Study of the Interaction between Adsorbed Hydrocarbon Molecules on Graphitized Carbon using the Chromatographic Step-and-pulse Method VON RYBINSKI ALBRECHT H. FINDENEGG BY WOLFGANG MARTIN AND GERHARD Institute of Physical Chemistry Ruhr TJniversity D4630 Bochum West Germany Received 17th July 1980 The chromatographic step-and-pulse method has been applied to a study of the adsorption of pure gases and gas mixtures on graphitized carbon black. It is shown how the retention time of a small pulse of a vapour B on a step of preadsorbed vapour A is related to the parameters of an equation of state (two-dimensional van der Waals equation) for the adsorbed gas mixture. The method is used to study the adsorption of isomeric C6-alkanes cyclohexane and benzene at 343.2 K.The adsorption isotherms derived for the pure vapours are compared with results obtained by static (gravimetric) measurements. The retention results of B on preadsorbed A are analysed and the best- fit values of the interaction parameter CAB are derived. These values are found to be smaller than the geometric mean of uAand uBfor the pure gases. Gas chromatography has been used essentially in two ways to study the physical adsorption of gases and vapours on solid surfaces (i) the retention time of very small samples of the vapour yields the initial slope (Henry’s-law constant) of the adsorption isotherm. In the case of adsorbents with homogeneous surfaces these studies have been used to investigate the gas-solid interaction potential of isolated adsorbed mole- cules.(ii) Chromatographic studies at higher concentrations of the vapour in the mobile phase can be used to determine adsorption isotherms up to higher surface coverages. The elution method frontal-analysis method and the step-and-pulse method have been applied for this purpose. In all these cases the pressure gradient along the column represents a major problem in the quantitative analysis of the ex- perimental data at higher concentrations of the vapour.2 The application of gas- chromatographic methods to adsorption studies of single gases therefore remains of limited interest since these measurements can now be made accurately by static met hods. When turning to the adsorption of gas mixtures a different situation arises. Static methods of studying the simultaneous adsorption of two components involve the measurement of two quantities viz.amount and composition of the adsorbate as a function of two independent variables pressure and composition of the gas phase. So far only a few studies of this kind have been sufficiently accurate for a test of theoretical models of mixed-gas ad~orption.~.~ A gas-chromatographic method to measure the infinite dilution activity coefficients of an adsorbed gas in the presence of a preadsorbed gas was used by Sloan and mull in^.^ However their method is valid only for negligible pressure gradient along the column. The experimental conditions under which such low pressure gradients can be achieved (short column large particle size low flow rate) lead to poor accuracy of the results.6 Recently Dondi et aL7 used the step-and-pulse method to determine single-gas adsorption isotherms of benzene CHROMATOGRAPHIC STUDY OF MIXED GAS-ADSORPTION and cyclohexane on graphitized carbon black considering in detail the finite pressure drop along the column.In the present work we extend this formalism to the adsorp- tion of binary gas mixtures. The theoretical basis of our analysis is outlined in the next section. It is shown how the retention time of a pulse of component B on a step of component A is related to the parameters of a two-dimensional equation of state for adsorbed gas mixtures. The method has been applied to binary mixtures of isomeric C,-alkanes including cyclohexane and benzene adsorbed on a low-surface-area graphitized carbon black.The adsorbates were chosen in order to study the influence of molecular shape on the pure-gas and mixed-gas adsorption parameters. THEORETICAL GENERAL RELATION FOR THE RETENTION IN GAS ADSORPTION CHROMATOGRAPHY The general theory of the step-and-pulse method has been derived by Valentin and Guiochon.6 Its application to the determination of pure-gas adsorption isotherms was outlined and discussed by Dondi et aL7 Here we summarize the application to two-component gas adsorption. In the present work the column containing the solid adsorbent is equilibrated with an adsorbable vapour A which is added to the inert carrier gas. When a constant mole fraction xAhas been established throughout the gas phase the partial pressure pAdecreases along the column in proportion to the total pressurep.As a consequence points along the column correspond to different points of the adsorption isotherm of pure component A. When a small sample of a second adsorbable vapour B is now injected two chromatographic signals will appear at the column outlet. One of these signals (retention time tA) is caused by the perturbation of the adsorption equilibrium of component A and is essentially the same as in the case of pure component A. The other signal (retention time tB) arises from the ad- sorption of component B in the presence of component A. It is convenient to intro- duce the retention quantities RA and RB where tM is the retention time of an inert gas measured at the same experimental con- ditions as tA and tg,pis the volume flow rate of the mobile phase at column outlet and column temperature T,msasis the area of the adsorbent in the column and j is related to the ratio of inlet pressure to outlet pressure PI=pi/po,by .3Pi-1 J=zpj51' The retention quantity R is related to the derivative of the adsorption isotherm of pure component A W. VON RYBINSKI M. ALBRECHT AND G. H. FINDENEGG RBis related to the initial slope of the partial isotherm of component B at given partial pressure (and hence given surface coverage) of component A (arB/ @B)pA,pB+() by The significance of the quantities drA/dpA and (arB/ apB)pB-Fo is illustrated in fig. 1. In eqn. (3) and (4) P is the reduced column pressure P =p/po. Hence the experi- mental values of RA(xA)and RB(xA) are related to average values of the corresponding PA PA FIG.1.-Schematic diagram of the adsorption of a binary gas mixture.isotherm derivatives for the range of vapour pressures pa = xAp,from column inlet to column outlet. It is not possible therefore to derive the shape of the adsorption isotherms from the experimental values of RA(x)and RB(x),but the parameters of any given isotherm equation can be determined by a least-squares fit of experimental R(xA)values for a range of mole fractions xA. For this purpose the right-hand sides of eqn (3) and (4) have to be inverted by expressing P as a function of the surface con- CHROMATOGRAPHIC STUDY OF MIXED GAS-ADSORPTION centration of the adsorbed components.Analytical isotherm equations of pure gases are usually given in the form pi =f(t?,) where Bi is the fraction of surface covered by component i = ei = ri/rm,iriN,,. (5) Inserting the isotherm equation pi I=f(Si) into eqn (3) leads to the relation7 The corresponding expression for RB(xA) refers to the situationp -+ 0; thus we have Taking P2and dP from this relation we find with Eqn (8) and (9) may be used to derive best-fit values for mixed-gas adsorption isotherms when the partial isotherms of the two components are given in the form pi =f,(Si €Ij). VAN DER WAALS EQUATION FOR MIXED-GAS ADSORPTION Several two-dimensional equations of state have been proposed in the literature to represent the physical adsorption of pure gases on homogeneous chemically inert surfaces.8 The most widely used of these is the two-dimensional van der Waals equation where a = 1/NT is the area per adsorbed molecule a and /3 are the two-dimensional van der Waals parameters.For mixtures the following combining rules for these parameters are usually adopted where xi and xi = 1 -xi are the mole fractions of the two components in the adsorbed layer and uAB is a parameter characteristic of the A-B interaction. By sub- stituting these expressions into the equation for phase equilibrium between the adsorbed layer and an ideal-gas phase the following relations are obtained for the partial isotherms of the two components9 W. VON RYBINSKI M. ALBRECHT AND G. H. FINDENEGG Here Kiis the Henry's law constant of component i and we have used the relation In the step-and-pulse method the concentration of the second component (B) is kept much smaller than the concentration of the first component (A).If KA and KB are of similar magnitude this condition implies that x 4x1 N 1 and & < 0,; thus the partial isotherm for component A reduces to the isotherm equation of pure A In the expression forp we substitute &/x," = OAPB/PAX~and neglect x& as compared with x~~AB; this leads to the partial isotherm of component B in the limit pB-f 0 Eqn (16) represents the van der Waals expression forf(0,) which can be inserted into eqn (6) to derive best-fit values of the parameters KAand aA/PAfrom a set of experi- mental values of RA(xA). Information about the mixed-gas adsorption parameters can be extracted from the retention quantity RB(xA).Eqn (16) and (17) yield the following expression for Here the Henry's-law constants KAand KB have been expressed by the retention quan- tities of the pure components in the analytical limit; viz. which follows from eqn (3) (5) and (16); similarly RO,B= rm,JKB;furthermore in the exponential of eqn (18) rm,B/rm,A has been substituted for PA/PB. Eqn (1 8)can be inserted into eqn (8) to derive best-fit parameters for the adsorption of gas mixtures from a set of experimental values of RB(xA). The numerical procedure is outlined in the next section. EXPERIMENTAL CHROMATOGRAPHIC MEASUREMENTS The gas-chromatographic apparatus with diffusion cell and saturation cell for the adjust- ment of the concentration of vapour A in the carrier gas has been described previously.1° A known weight (m,N 1 g) ofgraphitized carbon black (Sterling FT-G a = 11.1 m2g-l) was packed into the adsorption column (2 mm i.d.length 0.4 m). Adsorbed impurities were removed by flowing helium through the bed for 24 h at 400 "C. Helium (purity 99.996%) was used as the carrier gas. The hydrocarbons used as adsorbates had a purity of at least 99%. A representative chromatogram is shown in fig. 2. At first the pure carrier gas is displaced by a mixture of carrier gas and vapour A (mole fraction xA). When a steady signal has been attained at the column outlet (step height hA,s)the detector signal is compensated to zero CHROMATOGRAPHIC STUDY OF MIXED GAS-ADSORPTION and the amplifier is set to highest sensitivity.A small amount of vapour B is injected at time r = 0; two positive signals are then obtained (retention times tA and tB). For all experiments the pulse heights hA,pand hB,pwere < 5% of the step height hA,s. The residence time of a pulse of inert gas tM,was determined using methane. The mole fraction xAwas calculated from the saturation pressure of A pL( F), at the temperature of the saturation h t h =O rB I 1 I I r=O +t FIG.2.-Chromatogram for a step of cyclohexane (A) and a pulse of n-hexane (B). cell Tsc,using the vapour-pressure equation of the pure vapour," and from the total pressure in the saturation cell The pressures at column inlet and column outlet pi and p" were measured by mercury manometers; the ratio PIwas always between 1.4 and 2.0.The error in the experimental quantities causes a maximum relative error in the retention R of &3.5%. Results obtained with different columns (particle sizes 0.1250.1 50 and 0.200.30 mm re-spectively) agreed within 2%. Details of the experiments are given elsewhere.12 NUMERICAL PROCEDURE FOR PARAMETER FITTING The experiments yield the retention times tA and tB for given concentrations xA; RAand RBare obtained from the measured quantities by eqn (1) and (2),respectively. From a set of experimental RA(xA) data best-fit parameters of the isotherm equation of pure vapour A [eqn (16)] can be calculated; from these and a set of RB(xA) data the additional parameters of the partial isotherm equation of vapour B [eqn (17)] are then obtained (see below).As the initial (xA-+ 0) value of the retention RO,Aand RO,B,can be measured accurately (plateau values in fig. 3) the constants KA and KB in eqn (16) and (17) are expressed by r,,m,A/RO,A and rm,B/RO,B, respectively. Hence the isotherm equation for pure A [eqn (16)] contains the two adjustable parameters K~ and Tm,A= 1/NPA. Once these parameters for pure components A are known the equation for B in the mixture eqn (17) also contains only two adjustable parameters uiz. NAB and I'm,B= l/NbB. The fitting procedure for these two sets of para- meters proceeds in an analogous manner. Let us denote the adjustable parameters of a given isotherm equation by u and Tm. The best-fit values of cr I? are found by calculating the sum of the square deviations S(a r,) = V'X [R~(xA) -RC(XA)'JZ (20) for a range of values a and rmand drawing contour lines of S(u r,) in the two-parameter diagram of a as a function of rm.I3 The minimum of the deviation function S(u,I?,) corres- W.VON RYBINSKI M. ALBRECHT AND G. H. FINDENEGG ponds to the best-fit values of the two parameters; the mutual correlation between cc and I?,, and hence the physical significance of the resulting best-fit values can be judged from the shape of the S(a,r,,Jcontour diagram. In eqn (20) RE(xA)denotes an experimental retention value as obtained by eqn (1) or eqn (2); the corresponding value RC(xA)is calculated by eqn (6) or eqn (8) respectively with the chosen parameter values a and Tm.In these equations the integration limits 6; and 62 are derived from the given partial pressures pA = xApat column inlet and column outlet by solving the implicit isotherm eqn (16) by an iterative procedure; for the resulting values of 6A the relative error in pAwas made less than The integration of eqn (6) and (8) was then performed numerically again with a relative error d low5. Details of the computer program are given elsewhere.lZ RESULTS AND DISCUSSION We have made a study of the adsorption on graphitized carbon black of the five isomeric C,-alkanes (n-hexane 2-methylpentane 3-methylpentane 2,2-dimethyl- butane 2,3-dimethylbutane) cyclohexane and benzene. The pure vapours as well as binary mixtures of two vapours in the limit of the step-and-pulse method (pB-f 0) have been studied.In this paper some of the results are presented with the emphasis on a critical evaluation of the new method. All of the results refer to a common temperature of 343.2 K. Fig. 2 shows a chromatogram for a step of cyclohexane (A) and a pulse of n-hexane (B). The first peak can be attributed to the displacement of adsorbed A by the adsorption of B; so long as the basic condition of the step-and-pulse method is met uiz. hB,p S 0.05 h,,, the retention time fA is found to be independent of the nature of component B and to agree well with the retention time of a pulse of A on a step of A. The negative signal following the peak of A is a consequence of the mass balance of this component. The retention time tB and the ratio of the peak heights of the two components hA,p/hB,p,depends on the nature of B and also on the step height hA,s.Table 1 shows that for a given component A (cyclohexane) the ratio of pulse heights TABLE 1.-RATIO OF PEAK HEIGHTS hA,p/hB,p,AS A FUNCTION OF STEP HEIGHT (MOLE FRACTION XA)FOR PREADSORPTION OF CYCLOHEXANE (A) AND TWO COMPONENTS B xA/1o-3 n-hexane (B) XA110-3 2,3-dimethyl butane (B) -1.1 0.9 2.4 1.4 2.2 0.23 5.5 1.6 5.2 0.28 9.7 1.6 8.9 0.36 13.4 2.9 12.3 1.4 18.1 11 - h,/h increases with increasing mole fraction x,; hence the amount of A which is dis- placed by a given amount of B increases with increasing surface coverages. This effect is more pronounced for n-hexane than for 2,3-dimethylbutane (which is less strongly adsorbed than n-hexane but more strongly than cyclohexane; cf.fig. 3). Fig. 3 shows the dependence of the retention R of all hydrocarbons studied on the vapour phase concentration of n-hexane. For low xAthe retention R(x,) attains a constant plateau value R which corresponds to the retention of B on the homogeneous patches of the graphite basal plane in the analytical limit. R is a measure of the strength of interaction of adsorbed molecules with the solid. With increasing concen- CHROMATOGRAPHIC STUDY OF MIXED GAS-ADSORPTION tration xAthe retention RB(xA)increases and passes a maximum. This behaviour can be attributed to attractive lateral interactions between the test molecule B and surround- ing adsorbed molecules A.For the branched C6-isomers and cyclohexane this maxi- mum is found at higher bulk concentrations xA than for n-hexane itself (B = A). The retention of benzene on preadsorbed n-hexane does not exhibit such a maximum; -0.3- -3 I& e” E 0.2-* N * I E c-( s -e I E \ % 0.1 --lo-& I o-~ 10-2 XA FIG.3.-Retention R(xA) of small samples of the isomeric hexanes cyclohexane and benzene as a function of mole fraction xAof n-hexane in mobile phase. The adsorbent is Sterling FT-G graphit-ized carbon black column temperature 343.2 K PI = 2.0. this finding indicates that the lateral interaction of n-hexane with the aromatic mole- cules are weaker than with the aliphatic C6-isomers.We now turn to a quantitative analysis of the retention data in terms of the two- dimensional van der Waals (Hill-de Boer) equation as outlined earlier in this paper. Fig. 4 shows the experimental retention data R:(xA) of n-hexane on Sterling FT-G (corresponding to the data of the top curve in fig. 3) and the function R2(xA)calculated by eqn (6) with the pure-gas van der Waals functionf(OA) given by eqn (16) and ex- pressing KA by Tm,*and the experimental RO,A value [eqn (19)]. The curve exhibited in fig. 4 corresponds to the best-fit values of xA and rm,A as listed in the first line of table 2. The deviation function s(2&A/pA has a well-defined minimum with nearly-circular contour lines which means that the values of the two parameters are not strongly correlated and can be determined with some confidence.The results for pure cyclohexane can also be represented by the two-dimensional van der Waals equation but some of the branched isomers do not conform to this model equally well.12 W. VON RYBINSKI M. ALBRECHT AND G. H. FINDENEGG xA/10-2 FIG.4.-Experimental and calculated retention data of pure n-hexane adsorbed on Sterling FT-G graphitized carbon black; parameters listed in table 2. The chromatographic adsorption results for the pure components have been com- pared with static adsorption measurements on the same systems'* which were obtained by a gravimetric 171ethod.l~ Fig. 5 shows the results of the gravimetric study of the n-hexane/Sterling FT-G system. The two curves represent two-dimensional van der Waals isotherms with best-fit values of aA and rm,A as derived from the gravimetric data (a) and from the gas-chromatographic data (b).In the fitting procedure of the static measurements allowance has been made for a small positive increment of rA at pA= 0 to account for adsorption on high-energy sites of the graphitized carbon. Correspondingly in the fitting procedure of the gas-chromatographic measurements I 20 LI 0" .E G 10 1 .o 2.o 3.O rA/1Od6 rnol m-2 FIG.5.-Comparison of the gravimetric (a) and chromatographic (b)results of the n-hexane/Sterling FT-G system. Parameters of curve (a) RO,*= 0.258 x rnol m-' mbar-' rm,A = 4.09 x mol m-2 uA= 3.19 x J m2,TA(pA= 0) = 0.05 x mol m-'. Parameters of curve (b) see table 2.CHROMATOGRAPHIC STUDY OF MIXED GAS-ADSORPTION the value of Rohas been identified with the plateau value of R(xA)around xA = and the higher values of R near xA = lop5were attributed to the heterogeneity of the adsorbent. The initial slope of the isotherm obtained by the static and dynamic methods then agrees within 3%; the best-fit values of aAagree within 1%; but the monolayer capacity rm,A which follows from the static measurements exceeds the chromatographic value by ca. 8% (cf. fig. 5). The reason for this discrepancy in Tm between the static and dynamic results (which has also been found for other adsorp- tives) is not known. The experimental retention data R;(xA) obtained for a pulse of B on a step of A were analysed in terms of eqn (8) with the function g(eA) given by eqn (18).The 0.080 p -x I I I I 0.050' 2 .o 4 .O 6.O 8.0 xA,40-3 FIG.6.-Experimental retention data R;(xA)of cyclohexane (B) on preadsorbed n-hexane (A) and the corresponding functions R;(xA)for the three choices (a)-(c) of the parameters listed in table 2. parameters Ro,Aand RO,Bwere again taken from the experimental R(xA)at xA2 and the parameters aA and rm,A from the analysis of the single-gas gas-chromato- graphic results. The parameters KAB and rm,B could then be derived from the R;(xA) data as outlined in the experimental section. Results of such an analysis with n- hexane as the preadsorbed component A are summarized in table 2. In this table three different sets of the parameters NAB and rm,B are listed for each component B viz.(a> QAB =.\/a,aB; rm.B = r<B (b) ~AB best fit; rm,B = rmlB (c) NAB and rm,B best fit. Here aiand rm*,i denote the best-fit values derived from the gravimetric study of the pure gases i. Fig. 6 shows the experimental retention data R;(xA)for cyclohexane (B) on preadsorbed n-hexane (A) and the corresponding functions R;(xA)for the three choices (a)-(c) of the parameters listed in table 2. The simple mixing rule (a)yields systematic deviations in RZ at large concentrations x,; taking the pure gas parameter rtB and fitting aABalone [curve (b)]yields an aAB value slightly lower than the geo- metric mean of aAand as. If both parameters are fitted [curve (c)] the resulting Tm,B TABLE 2.-PARAMETERS OF EQN (18) FOR THE RETENTION OF HYDROCARBONS (COMPONENT B) ON STERLING FT-G WITH PREADSORBED n-HEXANE (COMPONENT A) AT 343.2 K.PARAMETERS RO,AAND RO,BFROM EXPERIMENTS AT LOW XA; PARAMETERS XA XB AND rm,A FROM ANALYSIS OF PURE GAS ADSORPTION; PARAMETERS CAB AND rm,B BY THREE DIFFERENT PROCEDURES (a)-(C)(SEE TEXT). component B &,A RO B XA RB rm,A rm,B ~AB mol m-2 rnbar-l J m2 mol m-2 10-39 Jm2 kAB method n-hexane 0.253 3.17 3.77 2-methylpentane 0.253 0.145 3.17 2.87 3.77 4.43 3.02 -4.43 2.47 0.18 -1.82 4.72 3-methylpentane 0.253 0.129 3.17 3.12 3.77 4.45 3.14 -4.45 3.03 0.04 -3.04 3.72 2,3-dimethylbutane 0.253 0.102 3.17 2.89 3.77 4.74 3.03 -4.74 2.83 0.07 -3.64 3.33 2,2-dimethylbutane 0.253 0.062 3.17 3.42 3.77 4.67 3.29 -4.67 3.OO 0.09 6.04 2.78 -cyclohexane 0.253 0.060 3.17 3.56 3.77 5.21 3.36 -5.21 3.31 0.01 Z 4.12 3.71 -benzene 0.253 0.175 3.17 1.09 3.77 6.50 1.86 -6.50 1.78 0.04 z -Q 6.92 1.78 Q CHROMATOGRAPHIC STUDY OF MIXED GAS-ADSORPTION is lower and aAB is larger than in (a).However the differences between the three curves are insignificant at most of the experimental concentrations xA. Fig. 7 shows the same sort of graph as fig. 6 for benzene (B) on n-hexane (A). In this case the three sets of parameters (a)-(c) are rather similar (table 2). For both systems the deviation function S(aAB rm,B) exhibits a flat minimum in the direction of the para- meter indicating that the variation of the monolayer capacity rm,B has little * 0.101 I 1 2.0 4 I.O 6.0 xA/lo-3 FIG.7.-Similar plot to fig. 6 for benzene (B) on preadsorbed n-hexane (A). influence on the function hence for B on preadsorbed A the parameter Tm,* cannot be determined with significant accuracy. In order to obtain aAB from the experimental results the fitting procedure (b)is therefore preferred. The best-fit values of aAB obtained by procedure (b)are smaller than the geometric mean of aAand aB for all systems listed in table 2. The deviation from the geometric mean is usually expressed by a parameter kAB:5 The values of k, for the present systems are also given in table 2. Results of further static and chromatographic adsorption measurements for the pure vapours and chromatographic results for mixtures of these vapours will be published.l5 This work has been supported by a Research grant from the Minister fur Wissen- schaft und Forschung des Landes Nordrhein-Westfalen.A. V. Kiselev and Y. Y. Yashin Gas-Adsorption Chromatography (Plenum Press New York 1969). P. Valentin and G. Guiochon Sep. Sci. 1975 10 245 271 289. R. 0.Friederich and J. C. Mullins Ind. Eng. Chem. Fundam. 1972,11,439. P. G. Hall and S. A. Muller J. Chem. SOC. Faraday Trans. 1 1978 74 948 2265. E. D. Sloan and J. C. Mullins Ind. Eng. Chem. Fundam. 1975 14 347. P. Valentin and G. Guiochon J. Chromatogr. Sci. 1976 14 56 132. 'F. Dondi M.-F. Gonnard and G. Guiochon J. Colloid Interface Sci. 1977 62 303 316. A. Patrykiejew M. Jaroniec and W. Rudzinski Chem. Eng. J. 1978 15 147. W.VON RYBINSKI M. ALBRECHT AND G. H. FINDENEGG S. E. Hoory and J. M. Prausnitz Chem. Eng. Sci. 1967 22 1025. loW. von Rybinski and G. H. Findenegg Ber. Bunsenges. Phys. Chem. 1979 83 1127. B. J. Zwolinksi and R. C. Wilhoit Handbook of Vapor Pressures and Heats of Vaporization of Hydrocarbons and Related Compounds (College Station Texas 1971). Landolt-Biirnstein Zahlenwerte und Funktionen aus Physik Chemie Astronomie Geophysik und Technik (Springer Berlin 1960) 11. Teil Bandteil a. l2 W. von Rybinski Dissertation (Ruhr-Universitat Bochum 1980). l3 A. Piechocki Staatsexamensarbeit (Ruhr-Universitat Bochum 1977). l4 S. Bliimel Diplomarbeit (Ruhr-Universitat Bochum 1980). l5 W. von Rybinski and G. H. Findenegg to be published.