Research: Science & Education
Gas Chromatographic Measurement of Void Volume and Mobile-Phase Volume: Illustration of the Concepts of Excess and Total Adsorption Jon F. Parcher and Kwang S. Yun1 Department of Chemistry, University of Mississippi, University, MS 38677 A literature survey shows that a great number of papers have been published on chromatography and chromatographic separations in this Journal, but only a handful of these articles dealt with fundamental aspects of chromatographic processes or thermodynamic studies of adsorption and retention of sample components (1–6). In a classical paper in this series, E. F. Meyer (6) presented a cogent pedagogical discussion illustrating the thermodynamic concepts of reference and standard states for gases adsorbed on solid surfaces utilizing examples derived from gas–solid chromatography experiments. Full comprehension of such adsorption processes and thermodynamics is essential for our understanding of the surface structures of solids, catalytic processes, and chromatographic retention mechanisms. Any discussion of adsorption thermodynamics must, however, acknowledge the caveat that there are two distinct concepts regarding the definition of the amount of something adsorbed on a surface. The amount of material adsorbed can be described as either excess or total (also called absolute) adsorption depending on where an interfacial boundary (or plane) defining the limit of the bulk phase is drawn. One of the objectives of this essay is to present a simple discussion concerning the concept of an interfacial plane and to delineate the differences between excess and total adsorption using chromatographic examples and experiments. The unique role of gas–solid chromatography in the determination and illustration of total adsorption will also be discussed in detail. It is believed that chromatography is presently the only experimental technique that can be used to distinguish between the two types of adsorption, and thus offers a clear illustration of the abstract concept of the interfacial plane. Another interesting aspect of gas– solid chromatographic studies is the idea that the adsorbed layer volume obtained from experimental measurements can be directly correlated to a molecular size parameter of the gas adsorbed, that is, the van der Waals b constant (7, 8). This demonstrates an additional advantage of gas–solid chromatographic techniques as a promising tool for measuring physical properties of materials adsorbed in thin layers. Following Meyer’s approach, an overall discussion of the adsorption of solutes on a solid stationary phase will be presented to illustrate two distinct but often misunderstood adsorption models, namely, excess and total adsorption.
lecular forces such as van der Waals attractive interactions between adsorbate molecules and atoms of the solid at the surface. The chromatographic retention of adsorbate (solute) molecules can also be attributed to this attraction. In Figure 1, a simple model for adsorption is illustrated; the open circles in the upper portion represent adsorbate molecules and the lower portions of the figure show concentration profiles. The sequence of A → B → C in the figure illustrates increasing strength of adsorption such as that observed with decreasing temperature even with a constant concentration of adsorbate in the bulk phase. When an equilibrium state is established between the adsorbed molecules and those in the gas phase, the density of the gas at and near the surface of the solid stationary phase could become higher than that in the bulk phase, and the density may gradually vary from a higher value at the surface to a lower one in the bulk phase at some distance, zbulk, from the surface; this is depicted in Figure 1C. To study the thermodynamics of systems in which surface effects are significant, such as adsorption or distribution of solutes between two phases, J. W. Gibbs in 1873 (9) introduced the hypothetical concept of an interfacial region that has zero volume but nonzero values of other physical and chemical properties; such an interfacial region is called the “Gibbs dividing surface, or plane,” which defines the limit of the bulk (unadsorbed) phase. Our understanding of the Gibbs dividing surface, particularly its location, is important and necessary, especially for the study of adsorption in multicomponent systems, but it is not simple or easy to comprehend the concept of the Gibbs dividing surface at an elementary level of physical chemistry. Thus, in this paper a pedagogical approach of modeling the adsorption of gases on a solid surface relevant to the retention of
Adsorption and the Interfacial Plane Adsorption of gas molecules on a solid occurs primarily because of weak intermo1
Author to whom correspondance should be addressed.
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Figure 1. Adsorption models and concentration profiles. A. No adsorption, the concentration profile is constant. B. Monolayer adsoption with a sharp change in the concentration profile. C. Multilayer adsorption with a gradual change in the concentration profile.
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solutes in gas chromatography is discussed. As shown in Figure 1A, there will always be some adsorbate molecules located close to the surface (in the so-called adsorption layer), even when there is no attractive force between the adsorbate and adsorbent. This brings up the question of whether or not this residual amount of material should be described as “adsorbed”? Or should only the amount of material located in the adsorption layer that is in excess of this residue be counted as adsorbed material? These questions lead to the concepts of excess and total adsorption. Total and Excess Adsorption The surface excess amount of adsorbed substance defined by the IUPAC Commission on Colloid and Surface Chemistry (10) can be stated as: the surface excess amount or Gibbs adsorption of component i, niσ, is defined as the excess of the amount of this component actually present in the system over that present in a reference system of the same volume as the real system in which the bulk-phase concentration remains uniform up to a chosen interfacial surface or the Gibbs dividing surface.
The surface excess amount of i adsorbed on a solid, ni σ, can be expressed by a mathematical equation: ∞
(1) ni σ = As ∫0 (ci – ci bulk) dz bulk where As, ci, ci , and z are the surface area, the local concentration, the bulk-phase concentration, and the distance measured from the solid surface, respectively. To illustrate this quantity more clearly, Figure 2 is drawn as a duplicate of Figure 1C with the shaded area depicting the integral given in eq 1. In addition to eq 1 and Figure 2, the IUPAC definition of the surface excess amount, ni σ, can be illustrated by the rectangular area represented by the line at a constant ciads and the line drawn at zδ in such a way that the areas marked with !
Ci
Ciads
Cibulk
+
-
Zδ
Zbulk
Figure 2. Surface density profile and excess adsorption.
Z
and @ in Figure 2 are equal. The surface defined this way can be construed as an interfacial surface or the Gibbs dividing surface between the adsorbed and the bulk phase. Such a concentration profile model can be considered for a system having only one gaseous component, which undergoes adsorption on a solid surface and no absorption of gas molecules into the solid adsorbent. Therefore the system beyond the solid can be treated as having two phases: the bulk gas phase and the adsorbed phase having a higher concentration than the bulk phase. The surface excess amount of adsorption, ni σ, can be related to volumes measured in chromatographic experiments. The mobile-phase volumes of a column are routinely measured quantities in chromatography. This volume of a column with nothing adsorbed on the adsorbent phase can be easily measured in gas chromatography. It is called the dead volume of a system or the void volume of a chromatographic column. The volume is mathemati∞ cally defined by the expression V0 = As ∫0 dz. On the other hand, if some component of the gas phase is adsorbed, the bulk-phase volume (or mobile-phase volume) measured up to the surface of the adsorbed layer or∞to the interfacial plane is given by the integral VM = As ∫zδ dz, where the lower integration limit, zδ, refers the location of the interfacial plane, or the Gibbs dividing surface. The latter volume is also a physically meaningful quantity. However, it is experimentally difficult to measure the bulk-phase volume beyond the interfacial plane because of the problem of locating the interfacial plane between the adsorbed and bulk phase—that is, the exact determination of a value for z δ. As stated before, to develop a clear understanding of these two different volumes and to show how these volumes are related to the two types of measurable adsorption are the primary objectives of this paper. A working definition of the surface excess amount of adsorption, niσ , can be derived from eq 1 with a constant ci bulk value: niσ = ni 0 – V0 ci bulk
(2)
where ni 0 represents the entire amount of component i in the system, which ∞is described by the first part of the integral in eq 1, As ∫0 ci dz, and V0 is the volume of the bulk phase up to the Gibbs dividing surface located at the surface of the solid adsorbent. If the volume of a system is measured from the bulk phase to the surface of the adsorbent solid surface, this type of adsorption is equivalent to the surface excess amount of adsorbed substance defined by IUPAC, niσ , shown in eqs 1 and 2. In this paper it is simply termed excess adsorption, niexcess . On the other hand, it is feasible to locate the Gibbs dividing surface at the interfacial boundary between the adsorbed and bulk phases. Then it becomes possible to define a new adsorption distinct from the IUPAC-defined surface excess amount, and the volume measured from the bulk phase to the Gibbs dividing surface located at the interfacial plane is different from V0, and the new volume is designated by VM; in this case the type of adsorption is called total adsorption (or absolute adsorption), ni total. A similar discussion was given by Sing (11) and Wittkopf and Brauer (12), and a brief summary of their discussions is presented here. Sing described a procedure for determining the adsorption of a gas at a solid surface incorporating both volumetric and gravimetric measurements. He discussed the concept of a quantity (na) that is equal to the surface excess amount of adsorbed substance (IUPAC-defined ni σ) or Gibbs adsorption when the surface of the solid is taken as the Gibbs dividing surface, and it is represented by the following equation using his
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notation: na = ∫
g V
0
0
(c – c ) dv = n – c V
g
(3)
where c is the local concentration of solute molecules, c0 is the bulk concentration, V g is the volume of the gas phase, and n is the amount of the gas admitted into the system. The quantities na, c, c0 , and V g are equivalent to those employed in this paper (niexcess, c i, ci bulk, and V0). An alternative measure of adsorption was also suggested by Sing: the “total amount of substance in the adsorbed layer”, ns. This “total amount” is given by the equation: ns = ∫V c dv a
(4)
where Va = τA s is the volume of the adsorbed layer, and τ corresponds to zδ. Sing stated that Va has to be defined on the basis of some appropriate model of gas adsorption that gives a value of the layer thickness. Wittkopf and Brauer introduced the methods of the thermodynamic adsorption analysis in terms of two basic concepts: (i) the two-phase approach, which considers the system to consist of the adsorbed phase and the remaining bulk phase and is an adequate picture of adsorption, and (ii) the one-phase approach, which introduces excess values for the thermodynamic description of adsorption. In contrast to liquid systems, it is easy to measure the volume of a gaseous eluent in a column with nothing adsorbed on the stationary phase, and most researchers define this volume as the void volume of a column. In this case, reported adsorption data commonly reflect the excess adsorption rather than the total adsorption. In nonchromatographic experiments, the void volume of the system to be studied is usually measured prior to the adsorption of any gas on the surface. Direct measurement of total adsorption is not feasible with volumetric or gravimetric techniques because the void volume measured using those methods represents only the volume of a system with nothing adsorbed (i.e., V0). Currently, chromatographic methods present the only way to obtain both to-
Ci
tal and excess adsorption, since the volume of the mobile phase (eluent) in a chromatographic column can be measured, under certain conditions, both with and without material adsorbed on the surface of the stationary phase (7, 8). In that case, the physical boundary between the stationary and mobile phases defines the position of the Gibbs dividing surface, zδ. The mathematical definition for total adsorption, nitotal, is given by the following equation and illustrated by the shaded area in Figure 3. ∞ ∞ nitotal = As (∫0 ci dz – ∫zδ ci bulk dz) (5) In order to elucidate the difference between total adsorption and excess adsorption, a clear distinction must be made between the void volume, a fixed quantity designated as V0, and the bulk-phase volume denoted by VM, which is the volume of the system measured from the bulk phase up to the interfacial plane at the surface of the adsorbed layer. The relation between these volumes is given by eq 6 (7): V0 – VM = Vads (6) where Vads may correspond to Va in Sing’s paper. Equation 6 implies that the void volume and the bulk-phase volume of a system are not equal if Vads > 0. The volume of the adsorbed phase, Vads , is identified as the volume bounded between the interfacial plane and the surface of the solid adsorbent. This is experimentally difficult to measure. The bulk-phase volume, VM, varies inversely with the amount of the material adsorbed. This volume cannot be measured directly with either volumetric or gravimetric methods, although it can be estimated if an accurate value of the molar volume or density of the adsorbed layer is available. Menon (13) described an approximate method of calculating the change in the bulk-phase volume, which he called a dead space. More recently, McCormick and Karger (14) introduced an iteration technique to calculate ni total from measured values of niexcess, ni 0, and V0. The difference between the two types of adsorption, total and excess, for component i can be derived from eqs 1 and 5: δ ni total – niexcess = As ∫0z cibulk dz = cibulk Vads (7) where ci bulk is assumed to be constant and Vads is the volume described in eq 6. Correct measurements of V0, VM, and the total amount adsorbed allow accurate determination of the molar volume, vi ads, of the adsorbed solute in gas–solid systems with a single adsorbate. The volume of the adsorbed layer, Vads, can be related to the molar volume by the expression: Vads = viads nitotal (8)
Ciads
Combination of eqs 6 and 8 gives (9) VM = V 0 – v iads nitotal Experimental measurement of VM as a function of ni total forms the basis for determining the molar volume of adsorbed species (7, 8).
Cibulk
Gas Chromatography
Zδ Figure 3. Illustration of total adsorption.
896
Zbulk
Z
The primary reason that absolute adsorption cannot be measured directly with static and volumetric methods is that no experimental method is available to determine the location of the interfacial boundary between the immobilized and free adsorbate or of the Gibbs dividing surface. Chromatography, on the other hand, requires the existence of two immiscible phases with one in motion relative to the other. The boundary between the two phases
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(mobile and stationary) can sometimes be distinct and the exact location of the phase boundary can be determined experimentally if the volume of either individual phase can be measured (15, 16). This is true even for systems in which the stationary and mobile phases are identical. In some cases, for example liquid chromatographic systems in which the eluent is adsorbed, it is not possible to determine experimentally the volume of the stationary or mobile phase (17). Nevertheless, in certain systems such measurements are possible, and such a system will be used to provide concrete examples of absolute and excess adsorption. This section describes exactly how a gas chromatographic technique can be employed for the measurement of the total and excess adsorption of solutes. In gas chromatography, a pure gas or mixture of gases, called the mobile phase or the eluent, is forced through a packed column that contains a known amount of adsorbent material. This adsorbent is called the stationary phase. Chromatographic separations of solutes are based on the differences in the extent to which the various solutes are distributed between the stationary and mobile phases. The distribution ratio of a solute between the stationary and mobile phases is an equilibrium constant (a thermodynamic quantity) that is directly related to the slope of the linear portion of an adsorption isotherm (a plot of the surface concentration of an adsorbate against the partial pressure of the adsorbate molecules in the mobile phase). The limiting slope, K, using Meyer’s approach (6), is represented by the equation K = σi @p i = (V0 @AsRT) (ns @ng)
(10)
where σi and p i are the molar surface concentration of the solute, i, in the stationary phase and the partial pressure of i in the mobile phase, respectively, and ng and ns represent the moles of solute in the bulk phase and adsorbed on the stationary mobile phase, respectively. Thus ns in eq 10 should correspond to ni total in this paper. At very low concentration, that is, in the linear region of the adsorption isotherm, the amount of solute adsorbed on the stationary phase is linearly related to the partial pressure. Thus the equilibrium constant or distribution coefficient remains fairly constant over a small range of σi and pi. The primary parameters measured in any gas–solid chromatography experiment are the retention times of various analytical solutes as well as some nonadsorbed solute called a dead-time marker. From these measured retention times, the corresponding retention volumes can be calculated using the measured flow rate of the carrier gas. As will be shown later in detail, the retention volume of the dead-time marker can generally be used to obtain the quantity known as the void volume of a chromatographic column. In other cases, it can be used to calculate the mobile-phase volume of a column. Usually, these two volumes (void volume and mobile-phase volume) are equal. However, in certain instances they are not equal, and these provide an example of the method used to distinguish between total and excess adsorption. The void volume of the column is defined as the total volume of the empty column minus the volume of the adsorbent. The void volume is a fixed quantity and is measured from the retention volume of a dead-time marker in a column with nothing adsorbed on the solid stationary phase. A dead-time marker can be any solute that is unretained by and unexcluded from the pores of a solid stationary phase. The void volume is then calculated from the eluent flow rate and the retention time of this solute. Unfortunately, this definition leads to the rather confusing concept of the “retention time of an unretained sol-
ute”; therefore, it is sometimes referred to as the “residence time” or “holdup time” of the unretained solute. Furthermore, the retention volume measured with such a dead-time marker may or may not be equal to the void volume. This is possible because the volume occupied by the mobile phase may vary with the amount of adsorbate condensed on a solid stationary phase if the eluent is excluded from the adsorbed layer. This information can be utilized (8) to characterize adsorption data as either total or excess and consequently specify the location of the Gibbs dividing surface. In this paper, as suggested by Zhu (15) the void volume minus the volume of the adsorbed phase, (eq 6) is defined as the mobile-phase volume of the column. The mobile-phase volume should decrease as more molecules are adsorbed on the surface, and measurement of the decrease in the mobile-phase volume can provide information about the volume, thickness, and density of an adsorbed layer. The retention time of a solute retained in the stationary phase is longer than the residence time of the deadtime marker, and the difference in their magnitude depends on physical and chemical properties of the individual solutes and adsorbents. The residence time, t 0, of a dead-time marker, simply called the dead time, can be defined in terms of the column length, L, and the average linear velocity of the mobile phase, um, as t0 = L/um . Likewise the solute retention time is defined as tR = L/kul, where kul is the average migration velocity of the solute in the column. Since it is retained by the stationary phase, the solute velocity kul is always less than or equal to um, thus t 0 ≤ tR. The retention time of a solute consists of two parts: the time for the migration of the solute with the mobile phase and the time due to the retention of the solute at the surface of the stationary phase. The relative ratio of these times, t R and t0, is equal to the mole ratio, ns/ng, as given by the equation: ns@ng = (tR – t0)@t0 (11) Substitution of eq 11 into eq 10 gives K = (V0 @AsRT) [(tR – t0)@t0)] = (F@AsRT) (tR – t0) (12) where F is the volume flow rate of the mobile phase, which is experimentally controlled and can be kept constant. The product of the flow rate and the retention time of the deadtime marker for a column with nothing adsorbed on the stationary phase gives the value of V0. It is interesting to note that the distribution coefficient, K, depends directly on the product of the flow rate of the mobile phase and the adjusted retention time of a component of the eluent, tR – t0. Thus the distribution coefficient can be written in terms of the retention volume as K = (1@AsRT) (VR – VM) (13) In adsorption systems it will be demonstrated that gas chromatography provides a simple and unique method for measuring the volumes V0 and VM and consequently the volume of the adsorbed layer and its variation with the amount of material adsorbed. Void Volume and Mobile-Phase Volume Measurement Detailed experimental procedures for measuring void and mobile-phase volumes are found in many references (15, 18, 19); therefore only a brief description is presented in this paper. Step 1. A column packed with a stationary phase is kept at a constant temperature. A carrier gas, which will not be adsorbed, is passed through the column at a known constant flow rate and a dead-
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time marker, which is unretained [e.g., 3He (8)], is injected into the column. From the corrected flow rate and the residence time of the dead-time marker the void volume, V0, can be obtained using the equation V0 = F t0 . Step 2. An adsorbate (which will be adsorbed on the stationary phase), mixed with the carrier gas, is introduced into the column with a known partial pressure. The column is kept at the same temperature as in step 1. After an equilibrium state for adsorption–desorption is established, a dead-time marker that is excluded from the adsorbed phase is injected and its residence time measured. From the residence time of the dead-time probe and the flow rate of the carrier gas, the volume of the column available to the dead-time probe solute with some species adsorbed can be calculated (i.e., the mobilephase volume, VM). Step 3. The same procedure is repeated with different partial pressures of the adsorbate to determine the mobile-phase volume of the column with different amounts of material adsorbed on the stationary phase. With a number of such measurements, the relations between the void volume, the volume of the adsorbed layer and the partial pressure of the solute can be established, and thus the variation of the mobile-phase volume as a function of the solute partial pressure.
A number of published papers have described techniques and methods for the measurement and determination of the retention volume of solute i, VR,i (4, 8, 10, 14, 15, 19). However, in order to determine the amount adsorbed from eq 2, the overall amount of component i in the column, ni 0, must be determined. One method to measure this parameter is tracer pulse chromatography because the retention volume of an isotopic tracer, i*, is directly proportional to ni 0 (8, 15, 19): VR,i* = ni0@cibulk (14)
Ci
Thus the experimental measurement of VR ,i* of an isotopic tracer solute i* at a fixed concentration in the bulk phase gives a direct measure of ni0. Combination of eqs 14 and 2, and also replacing ni σ and V0 by ni total and VM, respectively, gives nitotal = (VR,i* – VM) ci bulk (15) Equation 15 indicates that the determination of nitotal can be achieved with measurements of the retention volume and the concentration of component i in the mobile phase. When VM in eq 15 is replaced by V0, the amount adsorbed becomes the excess adsorption, ni excess, as given by niexcess = (VR,i* – V0) ci bulk (16) The difference between nitotal and ni excess, which is equal to cibulk (V0 – VM), has been given by eq 7 and it can be identified as the shaded area in Figure 4. Along with the bulk-phase concentration, which can be calculated from the column pressure and the composition of sample gases, the values of ni 0, ni total, and ni excess can be obtained from simple retention time measurements using eqs 14–16. Conclusions Chromatography, along with the concepts of stationary and mobile phases, serves as an excellent heuristic tool for the practical illustration of the rather abstract concepts of excess and total adsorption. In many instances, the two types of adsorption are indistinguishable—for example, when the term cibulk Vads in eq 7 approaches zero. However, in systems involving high pressure, low temperature, or a very high-surface-area adsorbent, the term cibulk Vads can be much larger than zero, and the conceptual distinction between excess and total adsorption therefore becomes important. Unfortunately, the two types of adsorption cannot be differentiated by classical volumetric or gravimetric techniques. On the other hand, the discussion presented here has shown how chromatographic methods can be used to measure both excess and total adsorption in a given experimental system. This paper also illustrates concepts of void volume and mobile-phase volume and the critical role of these volumes in differentiating excess and total adsorption. Acknowledgment Acknowledgment is made to the National Science Foundation and the donors of the American Chemical Society Petroleum Research Fund for support of this study. Also, the authors would like to thank Chaowei Zhu, Qingmei Bu, and Becky Edwards for their assistance in preparing the manuscript.
Ciads
Literature Cited 1. 2. 3. 4. 5. 6. 7. 8.
Cibulk
Zδ
Zbulk
Z
Figure 4. Illustration of the difference between total and excess adsorption.
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9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
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