HENDRICK C. VAN NESS
APPLIED THERMOD YNAMICS SYMPOSIUM
Therm ody n am ic Excess Propenties of Binary l i q u i d Mixiures The Role of Empiricism At present no generally satisfactory
ne might take the term empiricism to refer to all 0 knowledge derived from experience, however remotely or indirectly related to the topic of interest. It is
theory exists that provides a sound b a s i s
my purpose here to discuss the immediate empiricism that provides most directly the characteristic thermodynamic excess functions for binary liquid mixtures from the experimental measurement of macroscopic properties. I will also discuss the mathematical representation of these excess properties by empirical functions which have no known relation to molecular physics or to statistical mechanics. I t is not my purpose to derogate theoretical efforts. I merely accept the fact that at present no generally satisfactory theory exists that provides a sound basis for either the prediction or the correlation of thermodynamic data for binary liquid mixtures. Until such a theory appears, the incentive remains to develop empirical methods to their ultimate in refinement. There are at least three uses for experimental data on the thermodynamic excess properties of binary systems. The possibility of their direct application in chemical plant design and operation is obvious. Almost equally obvious is their use in testing theories or in guiding the formulation of theories which attempt to predict binary mixture properties from pure-component data. Because this objective may well be unrealistic, a use of perhaps far greater long-term value is to provide data for the evaluation of parameters characterizing interactions between unlike species, useful not only in the correlation of data for binary mixtures but also in the prediction of properties for multicomponent systems.
for either the prediction or the correlation of thermodynamic data for binary liquid mixtures, thus the incentive to develop empirical methods to their ultimate refinement.
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to provide the basic principles which explain in a coherent way the considerable body of empirical information now available. Experimental techniques have improved over the years, but in addition there has been a considerable advance in the processing of data, with far greater reliance on exact thermodynamic rela tion sh ips.
At a Discussion of the Faraday Society in 1953 on “The Equilibrium Properties of Solutions ofNonelectrolytes,” D. H.Everett gave an introductory paper on experimental work (2); reviewing especially the develop ments of the 17 years since the pr&ious discussion on a similar topic. Some 14 morcyears have now passed, and I should like to contrast the developments of the past 14 years with those reported by Everett. . . I n 1953 it was reported that %iostaking developments on the experimentalside had taken piace to compare with those that had revolutionize4 the~the&y of nonele%tw lyte solutions. Everett *ported that h y i 9 5 3 newly developed theories presented a chillenge to tbe, expcrimentalist to provide binary solution data “of such detail as to give a complete picture of the thermodynamic.quantities *kited With niixing.” He noted that in the 1936 Dkussion not a single graph of an e x e s property had been presented in spite of the earlier work of Scatchard (22). During the general discussion in 1953, A. E. Korvaee (73)suggqted that graphs presentlng the excess piopedea divided .by the,product of the mole fractions might be prefkable to the then function plots: However, no attention was given to the mathematical representation of data, and no mention whatever was made of tlie pcmible uses of automatic computers. My evaluation of the progress made since 1953,.is,in sharp contrast to Everett’s w s m e n t of the .developments between 1936 and 1953. The challenge to experimentalists was evidently accebed by a few.indirildualr, and now the shoe is on the other foot. No longer is it the experimentalist who is challenged to provide the data needed to test theixy; but rather it is the t h e i s t who is challenged to provide the hasic principles which explain in a coherent way the considdable bcdy of.empuica1 information now available. I doubt thzt anyone wouid characterize the changes on the experimental side as revolutionary. They reflect steady evolutionary prcgres, which the revolution & theory referred to ‘by Everett in 1953 has not yet matched. Everett reviewed b r i d y the developments in experimental techniques for obtaining several .types of data. With respect to vapor-liquid equilibria, he commented that both static and dynamic methods had improved over the years. I can report the same sort of progress, hut in addition there has been a considerable advance in the procesing.o€ data, with far greater reliance on exact thermodynainicrelationships. I n considering heatsf-mixing measurements, Everett pointed out that recent emphasis had been on the design ’
34
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
of calorimeters with little or no vapor space and that further development was needed. Much further development has indeed occurred and no one should now seriously consider reporting data taken with a calorimeter in which vaporization effects influence the results. Everett also stated that the importance of direct measurement of heats of mixing cannot be overstressed; such data are needed in a searching test of theory. This is at least as true today as it was in 1953, and has perhaps now found some acceptance. I t was also pointed out that heat capacity measurements are useful for an independent check on heat-of-mixing data. Moreover, Everett said that if the heat of mixing “is suspected of being temperaturedependent, then in principle one should measure heat capacities.” We now know that all heats of mixing should be suspected of being temperature-dependent, but it is by no means clear that this calls for the measurement of heat capacities. Everett clearly stated the practical problem existing in 1953 when he remarked that “measurements of the heat capacity of volatile solutions are not easily made, neither are reliable heat-of-mixing measurements at several temperatures.” Today, if it is the heat of mixing that one wants as a function of temperature, then the best procedure is to make direct measurements of this quantity at several temperatures. Heat-of-mixing calorimetry has developed during the past 14 years to the point where accurate data may be accumulated rapidly and routinely (77, 21, 26). Heat capacity measurements are still made by adiabatic calorimetry, and high accuracy requires exacting work. Moreover, a separate experiment is necessary for every C, determination. Thus, data of this kind are not collected rapidly. If one is interested in the accuracy of the excess heat capacity itself, then differentiation of high quality HE data with respect to temperature introduces no greater uncertainty than the differencing necessary to the calculation of CpEfrom measured C, values. The volume change of mixing was reported by Everett to require painstaking work by classical techniques. More recent work has led to the development of dilatometers for the direct measurement of AV, and accurate values for this quantity can now be readily determined ( 7 8 ) . They are not, however, of great practical interest, and their role in theory seems less clear than was once supposed. Gibbs Free Energy and Enthalpy
The two excess properties of greatest interest from both the practical and theoretical points of view are the excess
Gibbs free energy and the excess enthalpy. The excess entropy may of course be calculated from these, and it is to these that I shall devote the remainder of this paper. The single aspect of solution thermodynamics which has received the greatest attention is vapor-liquid equilibrium. The literature abounds with data for binary systems. A critical assessment of these data would be impossible, although it is clear that many sets are too imprecise for careful thermodynamic analysis. Many data sets have been taken at constant pressure, often at atmospheric pressure. Although data taken at constant temperature are preferred from the thermodynamic point of view, constant-pressure data, if accurate, still provide a valuable entry into the thermodynamic network whenever heat-of-mixing data are also available. For data at low pressures the equilibrium still remains the most popular experimental device. There are many designs from which to choose, and the best ones are capable of producing quality data both rapidly and routinely. The still can be operated at constant pressure or at constant temperature and over a considerable range of conditions. One normally measures temperature and pressure, and in addition analyzes the vapor and liquid phases to determine composition. These analyses are critical to the production of reliable data, and make difficult the determination of accurate results for the composition regions outside the range from 10 to 90 mole yo. The measurement of both vapor and liquid compositions provides more data than are actually needed for the specification of the system. This is to say that thermodynamics provides a relationship, the Gibbs-Duhem equation, which could be used for the calculation of one of the compositions. If this equation is not used, then it provides a test of the thermodynamic consistency of the data, and data taken with stills are readily tested in this way. This overdetermination of the system variables is often cited as being advantageous. As a general rule this is, in my view, open to serious question. I n the first place it serves to require unnecessary work, and I oppose this on principle. Moreover, it introduces an unnecessary source of experimental error, and perhaps worst of all it may condition the design of apparatus in a disadvantageous way. As an example, I cite the fact that circulating equilibrium stills were developed to provide, for purposes of analysis, a reasonable quantity of liquid having the composition of the equilibrium vapor. This composition measurement is not needed, and insistence on making it complicates the problem of taking accurate phase-equilibrium data. It may merely introduce the VOL. 5 9
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errors that consistency testing is designed to demonstrate. A proper aim of the experimentalist is to minimize errors by measuring the least number of variables in the most direct manner possible. A properly designed apparatus is one which yields experimental data rapidly and routinely and which, once developed and tested, can be counted upon to produce results of such accuracy that consistency tests are meaningless. A static method for the determination of phase equilibrium lends itself well to these aims. Such a method requires no analyses for composition, because the liquid phase can be made up gravimetrically to known compositions, and the vapor-phase composition is calculated by rigorous thermodynamic equations (23). One need merely ensure that the mixture is thoroughly degassed and that the temperature and pressure measurements are accurate. I look for the development during the next few years of standard equipment to provide vapor-liquid equilibria by this method. I have been discussing data taken at relatively low pressure, or in any event, at conditions well removed from the critical. Phase-equilibrium data at higher pressures, indeed in the critical region, are of considerable interest. Such data may be taken by a number of methods, but I would single out the method which employs small-bore tubes as having special merit because it appears to meet my criteria of accuracy and productivity. This method has been developed and exploited for 30 years by W. B. Kay at Ohio State University (5--10,20). I t produces not only P-T-x-y data for the phase boundaries but also the densities of the saturated phases. High pressure data such as these have not usually been subj ected to thermodynamic analysis, primarily because of the difficulty of taking into account vapor-phase nonidealities; however, this is definitely called for in the future. Temperature Dependence
The temperature dependence of the excess properties has long been given too little attention. Once its importance is accepted, one can take advantage of exact thermodynamic relationships to effect a considerable economy in experimental effort. Basic equations are :
-R= [ d ( Hb ET / R )]
CpE
P,*--
-HE
-=.[ RT
b (GE/RT ) dT
]
Pp-
The excess heat capacity for a mixture of given composition invariably shows a simple functional dependence on temperature. T o illustrate the utility of these equations, 36
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
we may assume that CpE is given by a linear function of absolute temperature :
CpE/R = A f B T
(3)
Integration of Equation 1 then gives
HE/RT
=
A
+ ( B / 2 )T + C / T
(4)
where C is a constant of integration. Integration of Equation 2 > ields G E / R T = - A In T - ( B / 2 ) T
+C/T +Z
(5)
where Z is another integration constant. I n principle, equations such as these allow determination of the constants in several ways. One might, for example, rely entirely on vapor-liquid equilibrium data taken at a series of temperatures to provide G E / R T as a function of T from which by curve fitting one could determine all of the constants A, B , C, and I . Once done this would provide equations for all of the excess properties being considered as functions of T . The disadLantage inherent in this procedure is that it amounts to a first and second differentiation of the GE data to produce HC and CPE. How ever this is done, each differentiation results in a loss of accuracy of one order of magnitude. In the present example this is reflected by the fact that four constants must be determined from the GE data. whereas only three are required for H E and two for CpE. In addition, the calculation of GE from phase equilibria requires auxiliary data, often of uncertain accuracy, and what is worse, a sufficiently high standard of accuracy in the basic vapor-liquid equilibrium data is difficult to maintain One might proceed in the opposite direction and measure heat capacities as a function of T . This would provide CPEvalues to fit as a function of T , and would yield in this example values of A and B of an accuracy consistent with the accuracy of the C p Evalues. However, one would then need a set of H E data at a single temperature to allow calculation of values for the integration constant C and in addition a set of GE data to permit calculation of values for the second integration constant I . The inherent disadvantage of this procedure is that it requires taking data of three different types in three different pieces of equipment. I n addition, no method for the direct measurement of CPE has been developed; thus, the values of CPE depend on taking the difference between
C.Van X e s s is Union Carbide Professor in Chemical Engineering at Rensselaer Polytechnic Institute, Troy, N . Y . T h i s paper introduced the fourth state-of-the-art summer symposium.
AUTHOR Hetzdrick
I
'
of miXing as afunrrion of moIt fraction
Figw T. A.
-
1.
J. a
C%kmfom(lFl,Cdimarr(2)at 30O e. (4) M/*m -4909.1 5285.1n 595.hi.9 8505.6n* (@ = 0.19%) l)-ouahydm yTan(2) at 300 e. (4) --El30 9 7&.ln 6310.2n' (c = 0.94%)
-
-
& rml-
&*dnlnnarm.
+ +
spare hrmw h i m n
-
y
+ 7937.5~14
& rxpmmw palrr jrm
two much larger numbers determined by the demanding experimental procedure of adiabatic calorimetry. A compromise is best suited to the capabilities of present-day experimental techniques. The direct determi nation of heats of mixing by isothermal dilution dorim. etry is now a highly developed art which can provide accurate values for H' as a function of T. Such data allow determination of all the constants in Equations 3,4, and 5 except I. All that is required in addition is a set d vapor-liquid equilibrium data at a single conditioneither at constant Tor at constant P-to provide the G" data nccearary to the determination of values for the constant I. The equation obtained for Cp' in this way mula implicitly from dflerentiation of the HEdata; however, all available evidence (24, 25) indicates that the CpJ values so obtained are at least as accurate as CpE values determined from heat capacity measurements. Furthermore, the excess heat capacity has so far been of limited practical or theoretical interest. This procedure places greatan emphasis on heat-ofmixing data and allows use to be made of phasxquilibrium data taken at any single condition, without plac; ing too great a r e l i c e on them. Accurate determination of heats of mixing as a function of temperature is essential, but the practicability of the method has been amply demonstrated (24, 25). If one takes any extra data, such as an additional set of phaseequilibrium data, then the thermodynamic consistency ofthe original data set is tested. With proper wcperimental techniques this should not generally be necessary. The availability of the temperature dependence of the excess functions obviously enhances the practical usefulness of such data. Momover, it places an additional demand on theory, for any model of solution behavior which makes pretense of being realistic should have built into it the means to account for the temperature dependence of the properties it is intended to predict. This may be d i n g a lot of theory, but surely it should be part of the long-range goal. No impediment on the experimental side stands in the way to delay the production of any needed data. The sort of data already available to theorists is illustrated in Figure 1, which shows heats of mixiig for a number of systems. There are two distinctions between the data shown a t the top and those a t the bottom. First, the data at the top are for systems in which hydrogen bonding presumably does not occur, whereas the systems qrrsena at the bottom aFe p-med to exhibit hydrogen bonding between unlike molecules. The sccond dictinction is that the data a t the top were, with the excep tion of the bottom curve (F), not taken by isothermal diluI O L 5 9 NO. 9 S E P T E M B Z E 1 9 6 7
37
As no theoretical model of been devised, we still often must rely upon arbitrarily chosen mathematical representations. :_
tion calorimetry, whereas all data a t the bottom were. Thus, we may observe two kmds of differences with respect to these data4ifferences in shape and location of the curves And differences in the scatter ofthe data points with respect to the lines representing them. There are four new, unpublished sets of data here: tetrahydropyran-carbon tetrachloride, tetrahydropyran-dichloromethane, 1,4-dioxane-chloroform, and tetrahydropyran-chloroform. We have previously published data for 1,4-dioxanedichloromethane (26). These data, shown with one exception at the bottom of Figure 1 are data of the quality I had in mind when I earlier implied that maximum use should be made of heat-of-mixing data. Although temperature dependence is not shown here, it has been illustrated in earlier publications (24-26). Even without consideration of temperature dependence, the data shown in Figure 1 present a challenge to any theorist. The system l+dioxanechloruform has been considered before by McGlashan and Rastogi ( 7 4 , and their six data points taken at 50' C. scatter a bit above the new data at 30' C. shown in the figure. They developed an equation based on a model which attributed the entire heat of mixing to the heats associated with the formation of two complexes in solution: an AB complex and an AB9 complex of dioxane (A) with chloroform (a). By curve fitting they determined values for two heats of reaction which resulted in an excellent correlation of their data. Satisfying as this may appear, it cannot in the light of all the data shown in Figure 1 represmt the whole story. There must, for example, be some kind of molecular interaction between dioxane molecules which accounts for the appreciable endothermic heat of mixing observed for dioxane-cyclohexane, and which must also be operative in the exothermic dmxane-ehloroform system. I could point out other theoretical problems raised by these data that do not appear to be readily resolved, but this example will m e as an adequate illustration. ProsenloHon d Data
Once one has accumulated data such as the new heatof-mixing data presented in Figure 1, the question arises as to how best to present them in a scientific publication. The evident choices are tabular, graphical, and mathematical representation. My own choice is the mathematical, provided that the mathematical expression is reasonably low ordered, correlates the data to withim ex38
INDUSTRIAL AND ENGINEERING CHEMISTRY
r, and provides realistic first derivatives upon differentiation. If these criteria are satisfied, then one has at once as complete a representation and as concise a presentation of the data as are possible. I t is of course essential that some statistical measure of goodneas of fit accompany any mathematical correlation of data. The prejudice of some editors and of many users ofdata for the experimental points themselves is an unfortunate holdover from the days when halfa dozen badly scattered data points for an excess function were relied upon to establish the composition dependence of the function. Examples appear in Figure 1. Be that as it may, it is absurd to maintain that smooth sets of data points taken, for example, by isothermal dilution calorimetry as shown in Figure 1 provide any information not equally well given by the mathematical expressions for the c w e a shown passing through the points. Today, if an experimentalist dealing with the standard properties at ordinary conditions cannot make this claim, he ought to redesign his equipment and withhold publication until he can. In the new era of experimentation where highly accurate data are produced rapidly and routinely, it is nonsense to fill journal pages with tables of raw experimental data when a fewshort mathematical expressions serve the purpose better. For purposes of presenting data it does not matter whether such equations are based on some theoretical model or not, provided they meet the criteria given. Because no theoretical model of any general validity has yet been devised, we still often must rely upon arbitrarily chosen mathematical representations; two formulations have recently been proposed by Klaus and Van Ness (77, 72). The first is a representation by orthogonal polynomials, subject to restraints imposed by thermodynamics and by common sense. The excess properties for many systems, including those shown in Figure 1, are well repnsented by orthogonal polynomials. An advantage of the particular orthogonal polynomials described by Klaus and Van Ness is that they permit immediate resolution of total exceas properties into partial properties. The propex order of a polynomial fit is determined by statistical tests, by an error analysis and by an inflection point analysis. Data for a few exceedingly nonideal systems are not well represented by low ordered polynomials. In this event, a spline fit of the data is advantageous. The basic spline-fit procedure assumes a separate cubic equation to connect each adjacent pair of data points. This
set of equations is subject to the following constraints: first, that the analytical expression represented by the collection of cubics pass through every data point and, second, that the first and second derivatives of adjacent cubics match at the data points where they join. The basic spline fit has the inherent disadvantage that it faithfully represents scatter in the data, sometimes exaggerating it, and thus does not provide for any smoothing of the data. A new version of the spline fit has, therefore, been proposed by Klaus and Van Ness (77). This procedure fits separate cubics not between each pair of data points but within arbitrarily selected intervals. Because each interval may contain a number of data points, and because the analytical expression need not pass exactly through the data points, smoothing of the data becomes possible. The constraints placed on the cubics are: that the’adjacent cubics give the same value for the dependent variable and for its first two derivatives at each interval boundary, and that the sum of squares of the deviations between experimental points and the analytical representation be minimized. The analytical representation is then expressed by a table of constants consisting of 2(n 1) entries, where n is the number of intervals. From these one can calculate the value of the function and its first derivative at any composition. Both of these fitting procedures require the use of a digital computer, but once programmed, the computer does all the work, and with good data and reasonable judgment on the part of the individual processing the data, both methods produce analytical representations of the data which are concise, continuous, and differentiable. I t would be unreasonable to ask for more. The surprising thing is that many still ask for less-namely, just the raw data.
+
The Status of Empiricism
What, then, is the status of empiricism in solution thermodynamics? With respect to the usual excess properties of binary systems at normal conditions, methods have been fully developed to take accurate data over the entire composition range and over a considerable range of temperature. I t has become just a matter of routine labor. The development of these methods has taken place to a large extent in universities, and it is one of their legitimate research functions to devise and to demonstrate the means for taking data. The routine production of data, however, is another matter, and I would think that this should not normally be considered one of their proper pursuits. This work then devolves upon governmental research laboratories and industrial laboratories. One can only hope that the latter will encourage publication. Phase-equilibrium data at higher pressures are also readily obtained. If heat-of-mixing data at the same conditions are also wanted, and they should prove valuable, there is no reason why the method of isothermal dilution calorimetry cannot be adapted to work well at these conditions There is certainly no limit to the amount of information that may ultimately be required for practical pur-
poses. This is the reason that the development of theory is so important. A suitable theory would provide an unlimited amount of information, something that no reasonable experimental effort can do. However, it is clearly feasible to measure data for binary systems as the need for such data arises. This is not the case for multicomponent systems, because the amount of data required would be enormously expanded. Thus, the problem in greatest need of theoretical solution is that of predicting the properties of multicomponent systems, not from the properties of the pure components, but from the properties of the constituent binaries. This problem is difficult enough, but it does have some prospect of solution. Its adequacy rests on the now well developed empirical methods for binary systems. Some data for ternary and higher ordered systems must be available for testing the results of theoretical predictions of the properties of multicomponent systems. Data do exist for a few ternary systems, but beyond that little is known. The experimental methods successful for binary systems can also be applied to multicomponent systems for production of the limited amount of data needed for theory testing. I would urge in this regard that full attention be paid to heat-of-mixing data by both experimentalist and theorist. For the experimentalist, the proved possibility of direct measurement of the desired quantity provides the opportunity for both accuracy and speed. For the theorist, such data present an honest challenge, because they yield much less readily to supersophisticated curve-fitting techniques purporting to represent theory than do data for the excess Gibbs function as derived from more commonly available phase-equilibrium measurements. Having been challenged to produce high quality data for the purpose of testing theory, and having done so, experimentalists now look to the theorists for predictions, not in qualitative, but in quantitative agreement with the data. Perhaps that asks too much; I hope not. LITERATURE C I T E D (1) Adcock, D. S., McGlashan, M. L., Proc. Roy. Sac. (London) A226, 266 (1954). (2) Everett, D. H., Discussions Faraday Sac. 15, 126 (1953). (3) Findla T. J Keniry, J. S . , Kidman, A. D., Pickles, V. A., Trans. Faraday Soc. 63 846 8967):’ (4) kughes, G. N., B.S. thesis, Rensselaer Polytechnic Institute, Troy, N.Y., 1967. (5) Jordan, L. W., Jr., Kay, W. B., Chem. Ens. Progr. Symp. Ser. 5 9 (44), 46 (1963). 30, 459 (1938). (6) Kay, W. B., IND.ENa. CIIEM. (7) Kay, W. B., Donham, W. E., Chem. Eng. Sci. 4, 1 (1955). (8) Kay, W. B., Fisch, H.A., Warzel, F. M., A.I.Ch.E. J.4,293 (1958). (9) Kay, W. B., Nevens, T. D., Chem. Ens. Progr. Symp. Ser. 48 (3), 108 (1952). (10) Kay, W. B., Rambosek, G. M., IND.ENa. CHEM.45,221 (1953). (11) Klaus, R . L., VanNess, H. C., A.I.Ch.E. J., in press. (12) Klaus, R. L., Van Ness, H. C., Chem. Eng.Progr. Symp. Ser., in press. (13) Korvezee, A. E., DiscussionsFaraday Soc. 15, 255 (1953). (14) McGlashan, M. L., Rastogi, R. P., Trans. Faraday Sac. 54, 496 (1958). (15) McKinnon, I. R., Williamson, A . G., Ausl. J . Chem. 17,1374 (1964). (16) Moelwyn-Hughes, E. A,, Thorpe, P. L., Proc. Roy. Soc. (London) A277, 423 (1964). (17) Mrazek, R. V., VanNess, H. C., A.I.Ch.E. J. 7, 190 (1961). (18) Pardo, F., VanNess, H. C., J.Chem. Ens. Dnla 10, 163 (1965). (19) Prengle, H . W., Jr., Worley, F. L., Jr., Mauk, C. E., Ibid., 6, 395 (1961). (20) Rebert, C. J., Kay, W.B., A.I.Ch.E. J.5,285 (1959). (21) Savini, C. G., Winterhalter, D. R., Kovach, L. H., Van Ness, H. C., J. Chem. Ens. Data 11, 40 (1966). (22) Scatchard, G., Chem. Reo. 8,321 (1931). (23) Van Ness, H . C., “Classical Thermodynamics of Nonelectrolyte Solutions,” pp. 136-47, Pergamon Press, London, 1964. (24) Van Ness, H. C . , Soczek, C. A., Kochar, N. K., J. Chem. Eng. Data 12, 346 (1967). (25) VanNess, H. C., Soczek, C. A , , Peloquin, G. L., Machado, R. L., Ibid., p. 217. (26) Winterhalter, D. R., Van Ness, H. C., Ibid., 11, 189 (1966).
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