h - dz
(7)
The original IUPAC symbolgfor excess adsorption was np(”, and the defined position of the GDS is equivalent to the UNA convention of Redo and Kovats.lo The UNAconvention is based on the assumption that “nothing is adsorbed in terms of volume” and the position of the GDS coincides with the physical surface of the solid. Excess adsorption is zero for singkomponent, liquid systems, because ci = C P at~any z, but may be nonzero for singlecomponent gas or supercritical fluid systems. In multicomponent, liquid systems, the sum of the excess adsorption of all components must be zero, i.e., Zniacm = 0. This restriction implies that npess for at least one component must be negative unless nacessis zero for all components. This condition is not true for gas or supercritical fluid systems because of the existence of empty space in the column. Dannerl1-l3 has developed an adsorption theory in which vacancies-defined as vacuum entities of the same size as the adsorbate-are treated as an additional component in the system. In this case, the condition Cnjexcess= 0 would again be valid with the vacancies included in the summation. Excess adsorption is an experimentallyaccessible quantity, and it has the added advantage that no model is required for the interpretation of the experimental data. That is, the data can be obtained experimentally without resorting to any abstract boundary concept, such as the Gibbs dividing surface. Total Adsorption. Convention (ii), which is equivalent to Redo and Kovats’sloJNA convention (where component J is an unretained and unexcluded probe solute), and convention (ii) are less commonly used because of the dficulty of establishing the exact position of the adsorbed film/bulk fluid interface or the mobile/stationary phase boundary. However, the type of adsorp tion derived from these conventions, which is usually designated as total or absolute adsorption (n?“ = np, as defined by eq l),is more relevant to chromatographic theory than npxcess, especially if the adsorbed film acts as the stationary phase in a chromatographic column. The possible correlation of the GDS and the interface between the mobile and stationary phases, Le., convention (iii), is a unique feature of chromatographic systems. However, such an association requires careful and exact definitions of both the void volume and the mobile phase volume in any column. Wittkopf and Brauer14 discussed the different conventions regarding the GDS in terms of a one-phase or two-phase model for thermodynamic expressions of the excess and total adsorption concepts. The one-phase model, in which the GDS and the solid surface are coincident, was used to illustrate excess adsorption, whereas a two-phase model, with the phases differing in composition and/or density and the GDS located at the interface of the two phases, was used to interpret total adsorption. From eqs 2 and 5, the two types of adsorption, excess and total, can be related by the expression (11) Suwanayuen, S.; Danner, R P. AIChE]. 1980,26, 68-76. (12) Suwanayuen, S.; Danner, R P. AIChE J. 1980,26, 76-83. (13) Cochran, T.W.; Kabel, R L.; Danner, R P. AIChE]. 1985,31, 20752081.
(14) Wittkopf, H.; Brauer Adsorpf. Sci. Tech. 1986,3, 271-291.
Because cPa is a constant, eq 8 reduces to
where
is the volume element bounded by the Gibbs dividing surface and the surface of the solid adsorbent. The term (VgdscCpd} in eq 9 represents a hypothetical quantity equal to the amount of component i that would occupy the volume Pads $the concentration of i in this volume were cpUlk. This hypothetical amount may be less than, equal to, or greater than the actual amount of component i in the volume Vgds of a real system. Thus npxcess may be positive, zero, or negative (ii ci < cPa for z IZS), whereas nit.“ is always nonnegative. In most gas-solid systems, cPkis very small, and the two types of adsorption are practically equivalent in magnitude. The symbols and terminology used by various authors differ widely with regard to the expressions for excess and total adsorption. The original IUPACgsymbol for the total amount of material adsorbed was np; unfortunately, the type of adsorption was defined in terms of the excess of component i in a real system over that of a reference system (eq 1). In addition, several authors1J5J6referred to this kind of adsorption as “absolute”,while others17use the term “total“for the same type of adsorption, Le., eq 1. In this work, the term total adsorption will be used with the symbol ntod but will still be defined by eq 1. With regard to excess adsorption, the symbol nexcesswill be used throughout the present paper, and the definition expressed in eq 5 will apply. CHROMATOGRAPHIC COLUMN VOLUMES
Void Volume. The void volume of chromatographic columns and strategies for measuring this parameter have been discussed extensively in the literature over a long period of time. Liquidsolid systems, especially those with species chemically bonded to the surface, present special difficulties if one tries to distinguish between the void volume and the volume of the mobile phase.lJ0J8 In these liquid systems, the mobile and stationary phases differ only in composition if at all. That is, the stationary phase is often composed mostly of mobile phase components. Because of this complexity, precise definitions of the terms mobile phase volume and void volume are imperative. It has been recommended by several authors3J6J8that the term void volume be defined as the total volume of eluent in the column, as defined by eq 7, without regard to the concepts of mobile and stationary phases. This defhition is most meaningful for liquid systems because the molar volumes of the eluent components are constant whether the component is present in the stationary or mobile phase. This condition is not true for gas- or fluid-solid systems; however, the definition is still valid, and the void volume in these systems (15) Haydel, J. J.; Kobayashi, R Ind. Eng. Chem. Fundam. 1967,6,546-554. (16) Martire, D. E.; Alvaredepeda, A ]. Chromafogr. 1991,550, 285-300. (17) Liu,Y.; Liu,L.; Yun, K S.; Zhu,C.;Yu, B.; Parcher, J. F.Ana1. Chem. 1994, 66,2852-2857. (18) Knox, J. H.; Kaliszan, R]. Chromatogr. 1985,349, 211-234.
Analytical Chemistry, Vol. 67, No. 3, February 7, 7995
615
would equal the mobile phase volume with nothing adsorbed or absorbed. This void volume, designated herein as VO,has been called the “thermodynamic dead volume” by Knox,l*the “Gibbs gas volume” by Kobayashi,’s the “maximum column holdup volume” by Zhu,’ the “total” or “holdup column volume” by Martire,16 and the “geometric void volume” by Hennion and Rosset.lg The void volume defined this way results from the application of convention (i) for the placement of the GDS, viz.,the GDS is coincident with the adsorbent surface. Equation 6 then provides the basis for a more specific working definition of excess adsorption, involving the void volume by replacing V in eq 6 with
vo:
n
y=
-
vOCP‘
(11)
Thus, nexcesscan be determined from experimental measurement of the amount of component i in the column, nP, and the void volume, VO,which is a k e d quantity for any given column. Mobile Phase Volume. The simplest dehition of the mobile phase volume is the volume of moving phase in a chromatographic column regardless of the nature, density, or composition of the material in that volume. While simple to define, this quantity is often d a c u l t to measure and may or may not be equal to the void volume as deiined herein. Moreover, this definition does not take into account the fact that much of the eluent in the column may occupy pores or interstices which are unswept by the moving portion of the eluent. However, if convention (iii) is applicable, then the mobile phase volume, Vm, can be defined in terms of the GDS by eq 4. That is, the mobile phase volume is the volume of eluent (stagnant or moving) in the column up to the GDS. In an alternative approach, Zhu’ has defined the mobile phase volume as the difference between “the maximum column holdup volume” (fias defined herein) and “the volume of the adsorbed phase”. Thus, if the volume of the adsorbed phase, Ps, is identified with the volume bounded by the GDS and the surface of the solid adsorbent (eq lo), then the void volume and the mobile phase volume are not equivalent in systems in which P d s > 0. In particular,
v -pas
I/m = O
(12)
This definition avoids the ambiguities of the kinetic states of different domains of the chemically isotropic mobile phase. If, however, the compositions of the mobile and stationary phases were equivalent in a particular system, it would be impossible to distinguish between the two with any chromatographic probe solutes. The volume of the moving phase and the amount of component i in the stationary phase, nibM, are related by a modified version of eq 3: ni
total
=
- T/
C.bu‘
mi
(13)
Equations 11and 13 are superkially similar but critically different, and the type of adsorption measured depends upon the use of (19) Hennion, M. C.; Rosset, R Chromatogmphia 1988,25,43-50.
616 Analytical Chemistry, Voi. 67, No. 3, February 1, 1995
either VOor V,. V, can be determined experimentally from the holdup or retention volume of a probe solute which is excluded fiom the stationary phase. Adsorbed Phase Volume. The condition VO t V, will hold if the GDS is defined to be located at the stationary/mobile phase boundary and Pdsis equated with the volume of the stationary phase. Experimental measurement of Vm = VO- vdds has proven impossible in classical gravimetric or volumetric methods for measuring physical adsorption. It is difficult, although by no means impossible as suggested by some authors, to measure V, in dynamic, chromatographicsystems. Experimental methods for the determination of VO,Vm, and Ps will be discussed in a later section. Correct determination of VOand V m along with the total amount adsorbed allows accurate measurement of the molar volume of the adsorbed phase for gas- or fluid-solid systems with a single adsorbable component from the relation
where e t d s is the molar volume of component i in the adsorbed phase and ‘ni is the amount of i adsorbed in the region bounded by the GDS and the surface of the solid. It has been shown that a plot of experimentallymeasured Vm as a function of aibd is linear, with a slope of -@,As.This approach has been used to measure the molar volume of numerous adsorbates.15J7 In liquid systems with only one adsorbable component, etds = l/c?d, where cpUk is the molar density of the bulk liquid component i, and eqs 13 and 14 can be combined to give
This relationship was used by McCormick and Kargel.2°,21as the basis for an iteration scheme to calculate nib“ from measured values of npxces, n?, and VO. In the iteration algorithm, the measured value of npxcesswas used as the initial guess for nib“ on the right-hand side of eq 15. The iteration process was used successfully to calculate the total adsorption of organic modifiers in WLC systems with water/moditier mixbxes as mobile phases.~~l The volume of the adsorbed phase was determined from the measured values of nito“ and cPk. The restriction to a single adsorbable components was fulfilled in these systems by assuming negligible adsorption of water for the n-alkyl-bonded phases used in the investigation. RETENTION VOLUMES The definitions and conventions discussed thus far are not dependent upon any chromatographic theory and are completely general. Chromatography is, however, a very useful method for the experimental determination of some of the parameters discussed above. In particular, n?, VO,and V, can all be measured chromatographically,and the primary experimental parameter is the measured retention volume of a probe solute. Experimental Determination of the Amount of Component i in the Column, nio. In order to measure the amount of material adsorbed in a chromatographic column, the concentration of adsorbate in the eluent must be controlled and kept constant. That (20) McCormick, R M.; Karger, B. L. Anal. Chem. 1980,52,2249-2257. (21) McCormick, R M.; Karger, B. L. J. Chromatogr. 1980,199, 259-273.
is, the adsorbable component(s) must be present in the eluent at a known, fixed concentration. This condition is commonly observed for RF'LC with organic components in an aqueous mobile phase, SFC with polar modifiers in the mobile phase, and GSC at low temperaturewith adsorbable components in the mobile phase. Under these conditions of tinite concentration chromatography, there are two common experimental methods for measuring the amount of a component adsorbed: the minor perturbation method,22also known as "elution-on-a-plateau", and tracer pulse meth0ds,2~-~~ with a distinguishable isotopic probe solute. Experimentally, both methods involve equilibration of a system with a k e d concentration of an adsorbable component in the eluent followed by injection of eluent components (perturbation) or distinguishable isotopes of one or more components of the mobile phase (tracer pulse). These two methods are complementary, and both have been discussed extensively in the literatureJ2 The distinguishing difference in the techniques is the mathematical form of the applicable retention volume equations,
or a fluid can be achieved by measuring the retention volume of an unretained probe with nothing adsorbed. In this case, VO= V,. This "nothing adsorbed" condition is impossible with liquid systems, so another approach is required. The method is based on the assumption that, for liquids, the molar volume of any solute is the same in the adsorbed or bulk phase, i.e., n;lds = !jib&. Also, eqs 13 and 18 can be combined, along with the definition cpUk = nPk/Vm, to give a retention volume equation ai""
Vii = Vm + -Vm
(19)
ni
For noncompressible liquids, the retention volume can be related to the volume fraction of component i in the bulk phase,
minor perturbation method where @i = K / V , Rearrangement and summation over all components of the eluent give
where no is the entire amount of eluent in the system and yi is the mole fraction of component i in the mobile phase. If the amount of i in the system is vanishingly small, then eq 16 simplifies to
It is difficult to obtain isotherm data from either of these differential equations. The retention volume equation for tracer pulse chromatography is, however, not in a differential form and can be solved analytically,
Vo= p i V R j Thus, VO can be obtained for liquid systems by measuring the retention volumes of distinguishableisotopes of all of the components of the eluent. This method was proposed by Knox18 in a study involving a ternary eluent (water, acetonitrile, and carbon tetrachloride) with radiolabeled isotopes. In systems with a single adsorbable component, measurement of the retention volume of a tagged sample of that component gives VO. The equivalent expression for supercritical fluids or gases is
tracer pulse and frontal methods
where V,i is the retention volume of a concentration perturbation or an isotopic tracer. This equation has been derived by a number of and is applicable to gas, liquid, and supercritical fluid systems. The tracer pulse techniques are the simplest and most commonly used, even though they require an isotopespecific detector because of the mathematical simplicity of the retention volume equationeZ6Experimental measurement of VR,~ of a tracer solute at k e d concentration of component i in the bulk phase gives a direct measure of n?. Experimental Determination of the Void Volume, VO. Experimental measurement of Vo for columns containing a gas (22)Conder, J. R;Young, C. L. Physicochemical Measurement by Gas Chromatography: Wiley-Interscience: New York, 1979. (23)Masukawa, S.;Kobayashi, R J. Cas Chromafop. 1968,6, 461-465. (24)Parcher, J. F.;Selim, M. I. Anal. Chem. 1979,51,2154-2156. (25)Findenegg, G. H.; Koster, F. J. Chem. Soc., Faraday Trans. 1 1986, 82, 2691-2705. (26)Hufton, J. R;Danner, R P. Chem. Eng. Sci. 1991,8, 2079-2091.
Fixperimental Determination of the Mobde Phase Volume, V,. Tracer pulse chromatography thus provides simple experimental methods for the determination of n?, no, and VO. In addition, the mobile phase volume of any system can be determined from the retention volume, Vw of any solute j for which npM= 0 (eq 19). That is, any component of the eluent which is not adsorbed and resides only in the mobile phase will give a true value of V,, even with some other component adsorbed. Zhul and MartirelGhave used this method to determine V, in RPLC columns with alkane-bonded silica adsorbents. The solute used was deuterated water in an organic-rich binary aqueous eluent. The same approach has been used for gas-l5'l7 or supercritical f l ~ i d - s o l i d ~systems ~ , ~ using helium, the helium-3 isotope, or neon as the probe solute. In gas- or fluid-solid systems, a holdup time probe solute such as helium or neon that does not interact with the adsorbed stationary phase, i.e., does not penetrate the GDS, has been used to directly measure the volume of mobile (27)Strubinger, J. R;Song, H.; Parcher, J. F. Anal. Chem. 1991,63,104-108. (28)Hagege, A: Rocca, J. L.; Djerki, R Chromatographia 1994,38, 373-380. Analytical Chemistry, Vol. 67, No. 3, February 1, 1995
617
phase. KobayashiBfirst showed the correlation between the type of adsorption (total or excess) and the void or mobile phase volume. In later studies, tracer pulse chromatography with helium as the dead time probe was used to measure the void volume and the mobile phase volume with various materials adsorbed in gas-solid Other studies involving liquid-solid systems with alkane-bonded stationary phaseslJ6 have taken advantage of the nonwetting properties of bonded WLC packings to measure V, with organic-rich eluents, in which water was assumed to be nonadsorbed. Thus, it is possible to measure total adsorption in chromatographic systems if the experiments are designed to allow the reliable evaluation of VOand Vm or PdS. Experimental Determination of the Amount(s) of Component i Adsorbed, n i m e M and Parallel sets of retention volume equations expressed in terms of the void volume and the excess amount of solute adsorbed, niexcess, or the mobile phase volume and the total amount adsorbed are given by the combination of eq 18with eq 11or 13. These equations have been derived by several authors1J5J6for different types of chromatography:
These two equations form the basis for the experimental determination of phase distribution isotherms from tracer pulse chromatography. CAPACITY FACTOR
Equation 25 is a classical equation more commonly expressed in the form given in eq 19. Because the retention volume is directly proportional to the retention time, i.e., Vki = F&it either eq 19 or 25 can be rewritten in terms of retention times and a pseudoequilibrium constant, k':
tRj = t,
+ k't,
(26)
phase phenomenon. Straightforward combination of the residence times of any solute in the two phases results in the overall retention time for that particular solute. Experimentally, the capacity factor of a solute can be determined from the measured retention volume and volume of the mobile phase from the relation
or from the retention times
Equations 29 and 30 are probably the most misused and abused relationships in the entire domain of chromatography. The problem, of course, lies in the exact definition and measurement of tm and V,. If these quantities are not precisely defined and accurately measured, the resulting capacity factor will not represent an accurate assessment of the ratio n$"/n,b", and the simple residence time analogy expressed in eq 26 will be lost. Many authors fail to appreciate the subtle importance of the accurate determination of V, or t,. This uncertainty then brings into question the validity of any thermodynamic data produced from capacity factor measurements. Substitution of the experimentally convenient Vo for the thermodynamically sound V, in eq 29 results in the calculation of an "excess capacity factor", k", from the relations
F h ~ d e n e g gh~ ~t proposed the use of this form of capacity factor and derived the relationship between k" and niexcess:
where t, is the retention time of an unretained, dead time probe and k' is the capacity factor, defined as the ratio of the amount of component i in the stationary phase to that in the mobile phase, i.e.,
or
Equation 26 has the appeal of simple elegance, because the first term on the right-hand side represents the residence time of any solute in the mobile phase, while the second term gives the residence time in the stationary phase. The t, term is constant for all solutes because they all travel at the same velocity in the mobile phase; thus, eq 26 illustrates the basic premise of all types of chromatography that chromatographicselectivityis a stationary (29) Masukawa, S.; Kobayashi, R J. Gas Chromatogr. 1968,6,257-265. (30)Hori, Y.;Kobayashi, R J. Chem. Phys. 1971,54, 1226-1236.
618 Analytical Chemisrry, Vol. 67,No. 3, February 7 , 7995
This equation is very similar to eq 2 8 however, there are some significant d~erences. The easily accessible void volume, VO, replaces the hard-todetermine mobile phase volume, but the correlation of terms with the residence times of the solute in the mobile and stationary phases is lost. In addition, the adsorption data obtained from k" are excess adsorption, which is much less valuable in both practical and theoretical terms than the total amount adsorbed. Another way to define k" is the following:
(33)
The two types of capacity factors can be related by an expression derived from eqs 28 and 32: (34)
A conceptual problem arises with the use of k“ due to the possibility of negative excess adsorption, which would result in a negative excess capacity factor. This phenomenon can occur if < cP* (see F i i e 1and eq 1). This condition is not common for gas or fluid systems; however, it is a necessary condition for at least one component in liquid-solid systems with a multicomponent eluent if niexcess> 0 for any other component. Such negative adsorption has been 0bserved~9~~ with microporous silica systems. If negative excess adsorption occurs, Vki < VOin eqs 24 and 31, and this leads to some significant implications for the interpretation of capacity factor data and the relation between the capacity factor and the free energy of transfer of a solute between the stationary and mobile phases. Moreover, the idea of a negative value for a capacity factor raises the possibility of negative retention, which is even less intuitively acceptable than negative adsorption. THERMODYNAMICS
The link between chromatography and thermodynamics is the equilibrium constant (adsorption or partition coefficient) for the transfer of a solute between the adsorbed (stationary) and bulk (mobile) phases. The equilibrium constant is related to the capacity factor (as defined by eq 27) by the expression
Ki= K ( S )
(35)
The relationship to phase transfer thermodynamics is established through the classical equation for the free energy of transfer:
AGO = -RT
In Ki
(36)
Thermodynamic data for myriad systems have been measured from these relationships. In most cases, the integrity of the data has been firmly established. However, in certain cases, the types of capacity factor, types of adsorption, and exact types of volumes measured are ambiguous. Thermodynamic parameters derived from such experiments are questionable at best. CONCLUSIONS
RPLC systems with binary or more complex eluents are very similar to GC or SFC systems in which one or more components of the mobile phase may adsorb on a solid adsorbent. The mathematical models developed for frontal chromatography apply equally well for all such systems. Tracer pulse chromatography has been used successfully to measure both absolute and total adsorption in RPLC, SFC, and GC systems; however, very careful differentiation between the void volume of a column and the volume of the mobile phase in a column must be established. To date, chromatography has proven to be the only experimental method available for the measurement of total adsorption. This is possible because the position of the Gibbs dividing surface can be defined to coincide with the boundary between the moving and stationary phases in a chromatographic column. Chromatographyin its many forms has proven to be a valuable tool for the investigation of solid surfaces and species adsorbed on such surfaces. The use of molecular species rather than photons, ions, or atoms to probe the interactions of adsorbates (31)Groh, R;Halasz, I. J. Chromatogr. 1980,199, 23-34.
allows the investigation of very low energy molecular interactions in a unique manner. However, the integrity of the experimental results can be compromised without a full appreciation of the subtle but very significant distinctions between the various parameters emphasized in this discussion.
Acknowledgment is made to the National Science Foundation and to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. LIST OF SYMBOLS AND DEFINITIONS
solid surface area concentration of component i concentration of component i in the adsorbed layer bounded by the GSD and the solid surface concentration of component i in the bulk fluid Gibbs dividing surface capacity factor defined by eq 27 capacity factor defined by eq 33 equilibrium constant for the distribution of component i between the stationary and mobile phases IUPAC symbol for “the surface excess amount of adsorbed substance (Gibbs adsorption)” amount of component i in the system excess amount of i adsorbed as defined by eq 5 total amount of i adsorbed as defined by eq 2 number of moles of all eluent components in the system retention time of component i residence time of any component in the mobile phase residence time of any component in the mobile phase of a column with nothiig adsorbed, Le., Pds= 0 molar volume of component i in the adsorbed layer molar volume of component i in the bulk fluid volume as defined by eq 4 volume as defined by eq 7 volume of the adsorbed layer as defined by eq 10 void volume defined by eq 7 volume of the mobile phase as defined by eq 4 retention volume of component i mole fraction of component i in the bulk fluid distance from the solid surface distance of the GDS from the solid surface distance from the solid surface at which ci = cpUk standard free energy for the phase transfer of component i molar density of component i in the bulk fluid volume fraction of i in the bulk fluid
Received for review June 21, 1994. Accepted November 11, 1994.@ AC9406219 @Abstractpublished in Advance ACS Abstracts, December 15, 1994.
Analytical Chemistry, Vol. 67,No. 3,February 1, 1995
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