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Gibbsian Surface Excess for Gas AdsorptionsRevisited S. Sircar Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195-1501
The Gibbsian surface excess (GSE) represents the true experimental variable for measuring pure and multicomponent gas adsorption equilibria and kinetics by all conventional adsorption methods (volumetric, gravimetric, chromatographic, column dynamic, total desorption, isotope exchange, etc). The GSE can be used as primary variables to formulate the general thermodynamic and kinetic models for adsorption on energetically homogeneous and heterogeneous adsorbents. Thus, practical thermodynamic properties like surface potentials, isosteric heats of adsorption of components, and heat capacities of adsorption systems can be estimated using GSE. Several thermodynamic consistency tests (differential and integral) for binary adsorption systems can also be formulated using GSE. A mathematical framework describing the isothermal and nonisothermal column dynamics for adsorption of multicomponent gas mixtures can be developed using GSE. It can be used to simulate practical process design models (pressure and thermal swing adsorption) for gas mixture separation and purification. There is no need to estimate actual amounts adsorbed from the measured GSE variables by making ambiguous assumptions about the size of the adsorbed phase. The experimental GSE can be used to completely describe all practical thermodynamic, kinetic, column dynamic, and process design aspects of gas adsorption technology. Introduction When an adsorbate gas mixture is contacted with a solid adsorbent, the gas-solid intermolecular forces of attraction create a region near the solid surface where the local adsorbate density and composition are different from those in the homogeneous bulk gas phase. The region is called the adsorbed phase. It may extend to a distance of several diameters of the adsorbate molecules from the solid surface. The creation of such an adsorbed phase forms the basis of all practical gas separation and purification processes like pressure swing and thermal swing adsorption.1 Unfortunately, the size and structure of the adsorbed phase as well as the actual density and composition profiles of the adsorbates within the adsorbed phase cannot be experimentally measured. These properties are unknown functions of the bulk gas phase pressure (P), mole fraction of component i in the bulk gas mixture (yi), and system temperature (T). Figure 1 shows a diagram of an adsorbed phase for a binary gas mixture (i ) 1, 2). The local adsorbate density is highest near the solid surface (assumed to be flat) and then it gradually decreases to the bulk gas phase density (F). The local adsorbed-phase mole fraction of the more selectively adsorbed component (component 1) of the gas mixture is higher than that of the bulk gas phase (y1). The local adsorbed-phase mole fraction of the less selectively adsorbed component (component 2) is smaller than that of the bulk gas phase (y2). The actual volume of the adsorbed phase is va. The average adsorbate density and the mole fraction of component i in the adsorbed phase are Fa and xai , respectively. The actual amount of component i adsorbed is nai ()vaFaxai ). For adsorption of a pure gas (yi ) xai ) 1), the actual amount adsorbed is given by na()vaFa). The quantities va, Fa, nai , na, and xai ()nai /∑nai ) are not experimental variables. Only the bulk gas properties like F and yi (or P, T, and yi) can be measured.
Figure 1. Schematic diagrams of adsorbate density and composition profiles in the actual and Gibbsian adsorbed phases.
The real physicochemical structures of microporous/ mesoporous adsorbents used for practical gas separations are very complex. The amorphous adsorbents, such as activated carbons, silica and alumina gels, and the polymeric adsorbents consist of an interconnected network of micro- and mesopores of various sizes and shapes. The walls of these pores generally have different surface chemistries from region to region. It is not possible by today’s technology to quantitatively characterize the heterogeneous pore structure and surface chemistry of these materials. The pore structures of crystalline adsorbents like alumino-silicate zeolites can be precisely measured but they exhibit adsorption heterogeneity due to structural defects, existence of a distribution of silica-alumina ratio in the crystal framework, presence of bare and hydrolyzed cations of different types at different locations within the framework, presence of trace quantities of water, heterogeneous structure of the binder material, etc. These
10.1021/ie9900871 CCC: $18.00 © 1999 American Chemical Society Published on Web 08/19/1999
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nm j ai - vj aFyi] ) vj a[Fjaxjai - Fyi] ) [n j i - v0Fyi] (2) i ) [n For the adsorption of a pure gas, the GSE (nm*)is defined by
j a - vj aF] ) vj a[Fja - F] ) [n j - v0F] nm* ) [n
Figure 2. Schematic diagram of the Gibbsian surface excess model.
zeolitic heterogeneities also cannot be quantitatively measured by today’s technology. However, the adsorbedphase properties on these materials are greatly influenced by their physicochemical structures. Thus, it is not expected that technology will evolve in the foreseeable future which will allow quantitative characterization of adsorbed phases in practical adsorbents. Gibbsian Surface Excess Model To circumvent the previously described problem in characterizing the adsorbed phase, J. W. Gibbs proposed an elegant model to represent the gas-solid adsorption system about 70 years ago.2 Figure 2 shows a schematic representation of the Gibbsian model. It consists of a multicomponent gas phase at P, T, and yi in contact with a unit amount of the adsorbent. The Gibbsian adsorbed phase is defined by an interface which is arbitrarily located within the bulk gas phase. The volume, the average density, and the average mole fraction of component i for the Gibbsian adsorbed phase are given by vj a (cm3/g), Fja (mol/cm3), and xjai , respectively. The volume and the density of the corresponding gas phase are given by v (cm3/g) and F (mol/cm3), respectively. The total void volume (cm3/g) for the entire system is v0 ()vj a+v). The quantity (mol/g) of component i in the Gibbsian adsorbed phase is n j ai ()vj aFjaxjai ). For adsorption of a pure gas, the quantity (mol/g) of the adsorbate in the Gibbsian adsorbed phase is n j a ()vj aFja). Figure 1 shows a diagram of the Gibbsian adsorbed phase similar to that for the actual adsorbed phase. The properties of the two adsorbed phases are identical if the location of Gibbsian interface coincides with that of the actual adsorbed phase. The total amount of component i in the adsorption system of Figure 2 is n j i (mol/g). It can be measured either by knowing how much of each adsorbate was added to the adsorbent to create the system of Figure 2 or by desorbing the entire content of the system of Figure 2 and measuring the total amount of each component desorbed. A simple material balance for component i gives 0 n j i ) nm i + v Fyi
(1)
where the quantity nm i (mol/g), which is called the Gibbsian surface excess (GSE) for component i, is defined by3
(3)
where n j is the total amount (mol/g) of pure gas in the adsorption system of Figure 2. m* can be Equations 2 and 3 show that nm i or n j , v0, F, and experimentally obtained by measuring n j i, n yi. Thus, the value of GSE does not depend on the location of the Gibbsian interface as long as it is placed in a position where the adsorbate properties are the same as those of the bulk gas phase. There is no need to know the size, the structure, and the properties of the actual adsorbed phase to calculate the GSE. Therefore, they represent unambiguous and practical means of quantitatively describing the extents of adsorption for pure or multicomponent gas adsorption systems. The GSE can be used to describe both the thermodynamic adsorption equilibria and the adsorption kinetics. For the former, nm i will be described as functions of equilibrium gas-phase conditions (P, T, and yi). For the latter, eq 1 can be rewritten for the system under transient conditions: 0 n j i(t) ) nm i (t) + v F(t)yi(t)
(4)
where n j i(t), F(t), and yi(t) are transient values of these variables at time t in the adsorption system of Figure 2 and nm i (t) is the transient nonequilibrium GSE for component i. The rate of change of nm i can be measured by monitoring the rates of changes of n j i, F, and yi in the adsorption system. The Gibbsian equilibrium selectivity [Sm ij ] and ki(t)] for adsorption of component i netic selectivity [Sm ij over component j are defined as4
Sm ij )
nm i yj nm j yi
Sm ij (t) )
nm i (t)yj(t) nm j (t)yi(t)
(5)
Component i is selectively adsorbed over component j if Sm ij >1. It will be shown later that all conventional experiments for the measurement of adsorption equilibria and kinetics determine the equilibrium GSE of the components of a gas mixture, the Gibbsian selectivity of adsorption between the components, and the rates of changes of GSE of the components under the conditions of the experiments. j ai if Fjaxjai . Fyi and eq 3 Equation 2 shows that nm i ≈ n m* a a j if Fj . F. Thus, the GSE of a pure shows that n ≈ n gas is equal to its actual amount adsorbed if the adsorbed phase is much denser than the gas phase. That condition may be satisfied for adsorption of a pure gas (a) at low pressures and (b) at a temperature well below its critical temperature, where the adsorbed-phase density may be liquid-like.5 Otherwise, a large difference j a. For the adsorption of a may exist between nm* and n m a j i only when xjai . yi and Fja . F. gas mixture, ni ≈ n m j ai for low-temperature and Thus, ni may be equal to n low-pressure adsorption of a gas mixture where comm for ponent i is very selectively adsorbed, but nj*i co-adsorption of other components of the same gas
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Table 1. Estimated Differences between GSE and Actual Amounts Adsorbed for Equilibrium Adsorption of the Ethane (1) + Ethylene (2) + Methane (3) Gas Mixture on BPL Carbon at 301.4 K (y1 ) 0.52, y2 ) 0.25, y3 ) 0.23) m [(n j ai - nm i )/ni ] × 100]
GSE (mmol/g) pressure (atm)
component 1
component 2
component 3
component 1
component 2
component 3
20.56 6.82 1.30
3.858 3.135 2.090
1.475 1.043 0.651
0.765 0.356 0.058
7.8% 3.2% 1.4%
9.9% 4.6% 2.2%
17.5% 12.5% 14.6%
mixture under the same conditions may not be equal to a . In general, the GSE cannot be the corresponding n j j*i assumed to be equal to the actual amounts adsorbed. Equations 2 and 3 further show that the actual amounts adsorbed can be calculated from the measured values of GSE only if the adsorbed-phase volume can be estimated as functions of P, T, and yi. Obviously, this cannot be done rigorously for practical adsorbents because of the reasons given earlier. A common assumption made for theoretical calculation of actual amounts adsorbed by assuming models for gas-solid molecular interactions and simplified structures of the porous solids, in conjunction with statistical algorithms like Monte Carlo techniques, is that the entire pore volume of the adsorbent forms the adsorbed phase at all values of P, T, and yi.6 This approach may be valid when all the pores of the adsorbent are so small that all molecules residing inside the pore are within the force fields created by the adsorbent molecules. In that case, the local adsorbate densities and compositions are different than those of the bulk gas phase everywhere within the pore. Otherwise, such assumptions about the adsorbed-phase volumes will not be rigorous. For example, eq 2 can be rearranged to get
(n j ai - nm i ) nm i
)
vj aFyi nm i
(6)
It follows from eq 6 that the differences between nm i and n j ai for an assumed value of vj a become larger when the gas-phase density and its mole fraction for component i increases and when the selectivity of adsorption for that component decreases. Table 1 shows examples j ai for equiof percentage differences between nm i and n librium adsorption of ethane (1), ethylene (2), and methane (3) on BPL activated carbon at 301.4 K7 calculated by assuming that the adsorbed-phase volume is equal to the total pore volume of the carbon (0.70 cm3/ g). The selectivity of adsorption for the components of this mixture increases in the order ethane > ethylene j ai > methane on the carbon. It may be seen that nm i ∼ n only for ethane at low gas densities. Otherwise, the calculation of n j ai from experimental nm i values will strongly depend on the assumed value of va. It will be demonstrated in this paper that there is no need to estimate the actual amounts adsorbed from the measured GSE for comprehensive study of adsorption equilibria, kinetics, and heats, and for the use of these fundamental Gibbsian properties in describing adsorption dynamics in columns and ultimately for the design of adsorptive gas separation processes.
Figure 3. Schematic representation of a heterogeneous adsorbent and Gibbsian surface excess model.
ies) with or without different surface chemistries. The adsorbent (unit amount) is in equilibrium with a multicomponent gas phase at F, yi, and T. The local surface excess of component i in a type k pore (nm ik) is given by a a nm ik ) nik - vk Fyi
k ) 1, 2, ...
(7)
where vak and naik are, respectively, the adsorbed-phase volume (cm3/g) and the actual amount adsorbed of component i (mol/g) in a k-type pore in equilibrium with the gas phase at F, yi, and T. It follows from eqs 2 and 7 that the overall surface excess of component i (experimentally measured quantity) on the heterogeneous adsorbent (nm i ) is given by
nm i )
∑k nmik,
va )
∑k vka,
nai )
∑k naik
(8)
Equations 7 and 8 can also be derived for transient GSE on a heterogeneous solid. Thus, the overall GSE of a component of a gas mixture on a heterogeneous adsorbent composed of a group of different adsorption sites can be obtained by a simple sum of the GSE of that component on the individual sites. This additive property of GSE conforms with the “patchwise homogeneous” model of a heterogeneous adsorbent.8 A special case of adsorbent heterogeneity is where certain adsorbate molecules of the gas mixture are excluded (true molecular sieving) from some of the pores of the adsorbent. In that case naik ) 0 and nm ik is negative for that adsorbate in those pores. It should be, however, noted that nm ik is not an experimental variable. If the pore structure of an adsorbent completely excludes an adsorbate, then according to eq 3, n j for pure gas adsorption is less than v0F and nm* will be a negative quantity. GSE for Pure Gas Adsorption on Porous Solids
GSE and Adsorbent Heterogeneity It was mentioned earlier that the practical adsorbents can be structurally and chemically heterogeneous. Figure 3 shows the diagram of a heterogeneous adsorbent which possesses different types of pores (k variet-
Almost all of the adsorption sites for a gas are located within the micropores and mesopores of a practical adsorbent. Only a negligible fraction of adsorption sites may be located at the external surfaces of such adsorbents. Thus, there is a limit to how much adsorbate can
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Figure 4. Surface excess isotherms of N2 on alumina.
Figure 5. Surface excess isotherm of N2 on DAY zeolite.
be packed into the pores of the adsorbent at high pressures. Consequently, the density of the Gibbsian adsorbed phase (Fja) initially increases with increasing gas-phase density (or pressure) and then it asymptotically levels off at some maximum value (Fja). The gasphase density, however, increases monotonically and indefinitely with increasing gas-phase pressure when the system temperature is above the critical temperature of the gas. Thus, according to eq 3 for pure gas adsorption, the equilibrium GSE must be equal to zero at some high gas density value (F*) where Fja* becomes equal to F*. The GSE then becomes negative as F is further increased because the gas-phase density exceeds the adsorbed-phase density in that region. Consequently, the equilibrium surface excess isotherm (nm* against P at constant T) for a pure gas above its critical temperature monotonically increases with increasing P in the low-to-medium pressure regions, goes through a maximum value of nm*, followed by a region where nm* decreases with increasing P, and finally it becomes negative. Figures 4 and 5 show two experimental examples of such behavior. Figure 4 plots the equilibrium GSE for the adsorption of N2 on a sample of mesoporous alumina at three different temperatures.9 Figure 5 plots the equilibrium GSE for adsorption of N2 on a microporous zeolite at a single temperature.10 Both figures demonstrate that GSE for N2 goes through a maximum value at some intermediate high pressure. Figure 5 further shows that the N2 GSE becomes negative at a sufficiently high pressure. The other limiting behavior of the pure gas GSE isotherm in the region of very low gas pressures (Pf0) is given by Henry’s law:5,11
(b) volumetric methods, (c) piezometric methods, (d) a combination of gravimetric-volumetric methods, (e) total desorption method, (f) column breakthrough methods, (g) closed-loop recycle methods, and (h) isotope exchange methods. A brief description of the principles involved in these methods will be given to demonstrate that all of these methods measure the GSE of the components of a gas mixture under the equilibrium or transient conditions. (a) Gravimetric Methods.11,14 The simplest gravimetric experiment consists of exposing a clean adsorbent sample to a pure gas at constant P and T (mass density Fw, g/cm3) or to a gas mixture at constant P, T, and yi (mass density Fw, g/cm3) and then measuring the change in the mass (weight) of the adsorbent sample with time (t) until equilibrium is reached. Various designs of spring and microbalances are used for this purpose. The adsorbent is placed in a pan which is suspended from the balance arm inside a thermostated tube as shown by Figure 6a. The mass of the pan (wP) and the adsorbent (wS) are independently measured under vacuum. The experiment measures the mass of a control volume (VC) which includes the volume of the pan (VP), the volume of the solid matter of the adsorbent (Vs), the transient volume of the adsorbed phase on the solid surface [Va(t)], and the transient volume of the gas phase [V(t)]. Let w(t) and w be the transient state and equilibrium state mass of the control volume, respectively. Then, one can write
VC ) VP + VS + Va(t) + V(t)
(10)
(9)
w(t) ) VPFP + VSFs + Va(t)Faw(t) + V(t) Fw - VCFw (11)
where K*(T) and K(T) are called Henry’s law constants. They are functions of temperatures only. For an ideal gas [F ) P/RT], K* is equal to (KRT) where R is the gas constant. Equation 9 shows that the pure gas GSE isotherm is a linear function of the gas-phase density (or pressure) in the low-pressure region. Many experimental data have been measured to demonstrate the validity of eq 9.12,13
where FP and Fs are, respectively, the chemical densities (g/cm3) of the pan and the adsorbent materials. Fwa(t) is the transient mass density (g/cm3) of the adsorbed phase. The last term on the right-hand side of eq 11 accounts for the buoyancy correction. Equations 10 and 11 can be combined and rearranged to get
nm* ) K*(T).F ) K(T).P
constant T
w(t) ) wP + wS + wSnw(t) - (VP + VS) Fw (12)
Experimental Measurement of GSE Many different techniques have been developed for the measurement of pure gas and multicomponent gas adsorption equilibria and kinetics. Some of the frequently used methods include (a) gravimetric methods,
nw(t) ) Va(t)[Faw(t) - Fw]/wS
(13)
where nw(t) is the transient total specific mass surface excess (g/g) per unit amount of the adsorbent. For a
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Equations 12 and 15 can be combined to get S g S {nw(t)-[wg/ws]nw R } ) [{w(t)-w }-{wR-w }]/w
(16)
w Equation 16 shows that if nw(t).[wg/wS]nw R , then n (t) can be unambiguously estimated by measuring w(t), wR, wg, and wS using a gravimetric apparatus. The quantity wg/wS ()Fg/Fs) is generally near unity because the chemical density of most practical adsorbents are similar to that of the borosilicate glass (Fs ) 1.9-2.2g/cm3). It is generally accepted that the inequality [nw(t). w nR ] for the same gas-phase density (Fw) is satisfied because of the following.
(a) The surface area of nonporous glass (∼10 cm2/g) is negligible compared to that of a practical adsorbent (∼100-1500 m2/g). (b) The glass and the reference gases are nonpolar and therefore there are no specific adsorption forces between them. Helium is often used as the reference gas. (c) The temperature of the reference experiment (TR) can be substantially higher than that of the base experiment (T) which will further minimize the relative adsorption of the reference gas on the glass.
Figure 6. Schematic diagrams of apparatus used by various methods for measurement of adsorption: (a) gravimetric, (b) volumetric, (c) piezometric, (d) total desorption, (e) column dynamic, and (f) closed-loop recycle and isotope exchange. w mixed gas system, nw(t) ) ∑nw i (t), where ni (t) is the transient specific mass surface excess (g/g) of component i of the gas mixture. The equilibrium mass surface excess nw can be obtained by replacing w(t) with w in eq 12. The mass surface excess (nw, g/g) is related to the molar surface excess (nm, mol/g) by
nm ) nw/M
(14)
where M is the molecular weight of the adsorbed phase. For pure gas adsorption M is simply the molecular weight of the adsorbate. For mixed gas adsorption, one needs to know the adsorbed-phase composition to calculate its molecular weight. We now carry out a similar experiment using the same gravimetric apparatus where the adsorbent sample is replaced by a nonporous, nonpolar material of extremely low surface area like borosilicate glass having the same volume (VS) as that of the adsorbent solid matter of the previous experiment. The clean mass of the glass sample is wg ()wSFg/Fs), where Fg (∼2.3 g/cm3) is the chemical density of the borosilicate glass. The glass sample is then exposed to a very weakly adsorbing reference gas at a temperature of TR and pressure PR where the mass density of the gas is Fw. The weight of this reference system (wR) is measured. It follows that P S wR ) wP + wg + wgnw R - (V + V ) Fw
(15)
where nw R is the specific mass surface excess of the reference gas on the glass sample (g/g) at PR and TR (or Fw and TR).
Criterion (a) alone should be sufficient. For example, the ratio of the surface area of spherical particles (radius ) 0.1 cm) of borosilicate glass and that of a practical adsorbent (area ∼ 50 m2/g) of equal solid volume is only ∼2.8 × 10-5. By the same argument, the gravimetric method can be used to measure the surface excess isotherms for pure helium adsorption on practical adsorbents at different temperatures (nm* He as functions of P and T) and Henry’s law constants [K*He(T)] for helium can be estimated by eq 9. It will be shown later that the knowledge of K*He(T) is critical information needed for rigorous use of methods (b)-(h) in the calculation of GSE. The transient rate of change of mass GSE at constant P, T, and yi can easily be obtained by the gravimetric method. It follows from eq 16 that
dnw(t) 1 dw(t) ) s dt w dt
(17)
The gravimetric method is absolutely essential for obtaining K*He(T). It can also be conveniently used to measure equilibrium surface excess isotherms and rates of change of GSE for pure gases. It is not very useful for gas mixture studies because it only measures the total mass GSE. The total molar GSE or the individual surface excesses for the components of the gas mixture cannot be estimated from the Gravimetric data alone. (b) Volumetric Methods.5,11,12,14 The simplest volumetric method consists of expanding a pure gas or a multicomponent gas from a reservoir of known volume (VR) into an evacuated sample chamber containing a known amount (wS, g) of the clean adsorbent. The void volume of the sample side (empty volume - volume of adsorbent solid) is V0. The entire system is thermostated. Figure 6b shows a schematic diagram of the volumetric apparatus. The experiment consists of monitoring the gas-phase P, T, and yi of the sample chamber and the reservoir as a function of time (t). The initial gas conditions inside the reservoir are PR and T for the pure gas and PR, yRi ,
Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 3675
and T for the mixed gas. A transient molar balance for component i using eq 1 yields 0 VRFRyRi ) VRF(t) yi(t) + wSnm i (t) + V F(t)yi(t) (18)
where FR is the initial gas density in the reservoir and F(t) is the transient gas density in the entire system (assuming instantaneous gas mixing). Equation 18 shows that the specific transient molar GSE for comR ponent i [nm i (t), mol/g] can be measured by knowing V , R R S 0 F , yi , yi(t), w , F(t), and V , which are experimental variables for the volumetric test. Furthermore, the rate of change of molar GSE of component i is given by
d[F(t)yi(t)] dnm i (t) 1 ) - S(V0 + VR) dt dt w
(19)
The equilibrium GSE (nm i for component i at P, T, and yi) can be calculated using eq 18 and the final equilibrium gas density (F). The same equations can be used for a pure gas [yi(t) ) 1]. Equations 18 and 19 show that V0 must be measured accurately to use the volumetric method for the estimation of GSE. Estimation of the Void Volume An experiment similar to that described above can be carried at a temperature TR using a very weakly adsorbing gas like helium in the reservoir side (PRHe, TR) at the start of the experiment and measuring the final equilibrium pressure P at TR (density)FHe). Then, according to eqs 9 and 18, one has
V0 ) VR[FRHe - FHe]/FHe - wSK*He(TR)
(20)
It is assumed that the helium expansion experiment is done at a sufficiently low starting pressure and at a relatively high temperature so that the helium GSE isotherm is linear [eq 9]. Equation 20 shows that V0 can be accurately measured if K*He(TR) is independently measured by the gravimetric experiment. A very high value of TR can be used to minimize the value of K*He(TR). It is customary, in the published adsorption literature, to assume that helium at low pressures (near or below ambient) and moderate temperatures (∼300 K) is not adsorbed on practical adsorbents. Thus, the second term on the right-hand side of eq 20 is neglected. The quantity [{VR(FRHe - FHe)]/FHewS] is called the specific helium void volume [v0He, (cm3/g)] of the sample side. It is then assumed that (V0/wS) is approximately equal to v0He to calculate nm i (t) from the volumetric experiment data. In fact, eqs 18 and 20 can be combined to get R R R S 0 nm i (t) ) [V {F yi - F(t)yi(t)}]/w - vHeF(t)yi(t) +
K*He(TR)F(t)yi(t) (21) Equation 21 shows that the last term on the righthand side can be neglected only when F(t) is relatively small. That term may be comparable with nm i (t) at high gas-phase pressures, even though K*He(TR) is small. That is particularly true when nm i (t) is relatively small.
This is precisely the situation at very high pressures where the GSE of a pure gas approaches zero (Figures 4 and 5). The volumetric data of Figure 4 were calculated by assuming that helium was not adsorbed. Thus, the data in the high-pressure region should be re-examined. Very high pressure volumetric GSE data often show anomalies like crisscrossing isotherms at different temperatures and local minimums in the isotherms.9 They may be caused by ignoring the helium adsorption in the estimation of void volumes. The above discussions clearly demonstrate the value of knowing K*He(TR) for accurate estimation of GSE at high pressures by the volumetric methods. However, that can be achieved only by using a gravimetric apparatus as explained earlier. There are sporadically published data on helium adsorption on various adsorbents like activated carbons and zeolites.15,16 An extrapolation of Henry’s law constants (KHe) from these data by assuming that ln KHe is a linear function of reciprocal T15,16 showed that K*He at 600 K is in the range of 0.03-0.06 cm3/g for these adsorbents. Thus, for the adsorption of an ideal gas (say N2) at a pressure of 150 atm and a temperature of 600 K (F ) 3.0 mmol/cm3), the last term on the right-hand side of eq 21 is between 0.09 and 0.18 mmol/g, which is comparable with the nm* values of Figures 4 and 5 at that pressure. Thus, that term cannot be ignored for high-pressure GSE measurement, even if the helium void is measured at 600 K. Estimation of the Helium Density of the Adsorbent The helium expansion experiment described above is also used to get the helium density of the adsorbent FSHe [)wS/(VE - wSv0He)]. VE is the empty volume of the sample chamber. The true chemical density of the solid adsorbent FS is given by [wS/(VE-V0)]. Thus, it follows that
(1/FS) ) (1/FSHe) + K*He(TR)
(22)
Equation 22 shows that FS can be calculated using the data from the helium expansion experiment at TR and Henry’s law constant for helium adsorption on the adsorbent at TR (obtained gravimetrically). A key assumption in writing eq 22 is that helium is accessible to all the pores inside the adsorbent. The pores in the adsorbent which are not penetrated by helium (Lennard-Jones kinetic diameter ) 2.55 Å) are also not accessible for the adsorption of any other molecules because helium is the smallest known gas molecule.17 According to eq 22, those pores form an integral part of the solid matter of the adsorbent and are included in the definition of VS (and FS). (c) Piezometric Methods.9 A known amount of a clean adsorbent (wS) is put inside an evacuated thinwalled glass ampule. The ampule is placed inside a thermostated vessel consisting of a series of bulbs of known volumes which are connected to each other through capillary tubes as shown by Figure 6c. The total initial void volume of the vessel (V0) is measured before the ampule is placed inside it. A known quantity of pure gas (n j ) is then introduced into the vessel and the ampule is broken to contact the adsorbent with the gas. The equilibrium gas pressure (P) and density (F) inside the vessel is measured at constant temperature (T) as a
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function of the system void volume (V), which is varied by progressively introducing mercury into the precalibrated bulbs. Equation 1 can then be used to calculate the pure gas GSE isotherm [nm*(P,T)]:
nm*(P,T) ) [n j - VF]/wS
(23)
The adsorption kinetics cannot be measured by this method. It is most suitable for very high pressure GSE isotherm measurements. (d) Combination Gravimetric-Volumetric Methods.18 This method is used for measuring component GSE isotherms of a binary gas mixture without analyzing the final gas equilibrium mixture as needed by the volumetric method. It consists of expanding a binary gas mixture at PR, yRi , and T (density FR) from a reservoir of volume VR into an evacuated sample chamber of known void volume (V0). An adsorbent sample of known mass (wS) is suspended from a spring or a microbalance inside the sample chamber. The entire system is thermostated. The final equilibrium gas pressure in the system (P) and its density (F) are measured along with the weight change of the adsorbent due to adsorption. The total m specific molar surface excess nm ()nm 1 + n2 ) of the final equilibrium state can be calculated by using eq 18: R R 0 S m nm()nm 1 + n2 ) ) {V [F - F] - V F}/w
(24)
The total specific mass GSE at the equilibrium condition (nw) can be estimated by using eq 16: w m m nw () nw 1 + n2 ) ) [n1 M1 + n2 M2]
(25)
where Mi is the molecular weight of component i. Equations 24 and 25 can be simultaneously solved to m get nm 1 and n2 . The final equilibrium gas-phase mole fraction of component i (yi) can then be calculated using eq 18: R 0 yi ) (VRFRyRi - wSnm i )/(V + V )F
i ) 1, 2
(26)
This method is not suitable for kinetic measurements. (e) Total Desorption Methods.19 A thermostated packed adsorbent column of known void volume (V0) containing a known amount of adsorbent (wS) is saturated with a gas mixture at P, yi, and T (gas density ) F). The adsorbed components are then completely desorbed out of the column by heating and evacuating the column. The desorbed components are collected in an evacuated and thermostated chamber of volume VR. The gas pressure (PR), the density (FR), and the composition (yRi ) of the collected gas are measured. Figure 6d shows a schematic diagram of the apparatus for this method. The total amount of component i desorbed is given by n j i ()VRFRyRi ). The GSE of component i, nm i [P,T,yi], can then be calculated by using eq 1. This method is also not suitable for measuring adsorption kinetics. (f) Column Dynamics Methods. The simplest application of this method is to measure the equilibrium GSE of a trace (or dilute) component from a bulk gas. A thermostated packed adsorbent column of known void volume (V0) containing a known amount of the adsorbent (wS) is saturated with the pure bulk gas at P and T of interest. A binary feed gas mixture containing the trace component (mole fraction y0) and the bulk gas at P and T is then passed through the column at a flow rate of F0 (mol/s). The transient flow rate [F(t), mol/s]
and the mole fraction of the trace component [y(t)] in the column effluent gas is measured until the effluent gas composition of the trace component becomes equal to that of the feed gas [y(t) ) y0 at time t ) t*]. The column is saturated with the feed gas at this time. A mass balance for the trace component using eq 1 yields
F0y0t* ) wSnm + V0F0y0 +
∫0t*F(t)y(t) dt
(27)
where nm is the specific GSE of the trace component in the presence of the bulk gas at P, T, and y0 (density ) F0). Thus, eq 27 can be used to calculate (nm) from the column adsorption breakthrough data. A desorption experiment can also be done by (a) initially saturating the column with the binary feed gas mixture at P, T, and y0, (b) desorbing the entire amount of the trace component from the column by flowing pure bulk gas through the column at a flow rate of F0, and (c) measuring the transient flow rate [F(t)] and composition of the trace component [y(t)] in the effluent gas until the column is completely clean [y(t) ) 0 at t ) t*]. A mass balance for the trace component using eq 1 yields
wSnm + V0F0y0 )
∫0t*F(t)y(t) dt
(28)
Equation 28 can be used to calculate the specific GSE for the trace component (nm) at P, T, and y0 by using the column desorption breakthrough data. Figure 6e shows a schematic flowsheet of an apparatus for carrying out the column breakthrough tests. This method is not very practical for measuring adsorption kinetics. (g) Closed-Loop Recycle Methods.20 These methods are very suitable for measuring the equilibrium and transient GSE for adsorption of trace (dilute) components from a bulk gas. A known quantity of adsorbent (wS) is packed in a sample chamber. The chamber is placed in a closed-loop arrangement with a gas recycling pump in the line as shown by Figure 6f. The entire apparatus is thermostated. The void volume of the packed chamber (V0) and the empty volume of the loop (VR) are measured. For the simplest experiment in this category, the adsorbent chamber is filled with the bulk gas at P and T (density ) F). The loop is filled with the same bulk gas containing trace amounts (mole fraction y0i ) of other components at P and T. The measurement is started by circulating the loop gas through the sample chamber and measuring the transient [yi(t)] and equilibrium [yi] concentrations of the contaminants. A transient mass balance for the contaminant i using eq 1 gives
VRFy0i ) (VR + V0)Fyi(t) + wSnm i (t)
(29)
where nm i (t) is the transient specific molar surface excess of contaminant i. Equation 29 can be used to estimate (a) the equilibrium GSE [nm i ] of contaminant i at P, T, and yi in the presence of the bulk gas and other contaminants and (b) the rate of change of nm i (t) during the process. (h) Isotope Exchange Methods.21 This is a special case of the closed-loop recycle method. The sample chamber is initially equilibrated with a multicomponent gas mixture at P, T, and yi. The component i, however, consists of a bulk isotope and one or more trace isotopes of j types (mole fraction y0ij). The loop is initially filled with the same gas mixture at P, T, and yi. However,
Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 3677 Table 2. Comparative Advantages and Disadvantages of Various Experimental Methods adsorption equilibria methods
advantages
(a) gravimetric
simple
(b) volumetric
simple, multicomponent gas
(c) piezometric
ideal for high-pressure data no gas analysis needed, simple
(d) gravimetric-volumetric
(e) total desorption (f) column dynamic
(g) closed-loop recycle
(h) isotope exchange
multicomponent gas, final state controlled, easy to repeat good for trace adsorbates in bulk gas, relatively easy to repeat, constant P,T good for multicomponent trace absorbates in bulk gas, constant P,T multicomponent bulk or trace, final state under control, easy to repeat, constant P,T,yi
disadvantages pure gas only, no control over final state, difficult to repeat no control over final state, random data, difficult to repeat pure gas only
) (VFyi)
[ ] ∞ y** ij - yij
y∞ij - y0ij
dnm dyij(t) ij (t) ) -(VF) dt dt
advantages simple
disadvantages nonisothermal data, difficult to repeat nonisothermal data, complex, difficult to repeat not useful
binary gas only, random data, no control over final state, difficult to repeat not simple
not very useful, nonisothermal data, complex boundary conditions not very useful
requires precise flow rate and composition measurement
directly gives column dynamics, isothermal for trace
model-dependent analysis
no control over final state, difficult to repeat
isothermal, constant P,T
needs isotopes and their analysis
multicomponent gas, isothermal, final state under control, easy to repeat, constant P,T,yi
good for multicomponent trace adsorbates in bulk gas, difficult to repeat needs isotopes and their analysis
the initial composition of the jth type trace isotope of component i in the loop gas is y*ij(>
