4098
J . Phys. Chem. 1991, 95, 4098-4109
In the case of the acid alternate film, we also observed temperature dependence of intensities and frequencies of the C=O stretching bands for both constituents. Since, however, there were mutual overlapping between these bands, the situation was not so clear as in the case of the barium salt alternate films. Relationship between the pyroelectricity and structure of the carboxyl group in the acid alternate films may be a subject of further studies. For the alternate LB films with long-chain fatty acids and amine derivatives, Davies et al,lZl3have reported a relationship between pyroelectricity and degree of proton transfer from acid to amine head groups. Therefore, the head group structure to be one of the most important factors for Pyroelectricity of the alternate LB films. Furthermore, we consider the effect of the number of unit bilayers (n) of the alternate S(W,-Ba and s(s),films on their pyroelectricity and molecular orientation at room temperature. For the barium salt film, both pyroelectric coefficients and Orientation angles of the hydrocarbon chains of the constituents are almost independent of the n value (Figures 4 and 10). For the acid film, on the other hand, the pyroelectric coefficients decrease with increasing n value (Figure 4), and at the Same time, the Y values of DOPC and St-d,, increase with increasing n value (Figure 10). The increase in the y value reflects the decrease in polarity along the normal direction to the film Surface and, therefore, may result in the decrease in pyroelectricity. Finally, we compare the pyroelectric activities between the acid and barium salt alternate films. As is seen in Figure 10, the molecular orientations of DOPC in the S(PS), film and of DOPC-Ba in the S(PS),-Ba film are almost the same. However, the pyroelectric coefficient of the S(PS), film is about twice as large as that of the S(PS),-Ba film (Figure 4). It can be concluded therefore that if the molecular orientations are in the same order, the acid alternate film has larger pyroelectricity than the corresponding barium salt alternate film. In the former film, DOPC and St-d,, in the adjacent monolayers form the ring dimer through two hydrogen bondings. In the latter film, on the other hand,
DOPC and St-d,, anions in the adjacent monolayers form ionic bindings through the divalent barium ion. This may be a reason of the above-mentioned statement that the acid films are thermally less stable than the barium salt films. Therefore, the head group structure is more easily changeable on heating, and then pyroelectric activity becomes larger in the acid film as compared to that in the barium salt film.
.
Conclusions In this paper, we discussed the relationship between the pyroelectricity and molecular structure for the alternate LB films consisting of DOPC and St-d,, and of their barium salts. The conclusions are as follows. (1) Maximum pyroelectricity can be obtained around 40 OC, until which the molecules keep their highly oriented state. (2) The pyroelectricity is ascribed to a decay of the spontaneous polarization due to a structure modification of the polar head groups. (3) The positive current which increases rapidly above ca. 50 o c is caused by increased conformational disorder and thermal motions of the hydrocarbon chains. (4) The alternate film with more highly oriented molecules shows a larger pyroelectricity. ( 5 ) If the molecular orientations are in the same order, the acid alternate film has larger pyroelectricity than the barium salt alternate film. This may be due to weaker intermolecular interactions in the former film than in the latter film. These results may serve as a clue to the design for LB films that have larger pyroelectric coefficients.
Acknowledgment. The authors are grateful to Prof. H. Taniguchi of Kyushu University for his kind gift of the DoPC sample. Their thanks are also due to Dr. M. Sugi and Dr. K. Saito of the Electrotechnical Laboratory for their helpful advice on the pyroelectric measurements, to Prof. T. Hanai of this institute for the use of the picoammeter, and to Prof. S. Hayashi of this laboratory for his valuable discussion. This work was partly supported by the Grant-in-Aid for Special Project Research from the Ministry of Education, Science and Culture, Japan.
Determination of the Surface Energy Distribution Using Adsorption Isotherm Data Obtained by Gas-Sdid Chromatography Jeffry Roles and Ceorges Cuiwhon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996- 1501, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 -61 20 (Received: September 24, 1990; In Final Form: December 21, 1990)
A new numerical method for calculatingthe adsorption energy distribution function from adporption s io h te r mdata is introduced.
It is intended to be robust with respect to quasi-langmuir adsorption isotherms, the type most often observed in inverse gas chromatographic experiments involving energetically heterogeneous stationary phases. The stability of the method and the accuracy of its solutions is demonstrated by using simulated data. The determination of the energy distribution of diethyl ether on activated alumina is reported. It is composed of two narrow bands at 11.04 and 12.68 kcal/mol. The variances of these bands are 1.66 X 10-' and 1.00 X 10-a kcalz/mo12,respectively. The monolayer capacity is 5.36 X 1@ mol/g for the low-energy band and 2.53 X lod mol/g for the high-energy band.
Introduction Progress in many Of the sciences be by the development Of advanced methods 'f' the investi@tion Of processes taking place at boundaries between phases. Inverse chromatography, suggested by Guillet,' is a Dotentially important tool foistudfng t6;gassofid interface. ChromatogTo whom correspondence should be sent at the University of Tennarsee.
0022-3654/91/209S-4098$02.50/0
raphy is principally a separation method used for analysis.2 The components of a mixtuie are separated by elution on a column of known properties. Chromatography is also used to study retention mechanisms, Le., to investigate the nature and the thermodynamic characteristics of the molecular interactions between ( I ) Guillet, J. E. J . Macromol. Scl., Chem. 1970, A l . 1669. (2) Kicrelev, A. V.; Yashin, Ya. 1. Gus Solid Chromutogruphy; Plenum Press: New York, 1969.
0 1991 American Chemical Society
Surface Energy Distribution a known, homogeneous stationary phase and various compounds in order to better understand the physicochemical properties of these compounds. In other words, the emphasis is usually on determining the properties of the solutes. Inverse chromatography, on the contrary, is a method of study of the properties of the stationary phase. The interactions between this phase and some well-known probe solutes are determined by using the same procedures as in direct chromatography, but the results are used to derive properties of the stationary phase. Inverse gassolid chromatography provides information that is very different from that derived from the classical methods of surface analysis (e.g., Auger electron spectroscopy, ESCA, LEED) but that may complement their results. In these physical methods, the surface of the sample is irradiated with high-energy particle beams, under vacuum. The results obtained are an average of the contributions of the atoms contained in a layer whose thickness always exceeds a few atomic diameters. Inverse chromatography probes only the electrical field extending above the external layer of atoms at the surface. In practice the main contribution comes from the external orbitals of these atoms.2 The energy involved in gas-solid adsorption is usually between 25 and 100 kJ/mol, i.e., 0.25-1 eV/molecule. It is orders of magnitude smaller than the energy involved in the collision of the electronic beams and the solid investigated with the physical methods. Potential advantages of inverse chromatography include speed, relatively low cost, and simplicity. In addition, chromatographic techniques do not require analysis under high vacuum and may be performed at relatively high temperatures. Thus, the materials of interest may be used as stationary phases and characterized under conditions that are much more similar to their natural state than when physical methods are used. The results obtained so far by inverse chromatography have been reviewed recently by Gilbert.' Guillet has developed methods to measure the glass transitions of polymers,' to study the temperature dependence of their ~rystallinity,~' and to determine the Flory interaction parameter'*$and the Hildebrand-Scatchard solubility parameter.'*' Several studies have investigated various aspects of heterogeneous catalysis.* The fate of toxic solutes adsorbed on fly ash9 and diesel particulate matterlo has been studied. Other applications include measurement of the selectivity of liquid-chromatographic stationary phases" and sorption of water and naturally occurring organic molecules on minerals.I2 Although the concept of energetic heterogeneity of surfaca was introduced long ago by Langmuir,'' interest in studying this property as a means of material characterization did not develop until the past 20 years. Even in chromatography, where peak tailing is systematically blamed on the Occurrence of active sites on the surface of the adsorbent or of the support used in gas-liquid chromatography, few experimental investigations of the energetic heterogeneity of the surface of the material used have been made. The basis of these studies is the calculation of the adsorption energy distribution function of a series of probe compounds from their adsorption isotherms." These functions provide valuable qualitative information regarding the energetic heterogeneity of the surface. More importantly, it has been shown that thermodynamic quantities (e.g., monolayer energies) may be calculated from the energy distributions obtained from inverse gas-chro(3) Gilbert, J . E. In Aduonces in Chromatography; Giddings, J. C., Grushka, E., Brown, P. R., Us.; Marcel Dekker: New York, 1984; Vol. 23, p 199. (4) SmiQrod, 0.;Guillet, J. E. Macromolecules 1969, 2, 443. (5) Stein, A. N.; Guillet, J. E. Macromolecules 1970, 3, 102. (6) Gray, D. G.;Guillet, J. E. Macromolecules 1971, I , 129. (7) Stein, A. N.; Gray, D. 0.;Guillet, J. E. Brit. folym. J. 1971.3. 175. (8) Paryjczak, T. Gas Chromotography in Adsorption ond Corolysis; Wiley: Chichcster, NY, 1986. (9) Eiceman, G. A.; Vandiver. V. J. Armos. Enuiron. 1983, 17, 461. (IO) R w , M. M.; Risby, S.S.; Lettz, S. S.; Yasbin, R. E. Enuiron. Sci. Technol. 1902, 16, 7 5 . (1 1) Boudrcau, S.P.; Smith, P. L.; Cooper, W. T. Chromorogr. Forum 1987, I (June), 31. (12) Cooper,W. T.; Hayes, J. M. J . Chromatogr. 1984, 314, 111. (13) Langmuir. 1. J . Am. Chem. Soe. 1918, 40, 1361. (14) Jaroniec, M. Thin Solid Films 1983, 100, 325.
The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 4099 matographic experiments.Is Thus, the surface energetics may be parametrized, and the information obtained may be much more useful for the characterization of the material than, for example, the retention times of the probe compounds, their retention index, or even the adsorption isotherm data. We are interested in the characterization of powders (e.g., alumina) that are utilized in the processing of ceramic materials. The aim of this paper is a description of the method we have developed for the determination of the energetic heterogeneity of surfaces from the adsorption isotherm of probe compounds. In this paper the emphasis is placed on the derivation and validation of a new numerical method for calculating the surface energy distribution from adsorption isotherm data that may be applied to a wide range of adsorbents with high accuracy. It must be emphasized that an adsorption energy distribution is specific of the probe compound used to determine it. The characterization of a surface requires the use of a number of probe compounds selected for their physicochemical properties (e.g., nonpolar, electron donor, electron acceptor).
Theory It has been shown that the calculation of the energy distribution requires the solution of a linear first kind Fredholm equation that relates the energy distribution to the adsorption isotherm via a suitably chosen model.I6 We assume the adsorbent surface to be heterogeneous. Let f(E) be the adsorption energy distribution function over the energy range, fl, that we consider. f ( E ) dE is the amount of the probe component adsorbed per unit mass of the adsorbent sample, with an adsorption energy between E and E dE. The total amount of adsorbate on the sample surface, q(P), in equilibrium with a vapor pressure P,at a constant temperature, is given by the equation
+
In this equation, 8(E,P) is the local adsorption equilibrium isotherm of the adsorbate used on the adsorption sites having an energy within the interval (E, E + dE), while q(P) is the global adsorption isotherm. Only q(P) is accessible experimentally. Equation 1 states mathematically the unnormalized problem. The ill-ped nature of this problem has been discussed extensively in the l i t e r a t ~ r e . I ~Because -~ the problem is ill posed, its accurate solution is very difficult. There are several major problems to solve. Two of them are fundamental obstacles, consequences of the ill-posed nature of the problem; the third one is more practical. First, the problem is ill conditioned. Small fluctuations in the global adsorption isotherm (due to noise and experimental errors) cause large fluctuations in the calculated energy distribution function. Second, the problem does not have a unique solution. As Adamson has if an exact, physically realistic, solution exists within the solution space (defined by the choice of the energy range, Q),then an infinite number of exact solutions exist within that space. These solutions are oscillatory, however, and have no physical sense. Somehow, methods for solving the integral equation must allow only physically realistic (i.e., nonnegative, nonoscillatory) solutions. Third, there are several experimental problems to be solved. First, it is difficult to determine the point on the adsorption isotherm corresponding to complete monolayer coverage. This derivation is especially important when thermodynamic quantities are to be calculated from the energy distribution function. Because local multilayer formation may occur before a global monolayer has formed, traditional isotherm equations used to determine Bondreau, S.P.; Cooper, W. T. Anal. Chem. 1987, 59, 353. Sips, R. J . Chem. fhys. 1948, 16,490. Dormant, L. M.; Adamson, A. W. Surf Sci. 1977,62, 337. House, W. A. J. Colloid Interfoce Sei. 1978, 67, 166. Merz, P. H. J . Compur. Phys. 1980, 38, 64. Miller, G. F. In Numerical Solurions of Integral Equations; Walsh, J., Delves, L. M., Eds.; Clarendon: Oxford, 1974; Chapter 13, p 175. (21) Adamson, A. W.; Ling, I.; Dormant, L. M.; Oren, M. J . Colloid Interfoce Sei. 1966, 21, 445. (15) (16) (17) (18) (19) (20)
4100 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991
monolayer coverage may yield unrealistic results.I2 Second, great difficulty may arise experimentally if one attempts to measure the entire submonolayer isotherm by gas-chromatographic methods. Undesirable nonlinear effects (e.g., the sorption effectu) may become significant at solute partial pressures approaching the equilibrium vapor pressure. This effect is especially significant at high column-pressure drop, so porous layer open tubular columns must be used. Finally, the retention time of the selfsharpening front boundary of a quasi-Langmuirian peak and the gas holdup time must be measured independently. These two times are close at high sample sizes, i.e., for measurements carried out a t high partial pressure of the probe compound, causing a significant error on the isotherm. If they overlap, the isotherm calculated by the ECP method is oscillatory. A general method for solving eq 1 would permit the determination of the energy distribution function from any adsorption isotherm, regardless of the experimental method used to measure the isotherm data, of the experimental conditions, and/or of the adsorbate/adsorbent pair studied (although the function itself obviously depends on the nature of the pair selected). This goal is probably not realistic. A worthwhile goal, however, is to develop a method that may be utilized for the majority of adsorption systems used in the separation sciences (e.g., chromatography) and in various technologies (e.g., for the characterization of ceramic materials). Since the energetic heterogeneity of such materials tends to result in adsorption isotherms that have negative curvature, Le., are convex upward, we may restrict the applicability of our method to the case of quasi-Langmuirian isotherms without any significant loss of its usefulness. Since the choice of a model relating the energy distribution to the global isotherm is central to the development of the method used for the calculations, the next section is devoted to discussing various aspects of theory. In the following section, a brief review of the methods of solution that are most important to our specific problem is given. Finally, the details of the numerical solution we have developed and its testing and application to experimental data are described. I. Relationships between Equilibrium Isotherm and Surface Energy Distribution. Consider a surface made up of K types of adsorption sites, the ith type having n,,,, sites capable of reversibly adsorbing one molecule of solute with an energy E,. If nl denotes the number of molecules adsorbed on the ith type of site when the system is in a particular state of equilibrium (determined by the constant temperature, T, and the solute partial pressure, P), then the total amount adsorbed, k, is given by
Roles and Guiochon TABLE I: Statisticnl Moments Tlut Provide Tbermodyarrmic
P~nwtriutioaof Eacrpy-DisMbutioa Perks param symbol equation
distribution function
muation
a Empirical parameters: @,, normalization factor; EIvJ,average energy for gaussian peak having variance a& T ~ exponential , parameter (analogous to time constant); A,, positive shape parameter; ql,width parameter. E' is an integration variable.
The choice of analytical function used to relate the solute partial pressure to the relative monolayer coverage at a given energy, generally referred to as the local isotherm, defines the model. Since
our global isotherms are restricted to quasi-Langmuirian types, the Langmuir model is the obvious choice for the local i~otherm.'~ It is given by 8(Ef,P)= [ l (K/P)e-E{/RT]-l (6) where K is a constant related to the physical properties of the vapor considered. The assumptions underlying the Langmuir model are the following: (i) The compound studied behaves ideally in the gas phase. (ii) The molecules of the compound studied are adsorbed reversibly. (iii) Adsorbed molecules interact with adsorption sites on a one-to-one basis (i.e., only one molecule is adsorbed per site). (iv) Intermolecular attractive forces between adsorbate molecules are negligible. (v) The adsorbed phase is not mobile (Le., mass transfer occurs only between the adsorbed phase and the gas phase and not from sites to sites). N. Possible Methods of Solution. This section is not intended as a comprehensive review of methods for solving the integral eq 1 for the energy distribution function. It is meant to highlight the strengths and weaknesses of the existing methods, which may be used to solve the less general problem outlined earlier. It will also convey the background and motivation that led to the development of the method described in the next section. The specific reasons why the integral equation is difficult to solve, the ill-posed nature of the mathematical problem, have already been discussed. We have limited the scope of the problem by restricting it to one class of isotherms, those that are convex upward. At this point, we must consider two important attributes of calculation methods, accuracy and robustness. To compare the accuracy of solutions, we have chosen our figure of merit as the root-mean-squared relative error between the isotherm points calculated by integrating eq 1 and the experimental isotherm points. It should be noted that physical realism is another aspect of accuracy. It is usually implicit but is important. To yield physically realistic solutions, a method must exclude those solutions that are not realistic (e.g., negative or oscillatory solutions). As pointed out earlier, a unique solution is not possible, and any solution obtained must be regarded as one of a family of solutions with superimposed oscillations.21 Any method for solving the equation must employ some technique for coping with this problem and must be carefully tested to ensure stability. A robust method will deal with all types of global adsorption isotherm data satisfactorily and will tolerate experimental errors to a degree. Within the range of the problem we have outlined, a robust method would be able to calculate the best solution corresponding to any quasi-Langmuir global isotherm using the Langmuir local isotherm. The energy distribution derived from isotherms measured under slightly different experimental conditions on the same sample must be similar. Several r e v i e w ~ ~ ~ , ~ '
Bosanquet, C.; Morgan, G. D. In Vapour Phase Chromatography; Desty, D. H..Ed.;Butterworths: London, 1957.
245. (24) Jaroniec, M.Adv. Colloid Sci. 1983, 18, 149.
Since the local isotherm, 8(Ef,P),is equal to the proportion of occupied sites of energy E,, nl/nm,f,we have
If one assumes that the total number of sites is practically infinite and that the range of adsorption energies is given by Q, eq 3 is replaced by an integral. Dividing both sides of the equation by the mass m of adsorbent gives the usual isotherm, related to the unnormalized energy distribution function: (4)
with
(22)
+
(23) Braucr, P.;Fassler, M.;Jaronicc, M. Thin Solid Films 1985, 123,
The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 4101
Surface Energy Distribution
LEGEND ---- Expehmentd Simdoted
0.0
1.1
0.8
Retention Time (Minutes)
Flgm 1. Elution profile of diethyl ether on an open tubular column with the inner wall coated with calcined alumina. Column dimensions: length, I5 m;i.d. 0.53 nun. Mass of alumina in the column, 44 mg. Carrier gas, helium; flow velocity, 73.1 cm/s. Temperature: 60 OC. Sample size of diethyl ether, 0.460 fig. Solid line: calibrated experimental chromatogram. Dashed line: chromatogram calculated using the isotherm derived from the energy distribution obtained.
Figure 2. Adsorption isotherm of diethyl ether on calcined alumina at 60 OC. Solid line: isotherm derived from the elution profile in Figure 1. Dotted line: isotherm calculated from the optimized energy distri-
bution (Figure 3).
and books25926discuss methods of solution on a much more comprehensive level than possible here. One way to deal with the ill-posed nature of the problem is to restate the problem in a better posed form. The most widely used class of methods for solving the integral equation employs some form of the condensation approximation. An analytical solution is made possible by replacing the Langmuir local isotherm by some sort of discontinuous function. The assumption is made that energy sites are filled in order of decreasing energy. In the simplest form of the method, the actual local isotherm is replaced by a step function:z7J8
where P, is some characteristic condensation pressure. Hobson has suggested a more sophisticated method that is an asymptotically correct a p p r o ~ i m a t i o n . ~It~employs a local isotherm that has a Henry’s law region for pressures below P,:
[1
e (E, P) = P K f f ”
if PIP, if P > P ,
The errors that result from using these methods have been discussed.3w32 These methods are robust and stable if the effect of experimental noise is adequately dealt with. Although very inaccurate except at very low temperatures, they yield physically realistic results if the entire submonolayer range of the global isotherm may be determined. Because of their simplicity and ease of calculation, these methods have been used extensively with adsorption data measured by gas chromatography.’ 1~12~15,33-37 (25) Paryjczak, T. Gas Chromotography in Adsorption ond Cotolysis; Wiley: Chicheater, NY, 1986; Chapter 3. (26) Jaroniec, M.; Madey, R. Physicol Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, The Netherlands, 1988; Chapters 2-3. (27) Sips, R. J . Chem. Phys. 1950, 18, 1024. (28) Cerofolini, C. F. Surf. Sci. 1971, 24, 391. (29) Hobson, J. P. Can. J . Phys. 1965.43, 1934. (30) Harris, L. B. Surf.Scf. 1968, I O , 129. (31) VanDongen, R. H.; Broekhoff, J. C. P. Surf.Sci. 1969, 18, 462. (32) Morrison, 1. D.; Ross, S.Sur/. Sci. 1977, 62, 331. (33) Rudzinski, L. A.; Waksmundzki, A.; Leboda, R.; Jaroniec, M .
Chromotogrophio 1974, 7, 663. (34) Rudzinski, W.; Waksmundzki, A,; Leboda, R.;Suprynowicz, Z. J . Chromotogr. 1974, 92, 25. (35) Waksmundzki. A.; Jaroniec. M.; Suprynowicz, Z . J . Chromotogr. 1975, 110, 381.
11
I2
Adsorption Energy (KCol/hlole)
Figwe 3. Energy distribution function for the adsorption of diethyl ether on calcined alumina at 60 OC. Distribution derived from the experimental isotherm in Figure 2. TABLE III: Optimized Parameters for the Energy Distribution Corresponding to Activated b~umirur/DietbylEther at 60 OC
monolayer capacity. mol/g
pcak both I
av energy, variance, kcal/mol kcaI2/mol2
RMS error 8 5.16 X lo-’
5.36 X IOd 2.53 X IOd
2
1 1.04 12.68
1.66 X 10-5 1.00 X IOd
Another way to restate the problem is to use an integral transform. Integral transform methods give accurate solutions; however, they are generally not robust. SipsI6 solved the integral equation for several analytical forms (assumed a priori) of the global isotherm using the Stieltje‘s transform. The form of the energy distribution is determined by the choice of the analytical function used to fit the isotherm data. Therefore, one must have some a priori knowledge of the energy distribution in order to choose an appropriate isotherm equation. Jaroniec helped alleviate this “robustness” problem by deriving a transform method (based on the Stieltjes transform) that employs an empirical isotherm 0 I(36) AI
Suprynowicz, 2.;Jaronicc, M.; Gawdzik, J. Chromotographio 1976.
-, .-.. (37)
Boudreau, S.P.;Cooper,W. T. Anal. Chem. 1989, 61, 41.
4102 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991
Roles and Guiochon
n
W
v c
9
11
1; J
9
11
13
E
'g :
'g 4
IV
3-
II
4 9
11
13
equation that could fit a large variety of isotherm^.^^ Traditional optimization methods utilize numerical linear algebra techniques to determine the energy distribution that causes the minimum distance between the global experimental isotherm and the isotherm calculated by integrating the Fredholm eq 1. Because no unique solution exists, these methods tend to give solutions that are physically unreal is ti^.^^ The most popular technique for forcing the solution to be smooth is regularization. Several methods of regularization have been described in the These methods employ a dampening function to suppress the oscillations in the energy distribution. A tradeoff is reached between the goodness of fit between the experimental and calculated isotherms and the degree of oscillation suppression. The accuracy of solutions is decreased because the function minimized must be larger since it includes the dampening function. An interesting and unfortunate fact is that integral equations that have a smooth kernel (e.g., the Langmuir local isotherm) have a relatively greater trend toward oscillatory solutions and thus must have relatively larger dampening function^.'^ The most promising method of this class seems to be the CAEDMON algorithm of Ross and Morrison, which utilizes a multivariant NewtonRalphson method of ~ptimization.~' The Adamson and Ling (AL) method is a graphical algorithm that requires a priori knowledge of the global monolayer capacity.'2 (38) Jaroniec, M.Sur/. Sci. 1975. 50, 553. (39) Noble, B. In The State of the Art in Numericul Analysis; Jacobs, D., Ed.; Academic Press: New York, 1977; p 915. (40) House, W. A. J. Chem. Soc., Furuduy Trum. I 1978, 74, 1045. (41) Morrison, 1. D.; Ross, S. Sur/. Sci. 3975, 52, 103.
I
9
11
13
It has been applied by several authors.434s The iterative improvement scheme of the AL method was used in the computer program HILDA written by House and Jaycock.46 Besides converting the AL algorithm to numerical language, they made several improvements over the original algorithm. These improvements include (i) the provision of quadratic smoothing options, both for the global isotherm and for the energy distribution, (ii) the option for choosing one between several possible local isotherms, and (iii) an iterative calculation of the monolayer capacity. The program HILDA was applied exten~ively."-~ Since Adamson and Ling published their original method, numerous studies have been performed to compare the methods that utilize the AL iterative improvement scheme with other method^.^'-^^ This class (42) Adamson, A. W.; Ling, I. Ado. Chem. Ser. 1961, 33, 51. (43) Hsieh, P.Y.J. Cutul. 1963, 2, 221. (44) Sorrell, J. B.; Rowan, R. A d . Chem. 1970, 42, 1712. (451 Jackson. D.: Davis. B. J. Colloid Interfuce Sci. 1974. 47. 449. (46) House, W. A.; Jaycock, M. J. ColloidPolym. Sci. 1978; 256, 52. (47) Sidebottom, E.;House, W. A,; Jaycock, M.J. Chem. Soc.. Furuduy Trans. I 1976. 72. 2709. (48) House; W.'A.; Jaycock, M.J. J. Colloid Intetface Sci. 1977,59,266. (49) House, W. A.; Born, G.; Brauer, P.;Granke, S.; Henneberg. K.H.; Hofer, P.;Jaroniec, M. J. Colloid Interfuce Sci. 1984, 99, 493. (50) Leng. C. A.; Clark, A. T. J. Chem. Soc., Furuday Trow. I 1982,78, 3163. (51) House, W. A.; Jaycock, M.J. J. Chem. Soc., Furuduy Trans. I 1977, 73, 942. (52) House. W. A. J. Colloid Interfuce Sei. 1978, 67, 166. (53) House, W. A.; Jaroniec, M.;Brauer, P.;Fink, P . Thin Solid Films 1982, 87, 323. (54) Dormant, L. M.;Adamson, A. W. Surf. Sci. 1977,62, 337. ( 5 5 ) House, W. A.; Jaycock. M.J. J. Colloid Interface Sci. 1974.47. 50.
The Journal of Physical Chemistry, Vol. 95, No. IO, 1991 4103
Surface Energy Distribution n
D
'g Y
Ill U
.j gd
Y 9
11
9
13
11
13
E
ii
I
9
11
13
E F'igure 5. Difference between simulated (solid line) and optimized (dashed line) single-peak EMG energy distribution functions. In each case, qm = 1.O X lp ml/g and E," = 10.0 kd/mol. Under each distribution function, a plot of the difference between the simulated and the optimized distribution is given in a magnified scale.
of methods seems to be highly robust. Great care must be exercised however to prevent the possibility of generating spurious peaks (due to experimental noise) in the sol~tion?~ Also, the entire submonolayer isotherm must be determined. In summary, this literature survey has revealed several important mdts regarding the applicability of existing methodology to our problem. No single method stands out as the perfect solution. On the Contrary, it appears that it Could be possible to develop a very robust and highly accurate method by combining features taken from several present methods. The most salient results from this survey, which must be used as a basis for the development of a new method, are summarized here: (i) The most robust methods have used the AL iterative improvement scheme. (ii) Fitting the global adsorption isotherm to an analytical expression, as in the integral transform methods, is a valuable technique since a less-ill-posed problem may be solved. However, these methods apply only to a limited class of global isotherms. (iii) Condensation approximation methods and methods that use a traditional optimization scheme are less desirable because of poor accuracy and robustness. (iv) Since the studies that compared calculation methods were performed using data obtained from low-temperature vacuum experiments, they should be repeated using gas-chromatographic data before general conclusions regarding their performance may be drawn. I I I . Description of a New Numerical Method. Our method consists of two subalgorithms. The method evolved because we found that we could obtain much more accurate solutions by
applying the second subalgorithm, using the solution from the first subalgorithm as a starting point. In cases where the second subalgorithm is not applicable, one may acquire a fairly accurate solution from the first. Thus a highly robust method is possible. The two subalgorithms are described separately. ( I ) Subalgorithm I : The Quasi-Adamson (QA) Method. To allow the solution of a less-ill-posed problem, the global isotherm is replaced by a smooth function. The multi-Langmuir isotherm equation, first suggested by GrahamS6and used successfully by L a ~ b , ~is' used:
(9) where ak and bk are numerical coefficients, determined by best fit of the experimental data. Equation 9 should fit any submonolayer quasi-Langmuir global adsorption isotherm with reasonable accuracy. In most cases, a bi-Langmuir isotherm should be sufficient. Unless data are available in a very large range of partial pressures, the determination of three sets of coefficients (Ok, bk) in eq 9 cannot be made accurately. Furthermore, a high accuracy is not necessary in cases where subalgorithm 1 is used as a starting point for subsequent optimization. By use of eq 9 with the best-fit values of the coefficients, the energy distribution is calculated by using the AL iterative improvement scheme." The specific algorithm for the AL section is adapted from House et aI.* The range of energies considered (56) Graham, D.J. Phys. Chcm. 195J.57, 665. (57) Laub, R. J. ACS Symp. Ser. 1986, 297, 1.
4104
The Journal of Physical Chemistry, Vol. 95, No. 0, 1991
Roles and Guiochon
" l
'oa
1
h
W S
Y
9
,
0
9
11
11
,
13
E
U
9
ii
9
11
i
E
Figure 6. Difference between simulated (solid line) and optimized (dashed line) single-peak y energy distribution functions. In each case, qm = 1.0 X lVs mol/g and E," = 10.0 kcal/mol. Under each distribution function, a plot of the difference between the simulated and the optimized distribution is given in a magnified scale. (see next section for its determination) is divided into 400 equally spaced, discrete energies. The corresponding pressures are calculated, and the values of the best multi-Langmuir isotherm function are evaluated a t each of these pressures. The jth (1 < J < 400) isotherm value calculated in this manner is designated q(P,!ML. We define FK(E,)as the integral ofy(E,), where K is the iteration number:
The condensation approximation is taken as the starting point, with The iterative improvement is accomplished by using the following algorithm:
The iterations stop when the relative error function, 6, reaches an acceptably low threshold:
The energy distribution function is calculated on each iteration by fitting the integral distribution function, F(E), to the Akimas8 cubic spline. The spline function is then differentiated to obtain (E,). ( 2 ) Determination of the Range of Energy. The range, il,of adsorption energies that should be considered may be determined by the method originally suggested by Zeldowitchs9 and used by others.52*m A particular energy may be calculated by E, = -RT In (P,/K) (15)
Y+'
Jaroniec3* has suggested to take for the constant K the value given by K = p0eWRT (16) where Po is the vapor pressure of the probe compound and E, its heat of vaporization a t the temperature at which the adsorption isotherm has been determined. With that definition, the minimum energy used in the calculation (Le., E,) is that energy that corresponds to Po. The maximum energy (EoM)corresponds to the minimum solute partial pressure measured. ( 3 ) Subalgorithm 2 The Distribution Function Substitution (DFS) Method. In this method, the problem is better posed by replacing the energy distribution by an analytical function. In particular, a linear combination of unnormalized probability (58) Akima, H.JACM 1970, 17, 589. (59) Zeldowitch, J. J . Acrcr Physichim. 1935, 1, 961. (60) Berenyi, L. 2.Phys. Chem. 1920, 94,628.
The Journal of Physical Chemistry, Vol. 95, NO. 10, 1991 4105
Surface Energy Distribution
11 8
ii Adsorption Energy (KCuI/Molo)
Figure 8. Difference between simulated and optimized two-peak energy distribution functions. The simulated distribution (solid line) is similar to that shown in Figure 7, but the large Gaussian peak (Figure 7) is now replaced by a y function having the same monolayer capacity and average energy but a different variance and skew. The optimized distribution peaks (dashed line) are both Gaussian.
The relevant statistical moments are calculated for each peak, following the optimization. The equations used for each of the three distribution functions and a description of their parameters are given in Table 11. As an example, the function, 6, which is minimized in the determination of a twepeak-maxima energy distribution, using Gaussian peaks is given here: I
9
11
E
Figure 7. Difference between simulated (solid line) and optimized (dashed line) two-peak energy distribution functions, with Gaussian peaks. For the low-energy, qm = 1.0 X mol/g and E." = 10.0 kcal/mol. For the high-energy peak, qm= 1 .O X lod mol/g and E," = 12.0 kcal/mol. Under the distribution function, a plot of the difference between the simulated and the optimized distribution is given in a
magnified scale. distribution functions, one function per peak maxima, is used. Many such functions are known?' and if one is chosen wisely, an accurate and robust method can be developed. The probability distribution functions that have been used so far in our calculations are the Gaussian function, the exponentially modified Gaussian (EMG), and the y function. The parameters of the distribution functions are optimized by a downhill simplex routine until the best fit is obtained between the experimental and calculated global isotherms. Following is a stepwise description of the algorithm. Each peak of the AL solution is replaced by a distribution function that is chosen depending on the magnitude of the skew of the individual AL peak. At present we have performed optimization calculations only for energy distributions having one or two peak maxima. For AL peaks that are symmetrical, the Gaussian function is used, for slightly skewed peaks, the EMG, and for highly skewed peaks, the y function. For clarity, the lower order statistical moments that parametrize a given peak on an energy distribution function are given in Table 1. In the case when the optimization is camed out with a Gaussian function, these moments are used as the optimizable parameters in the simplex program. For the EMG and y function optimization, some of the optimizable parameters, though related to the statistical moments, are simply empirical "curve fitting" constants. (61) Abramowitz, M.;Steiger, 1. Handbook of Murhemuricul Functions und Formula; National Bureau of Standards: Washington, DC, 1964.
(17) where M is the number of isotherm data points. It should be noted that the actual isotherm data points, rather than the best multi-Langmuir equation, are used in this optimization.
Experimental Section Apparatus. Chromatographic data were obtained on a Perkin-Elmer 8500 (Norwalk, CT) gas chromatograph. The analog output from the flame ionization detector was digitalized and recorded on an IBM PC microcomputer (Bow Raton, FL). The computer was interfaced to the chromatograph by a 1/0 board (Data Translation, Marlborough, MA) that used 12-bit A/D conversion and was controlled by in-house-written software. Reagents. HPLC grade diethyl ether was obtained from Aldrich (Milwaukee, WI). It is preserved with 0.1 PPM BHT and was used as received. Dimethyl sulfoxide was HPLC grade from Aldrich. Stationury Phase. Alumina, dry bore milled to average particle diameter of 0.5-0.8 r M , was obtained from Malakoff Ind. (Malakoff, TX). The surface area measured by the BET method was 8-10 m2/G. It was dried overnight in a vacuum oven at 150 OC.
Column Preparation. The porous layer open tubular (PLOT) column was fabricated by using a dynamic method similar to the method described by Halasz and Horvath62 for making SCOT columns. The dried alumina was suspended (1% mass/volume) in DMSO. The silica tubing (length, 15 m, 530-pM, i.d., Alltech Europe, Nazareth, &Igium) was filled with the suspension. Then, one end was plugged and the other was drawn through a metal (62)
Halasz, 1.; Horvath, C. Nurure 1963. 197, 72.
Roles and Guiochon
4106 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991
LEGEND ----" SMULATED 0
0
d
B
I\
11
n
I
9
11
13
Adsorption hqgy ( ~ c ~ l / M o k )
Figure 9. Robustness of the calculation of an optimized two-peak energy distribution function. The simulated distribution is the same as that shown in Fire 7. The 'measured" isotherm is derived by a perturbation of the simulated isotherm achieved by damasing the a2parameter of the bi-Langmuir isotherm (correspondingto the high-energy peak) by 5%. tube into an oven where it is coiled on a rotating spool. The velocity of the tube is set so that the DMSO vaporizes and its meniscus stabilizes somewhere in the thermal gradient along the metal tube. The oven is placed under a hood so the vapors of DMSO are vented outside the laboratory. The thickness of the porous layer depends on the density of the suspension and is a few particles thick, on the average. Isotherm Determination. The chromatograph was operated isothermally, at an oven temperature of 60 O C . The pressure drop was 5.0 psi. The split/splitless injector and the detector were maintained at 150 O C . The tail portion of each chromatographic peaks was smoothed (fivepoint After ensuring that the shape of the chromatographic peak was due to the adsorption isotherm,64we determined the isotherm by the ECP method.6s
Results rad Discussion The nature and quality of the thermodynamic data obtained from inverse gas chromatography depend on several factors, both experimental and theoreticai. One may easily measure the retention times of probe solutes using conventional practices of analytical chromatography. These retention times may serve as an indication of the magnitude of the interaction with the surface. A logarithmic plot of these retention times versus the reverse of the absolute temperature supplies the adsorption energy at zero surface coverage? Alternately, the entire band profiles of highconcentration samples may be recorded. All the stages of the process, the experiment and the management and manipulation of the data, may be made highly sophisticated. Then information regarding the distribution of adsorption energy on the surface may be accessible. The degree of required sophistication is determined by the nature of the problem under investigation. Rapidly obtained retention time data may be highly valuable, especially if speed is important or if there is a paucity of other information available regarding the characteristics of a new surface. On the other hand, the development of a highly sophisticated methodology is necessary to extract all the data available in the concentration profiles regarding the properties of the material contained in the column. (63) Savitzky, A.; Golay, M. J. E. AM/. Chcm. 1964, 36. 1627. (64) Hukr,J. F. K.; Keulemans. A. I. M. In Gar Chromatography;Van Swaay, M., Ed.; Butterworths: London, 1962; p 26. (65) Cremer, E.; Hukr,J. F. K. Angew. Chcm. 1961, 73. 461. (66) Guiwhon, G.; Golshan-Shirazi, S.; Jaulmes, A. Anal. Chcm. 1988, 60, 1856.
,
I
9
11
13
E
Flgnre 10. Robustness of the calculation of an optimized two-peak energy distribution function. The simulated distribution is the same as that shown in Rgure 7. The 'measured" isotherm is derived by a perturbation of the simulated isotherm achieved by decreasing the b2 parameters of the bi-Langmuir isotherm (corresponding to the high-energy peak) by 5%. Under the distribution function, a plot of the difference between the simulated and the optimized distribution is given in a magnified scale.
The recent increase in the availability of high-speed computers makes a numerical approach to the solution of eq 1 an important but accessible goal. We present here experimental results regarding the adsorption of diethyl ether on a calcined alumina powder used to prepare alumina-based ceramics and a detailed investigation of the accuracy, reproducibility, and robustness of the method we have developed for the determination of the distribution of adsorption energy on a surface. As the validation of the numerical method used involves numerous calculations performed with the experimental data, we present them first, for the sake of clarity. (I) Experimental Study of the Adsorption of Diethyl Ether on Alumina. In Figure 1 we show the calibrated profile of the experimental chromatographic peak (solid line) corresponding to an injection of 0.46 pg of diethyl ether on an alumina wall-coated open tubular column at 60 O C . The response factor of the flame ionization detector was used to transform the recorded signal (detector voltage versus time) into a time profile of the diethyl ether partial pressure in the carrier gas at column outlet. Figure 2 shows the equilibrium isotherm (solid line) derived from the chromatographic elution profile in Figure 1 by using the ECP method.65 Finally, in Figure 3 we show the energy distribution of diethyl ether adsorbed on the alumina sample at 60 OC. This distribution function is made of two very narrow Gaussian peaks. The relevant optimized parameters are given in Table 111.
The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 4107
Surface Energy Distribution
0
0.6
1.2
1.8
i4 e
1
0
I
0.6
I
I
1.2
1.8
Retention Time (Min.)
I
2.4!
Figure 11. Effect of chromatogram noise on the determination of an energy distribution function. I: experimental elution profile (solid line) and chromatogram obtained with 5% superimposed noise (+). For the sake of clarity, only a small proportion of the data points are shown. 11: plot of the noise profile added to the chromatogram, in a magnified scale. Note that the experimental chromatogram corresponds to the experimental data given earlier (Figures 1-3).
It is not surprising to find a two-peak energy distribution for the adsorption energy of diethyl ether on calcined alumina. It is known that the alumina surface is covered with one type of proton-donating site, the surface hydroxyls, and three types of Lewis acid sites?' corresponding to A13+ and two different electron-deficient oxygen species. Diethyl ether can interact with the surface hydroxyls by hydrogen bonding. It can also interact with the Lewis acidic sites. Diethyl ether was chosen as the probe solute to be used in this study for several reasons. First, the goal of this paper is the presentation of a new procedure for the evaluation of the degree of homogeneity of the surface energy of an adsorbent. The use of data pertaining to a single probe is needed for this purpose and for the validation of the method. Second, diethyl ether has been shown to be an effective probe for the study of surface heterogeneity.' 1 The use of simpler, smaller, or less reactive molecules would have required the use of a sophisticated gas sample injection system and of subambient temperature gas chromatography. These experimental complications are not justified by the possibility to extract information on the degree of surface energy heterogeneity a t a smaller scale. For the investigation of the surface properties of ceramic powders, we need probes comparable in molecular size and properties to the solvents used to prepare the slurries from which the green bodies are molded. Finally, preliminary experiments have shown that only compounds in a (67) h w i n g , J.;
Monks, G. T.; Youll, B. J.
Caralysis 1976, 44, 226.
5I 11
12 E
Figure 12. Effect of chromatogram noise on the determination of an energy distribution function. Comparison between the energy distribution calculated from the elution profile without superimposed noise (solid line, note, same as in Figure 3) and from the elution profile with 5% superimposed noise. Under the distribution function, a plot of the difference between the two energy distribution functions is given in a magnified scale.
narrow range of chemical reactivity can be useful. Compounds with functional groups that are more reactive with respect to alumina (e.g., amines) elute from the column only at high temperatures and their interaction with the column participates of chemisorption rather than physical adsorption. On the other hand, less reactive compounds (e.g., thiophene) elute too close to the void time at room temperature. Results obtained with other probes will be reported and discussed shortly?* (IZ) Testing of the Numerical Method. We have performed two different series of tests of the method described above that permits the calculation of numerical solutions of eq 1. The first series estimates the reproducibility of the results obtained, the second studies its robustness. ( 1 ) Reproducibility of the Calculation Procedure. The aim of this study is to verify that the method is self-consistent. If an energy distribution is assumed, the global, macroscopic isotherm can be derived by the direct integration of eq 1, where both 0 ( E , P ) andf(E) are known. This global isotherm is in turn used in our calculation procedure to determine an estimate of the energy distribution. The initial and final functions are compared. Ideally, they should be identical. We simulated several energy distributions using three different distribution functions: Gaussian function, EMG, and y function. Figures 4-6 show the results obtained with six different mono(68) Roles, J.; Guiochon, G., manuscript in preparation.
4108 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991
20
1.0
0.0
P (Ah.)
*
Roles and Guiochon
3
I
Figure 14. Effect of temperature and flow rate fluctuation on the de-
termination of an energy distribution function. Energy distribution calculated from the specific corrected retention volume, VN, without superimposed noise (solid line, note, same as in Figure 7) and from VN with 3% superimped noise. Under the distribution function, a plot of the difference between the two energy distribution functions is given in a magnified scale.
I
I
.O
I
20
1.0
I
P (Atm.) Figwe 13. Effect of temperature and flow rate fluctuation on the determination of an energy distribution function. Experimental specific corrected retention volume, VN (solid line) with 3% superimposed noise (+). Under this plot, a plot of the noise added (difference between VN with and without noise) is given in a magnified scale. Note that the experimental VN corresponds to the experimental data given earlier (Figures 1-3).
modal distributions of increasing skew. Figure 7 shows the result obtained with one bimodal energy distribution having two Gaussian peaks. In all cases, there is so little difference between the initial energy distribution and that obtained as a result of the calculation that a direct graphical comparison is difficult or impossible. We show in the Figures 4-6, under each energy distribution, a plot of the difference between the initial and the final function, at a greatly magnified scale. This difference is significant only in the cases shown in Figure 6 (very unsymmetrical distributions), and then it does not exceed a few percent. Finally, using the energy distribution function shown in Figure 3, we calculated through eq 1 the optimized isotherm plotted in Figure 2 (dashed line), which is nearly indistinguishable from the original isotherm obtained by ECP. Finally, with knowledge of the sample size, the equilibrium isotherm, and the column efficiency, it is possible to calculate the band profile by using a finite differences simulation program.66 The calculation has been carried out assuming a column eficiency of 8000 theoretical plates, which agrees with the data obtained with the unretained methane peak. The agreement between the two band profiles is excellent, as shown in Figure 1. We conclude that practically all the information regarding the distribution of adsorption energy of diethyl ether on the alumina surface is contained in Figure 3 and that none has been spuriously introduced.
It is important to note that the number of terms used in the multi-Langmuir isotherm equation (eq 9) must be equal to or greater than the number of peak maxima in the energy distribution. All of the energy distribution data in this study has either one or two peak maxima and was fit to a bi-Langmuir equation. In the calculations we have performed so far with monomodal and bimodal simulated data, the quasi-Adamson energy distribution has always had the correct number of peaks. We can imagine, however, situations that might cause problems. For example, the resolution of the method depends much on the accuracy of the isotherm data. It is possible for two peaks to be so close together that they are mistaken for a single mode. These types of questions are related to the quality of the experimental data and especially to the reproducibility of the shape of the chromatographic elution profile. The need for a more rigorous deconvolution algorithm ensuring that a particular mode is not the superimposition of two modes has not yet been demonstrated. Before we proceed with a more rigorous testing of our algorithm, we must first perform an analysis of the reproducibility of experimental results and study the resolution between two energy distribution peaks afforded by the present method. These questions will be answered later.68 ( 2 ) Study of the Robustness of the Calculation Procedure. A two-peak energy distribution having a slightly skewed y function peak (Eav= 10.0 kcal/mol) and a Gaussian peak at 12.0 kcal/mol was simulated. The corresponding isotherm was calculated. Using this isotherm, the program was forced to optimize this system and calculate the best energy distribution using two Gaussian functions. The results are shown in Figure 8. There is an excellent agreement between the two distributions, insofar as a Gaussian peak may account for a slightly skewed function. The Gaussian peak has exactly the same position as the y function average and the same width. The importance of this optimization is 2-fold. First, it demonstrates that a good, physically meaningful solution is possible even if an exact solution is impossible. The number of peaks in the optimized solution, their monolayer capacity, and their average energy are much more important to the optimivltion than the exact shape of an individual peak. This demonstrates the validity (at least from the standpoint of curve fitting) of replacing the actual energy distribution by a combination of distribution functions. Second, the stability of the method with respect to a small perturbation in the global isotherm is demonstrated. Next, the stability of the solution was tested as follows: A two-peak distribution function was assumed, and the global iso-
J . Phys. Chem. 1991, 95, 4109-4113 therm was calculated. The global isotherm obtained was then fit to the bi-Langmuir equation (eq 9 with two terms, i = 1 and 2). A perturbed isotherm was constructed by calculating the residual corresponding to the bi-Langmuir fit, changing one of the four bi-Langmuir parameters slightly (i.e., reducing at or b2 by 5%), calculating the slightly changed bi-Langmuir isotherm, and adding the residual back to the slightly changed bi-Langmuir isotherm. Then, the energy distribution function corresponding to the new, perturbed isotherm is calculated. Figures 9 and 10 show the simulated and optimized distributions obtained for 5% perturbations of the a and 6 (respectively) bi-Langmuir parameters corresponding to the smaller peak. This small energy peak is slightly shifted in the former case, hardly changed at all in the latter. The determination of the adsorption isotherm from the experimental chromatogram of a large size sample, using the ECP method, may introduce experimental errors. A particular chromatographic characteristic point may be subject to two types of random errors: those that affect the detector response (flame current noise) and those that affect the retention time (carrier gas flow rate and/or column temperature fluctuations). To test the robustness of the method, we modified the experimental data discussed above (Figures 1 and 2), introducing higher and higher fluctuations, until a change was seen in the calculated energy distribution. First, the detector response was perturbed, holding the retention times constant. A random number generator with a rectangular distribution probability was used to superimpose a 5% random noise on each point (Figure 1111). The resulting chromatogram (symbols) is obtained by adding this noise to the data points (solid line). The noisy chromatogram is compared with the actual experimental chromatogram in Figure 111. The energy distribution calculated with the noisy chromatogram is compared to the energy distribution corresponding to the experimental data in Figure 12. The residual is extremely small, because of the smoothing effect of the integration calculation performed. In practice, the detector noise will have no effect on the determination of the adsorption energy distribution, as the signal noise for high concentration band profiles is much below 5%. Finally, the retention time of each characteristic p i n t was perturbed. Figure 131 shows the specific, corrected retention
4109
volume of diethyl ether plotted as a function of its partial pressure (solid line). A 3% random error on this volume was superimposed on each point (symbols), simulating the effect of a fluctuation of the carrier gas flow rate or of the column temperature (Figure 1311). The calculated energy distribution is compared to the energy distribution corresponding to the experimental data in Figure 14. The only effect is a very small shift, of the order of 1 X 1V2kcal/mol, in the average energy of the two peaks of the energy distribution function.
Conclusion The numerical method we have developed for solving the integral equation of the problem (eq 1) offers several potential advantages over existing methods when applied to gassolid isotherm data obtained by the ECP method. These potential advantages include an improved accuracy (compared to the condensation approximation and the existing numerical methods) and a higher degree of robustness (compared to integral transforms). We have also shown that the characteristics of the energy distribution function derived from experimental results are not affected significantly by fluctuationsof the experimental conditions within a range that exceeds the normal range of noise and drift expected from modern instruments. The major objective of this paper, the introduction of a new powerful and reliable numerical method, has been achieved. Further studies, involving the systematic application of the method to characterize the surface of a number of different samples of similar materials using several probe solutes, are required before definite conclusions can be drawn regarding the practical usefulness of our approach!*
Acknowledgmenr. We acknowledge fruitful discussions with W. H. Griest (ORNL, Division of Analytical Chemistry) and M. A. Janney (ORNL, HTML). This work was supported in part by Grant DE-FG05-86ER13487 from the U.S. Department of Energy, Office of Energy Research, and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. We acknowledge continuous support of our computational effort by the University of Tennessee Computing Center.
Entropies of Solvation of Solvated Electrons Sidney Goldent and Thomas R. Tuttle, Jr.* Department of Chemistry, Brandeis University, Waltham, Massachusetts 02254-91 10 (Received: September 13. 1990)
An expression is derived for the entropy of solvation of a solvated electron that can be evaluated directly from (1) an established correlation between the spectral energy and solvent energy of the solvated-electron system and (2) the entropy of the pure solvent. The theory predicts that entropies of solvation of solvated electrons should be positive in all pure solvents, under the usual standard conditions, in marked contrast with the negative values obtained for the hydration of conventional ions. This is supported by current experimental results: for NH', the theory yields a value of +I20 J/(mol-K), while the most the theory yields a value of +118 J/(mol.K), while recently reported experimental value is +I54 f 20 J/(mol.K); for HzO, the most recently reported experimental value is +118 f 20 J/(mol.K).
1. Introduction For Some time,' solvated electrons in liquid ammonia were thought to have a standard entropy of ammoniation that was positive while solvated electrons in liquid water were thought to have a standard entropy of hydration that was negative, the latter sign appearing to be characteristic for the standard entropies of 'Emeritus Professor of Chemistry.
0022-3654/91/2095-4109$02.50/0
hydration of conventional ions.z This situation persisted' until quite recently: when a new determination was found to yield a positive standard entropy of hydration for solvated electrons ( I ) Lepoutre, G.; Jortner, J. J . Phys. Chrm. 1972, 76, 683. Scientific: (2) Conway, New9. York. E. Ionic 1981. Hydration in Chcmisrry and Biophysics; Elsevier (3) Schindewolf, U: Ber. Bunscn-Ges. Phys. Chrm. 1982, 86, 887. (4) Han, P.; Bartels, D. M. J . Phys. Chem. 1990, 91, 7294.
0 1991 American Chemical Society
