摘要:

336 AUTHOR INDEX * Angell, F. G., 47. Barby, D., 255. Barker, K. H., 199. Barrer, R. M., 135, 162. Beaven, G. H., 325. Boldingh, J., 162. Brockmann, H., 58. Burstall, F. H., 179. Butler, J. A. V., 330. Cassidy, Harold G., 259. Claesson, Stig, 11, 34, 321. Consden, R., 329. Cook, C. D., 153, 240. Crisp, 1). J., 161. navies, C. W., 158, 241. Davies, G. R., 179. Davison, B., 45, 49. Duncan, J. F., 52, 104, 154, 1563, 158. Elbeih, I. I. M., 183. Flood, H., 190. Glueckauf, E., 12, 47, 50, 53, 154, 199, 238, 239, 240. Gordon, A. H., 128. Hale, D. K., 79, 160. Heymann, E., 161. Hirst, E. L., 268. Jones, J. K. N., 268. Jones, Tudor S. G., 285. Kitchener, J. A., 90, 155, 159. Kitt, G. P., 199. Klinkenberg, A., 151. Kressman, T. R. E., 90. Kunin, Robert, 114. Lawrence, A. S. C., 255, 327.Lederer, E., 328, 329. Leigh, T., 311. Lessing, R., 240. Lester Smith, E., 317. Levi, Alfred A., 124. Lister, B. A, J., 104, 154, 156, 237. Martin, A. J. P., 151, 328, 332. McOmie, J. F. W., 183. Mongar, J. L., 118. Myers, Robert J., 114. Nestler, F. H. Max, 259. Offord, A. C., 26, 51. Partridge, S. M., 296. Pepper, K. W., 331. Phillips, C. S. G., 241. Pollard, F. H., 183, 23s. Reichenberg, D., 79, 160. Reichstein, T., 305. Richardson, R. W., 159. Robertson, R. H. S., 163. Robinson, F. A., 168. Robinson, G., 195, 238, 239. Sacconi, Luigi, 173, 238. Schwab, George-Maria, 170. Shepard, Charles C., 275. Shoppee, C. W., 305. Smit, W. M., 38, 48, 240, 248. Spedding, Frank H., 214. Stewart, A., 65, 163. Strickland, J. D. H., 157. Synge, R. L. M., 163, 164. Tiselius,Arne, 7,151,162,239,275,328.Tompkins, Edward R., 232. Wassermann, A., 118, 160. Weiss, D. E., 142. Weiss, Joseph, 26, 52, 53. Wells, R. A,, 179, 238. Williams, K. A., 264. Woiwod, A. J., 331. Zechmeister, L., 54, 151, 239, 329. * The references in heavy type indicate papers submitted for discussion.336 AUTHOR INDEX * Angell, F. G., 47. Barby, D., 255. Barker, K. H., 199. Barrer, R. M., 135, 162. Beaven, G. H., 325. Boldingh, J., 162. Brockmann, H., 58. Burstall, F. H., 179. Butler, J. A. V., 330. Cassidy, Harold G., 259. Claesson, Stig, 11, 34, 321. Consden, R., 329. Cook, C. D., 153, 240. Crisp, 1). J., 161. navies, C. W., 158, 241. Davies, G. R., 179. Davison, B., 45, 49. Duncan, J. F., 52, 104, 154, 1563, 158. Elbeih, I. I. M., 183. Flood, H., 190.Glueckauf, E., 12, 47, 50, 53, 154, 199, 238, 239, 240. Gordon, A. H., 128. Hale, D. K., 79, 160. Heymann, E., 161. Hirst, E. L., 268. Jones, J. K. N., 268. Jones, Tudor S. G., 285. Kitchener, J. A., 90, 155, 159. Kitt, G. P., 199. Klinkenberg, A., 151. Kressman, T. R. E., 90. Kunin, Robert, 114. Lawrence, A. S. C., 255, 327. Lederer, E., 328, 329. Leigh, T., 311. Lessing, R., 240. Lester Smith, E., 317. Levi, Alfred A., 124. Lister, B. A, J., 104, 154, 156, 237. Martin, A. J. P., 151, 328, 332. McOmie, J. F. W., 183. Mongar, J. L., 118. Myers, Robert J., 114. Nestler, F. H. Max, 259. Offord, A. C., 26, 51. Partridge, S. M., 296. Pepper, K. W., 331. Phillips, C. S. G., 241. Pollard, F. H., 183, 23s. Reichenberg, D., 79, 160. Reichstein, T., 305. Richardson, R. W., 159. Robertson, R. H. S., 163. Robinson, F. A., 168. Robinson, G., 195, 238, 239. Sacconi, Luigi, 173, 238. Schwab, George-Maria, 170. Shepard, Charles C., 275. Shoppee, C. W., 305. Smit, W. M., 38, 48, 240, 248. Spedding, Frank H., 214. Stewart, A., 65, 163. Strickland, J. D. H., 157. Synge, R. L. M., 163, 164. Tiselius,Arne, 7,151,162,239,275,328. Tompkins, Edward R., 232. Wassermann, A., 118, 160. Weiss, D. E., 142. Weiss, Joseph, 26, 52, 53. Wells, R. A,, 179, 238. Williams, K. A., 264. Woiwod, A. J., 331. Zechmeister, L., 54, 151, 239, 329. * The references in heavy type indicate papers submitted for discussion.

ISSN:0366-9033    

DOI:10.1039/DF94907FX001

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

336 AUTHOR INDEX * Angell, F. G., 47. Barby, D., 255. Barker, K. H., 199. Barrer, R. M., 135, 162. Beaven, G. H., 325. Boldingh, J., 162. Brockmann, H., 58. Burstall, F. H., 179. Butler, J. A. V., 330. Cassidy, Harold G., 259. Claesson, Stig, 11, 34, 321. Consden, R., 329. Cook, C. D., 153, 240. Crisp, 1). J., 161. navies, C. W., 158, 241. Davies, G. R., 179. Davison, B., 45, 49. Duncan, J. F., 52, 104, 154, 1563, 158. Elbeih, I. I. M., 183. Flood, H., 190. Glueckauf, E., 12, 47, 50, 53, 154, 199, 238, 239, 240. Gordon, A. H., 128. Hale, D. K., 79, 160. Heymann, E., 161. Hirst, E. L., 268. Jones, J. K. N., 268. Jones, Tudor S. G., 285. Kitchener, J. A., 90, 155, 159. Kitt, G. P., 199. Klinkenberg, A., 151. Kressman, T. R. E., 90. Kunin, Robert, 114. Lawrence, A. S. C., 255, 327.Lederer, E., 328, 329. Leigh, T., 311. Lessing, R., 240. Lester Smith, E., 317. Levi, Alfred A., 124. Lister, B. A, J., 104, 154, 156, 237. Martin, A. J. P., 151, 328, 332. McOmie, J. F. W., 183. Mongar, J. L., 118. Myers, Robert J., 114. Nestler, F. H. Max, 259. Offord, A. C., 26, 51. Partridge, S. M., 296. Pepper, K. W., 331. Phillips, C. S. G., 241. Pollard, F. H., 183, 23s. Reichenberg, D., 79, 160. Reichstein, T., 305. Richardson, R. W., 159. Robertson, R. H. S., 163. Robinson, F. A., 168. Robinson, G., 195, 238, 239. Sacconi, Luigi, 173, 238. Schwab, George-Maria, 170. Shepard, Charles C., 275. Shoppee, C. W., 305. Smit, W. M., 38, 48, 240, 248. Spedding, Frank H., 214. Stewart, A., 65, 163. Strickland, J. D. H., 157. Synge, R. L. M., 163, 164. Tiselius,Arne, 7,151,162,239,275,328.Tompkins, Edward R., 232. Wassermann, A., 118, 160. Weiss, D. E., 142. Weiss, Joseph, 26, 52, 53. Wells, R. A,, 179, 238. Williams, K. A., 264. Woiwod, A. J., 331. Zechmeister, L., 54, 151, 239, 329. * The references in heavy type indicate papers submitted for discussion.336 AUTHOR INDEX * Angell, F. G., 47. Barby, D., 255. Barker, K. H., 199. Barrer, R. M., 135, 162. Beaven, G. H., 325. Boldingh, J., 162. Brockmann, H., 58. Burstall, F. H., 179. Butler, J. A. V., 330. Cassidy, Harold G., 259. Claesson, Stig, 11, 34, 321. Consden, R., 329. Cook, C. D., 153, 240. Crisp, 1). J., 161. navies, C. W., 158, 241. Davies, G. R., 179. Davison, B., 45, 49. Duncan, J. F., 52, 104, 154, 1563, 158. Elbeih, I. I. M., 183. Flood, H., 190.Glueckauf, E., 12, 47, 50, 53, 154, 199, 238, 239, 240. Gordon, A. H., 128. Hale, D. K., 79, 160. Heymann, E., 161. Hirst, E. L., 268. Jones, J. K. N., 268. Jones, Tudor S. G., 285. Kitchener, J. A., 90, 155, 159. Kitt, G. P., 199. Klinkenberg, A., 151. Kressman, T. R. E., 90. Kunin, Robert, 114. Lawrence, A. S. C., 255, 327. Lederer, E., 328, 329. Leigh, T., 311. Lessing, R., 240. Lester Smith, E., 317. Levi, Alfred A., 124. Lister, B. A, J., 104, 154, 156, 237. Martin, A. J. P., 151, 328, 332. McOmie, J. F. W., 183. Mongar, J. L., 118. Myers, Robert J., 114. Nestler, F. H. Max, 259. Offord, A. C., 26, 51. Partridge, S. M., 296. Pepper, K. W., 331. Phillips, C. S. G., 241. Pollard, F. H., 183, 23s. Reichenberg, D., 79, 160. Reichstein, T., 305. Richardson, R. W., 159. Robertson, R. H. S., 163. Robinson, F. A., 168. Robinson, G., 195, 238, 239. Sacconi, Luigi, 173, 238. Schwab, George-Maria, 170. Shepard, Charles C., 275. Shoppee, C. W., 305. Smit, W. M., 38, 48, 240, 248. Spedding, Frank H., 214. Stewart, A., 65, 163. Strickland, J. D. H., 157. Synge, R. L. M., 163, 164. Tiselius,Arne, 7,151,162,239,275,328. Tompkins, Edward R., 232. Wassermann, A., 118, 160. Weiss, D. E., 142. Weiss, Joseph, 26, 52, 53. Wells, R. A,, 179, 238. Williams, K. A., 264. Woiwod, A. J., 331. Zechmeister, L., 54, 151, 239, 329. * The references in heavy type indicate papers submitted for discussion.

ISSN:0366-9033    

DOI:10.1039/DF94907BX003

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

CHROMATOGRAPHIC ANALYSIS GENERAL INTRODUCTION BY ARNE TISELIUS Received 2znd September, 1949 It is probably well known to all those present how chromatography, although invented and applied for the first time as early as in 1906 by Tswett, was almost completely neglected for a period of 25 years, and how it was actually rediscovered in 1931 by Kuhn, Winterstein and Lederer, who applied it to the resolution of plant carotene into its components. Then followed a period of rapid development, the importance of which may perhaps best be illustrated by a quotation from Karrer, who in his Congress Lecture to the International Congress of Pure and Applied Chemistry in London in 1947 stated that “ . . . no other discovery has exerted as great an influence and widened the field of investigation of the organic chemist as much as Tswett’s chromatographic adsorption analysis.Research in the field of vitamins, hormones, carotinoids and numerous other natural com- pounds could never have progressed so rapidly and achieved such great results if it had not been for this new method, which has also disclosed the enormous variety of closely related compounds in nature.” As a matter of fact the progress in the field of chromatography during the last 5-10 years has been so striking both with regard to the method itself and to its scope of application that one is tempted to speak about a second discontinuity in the curve which would describe the development of this remarkable method and its importance in all fields of chemistry. The time chosen for this Discussion is therefore particularly suitable and I feel that we all would like to express our thanks to the Faraday Society for organizing a discussion of this matter just now, The number of communications in this field is rapidly increasing and there has hardly been time enough to compare experiences with alternative methods or to discuss the numerous new applications which now seem possible. This meeting should afford an opportunity for this and for other valuable discussions in the field.Some of the most important recent developments are : the development of partition chromatography, the introduction of efficient ion-exchange materials suitable for chromatographic work, the introduction of improved methods for following the process of separation by continuous physico- chemical measurements or by automatic sample-collectors and the micro- chromatographic methods made possible by using filter paper as a medium for partition chromatography.The combination of chromatographic and tracer methods is only in its beginning but has tremendous possibilities. Although not strictly belonging to chromatographic methods in a narrow sense I would like to add to these examples the counter-current extraction methods developed by Craig. If one would try to mention some particularly striking examples of newer applications, it is obvious that chromatography is no longer limited to the substances studied by organic chemists and biochemists. The recent successful separations of rare earths and fission products from the uranium piles are particularly good examples of the wide scope of the method.Even 78 GENERAL INTRODUCTION if it is obvious that the separation and purification of organic substances of native or synthetic origin is the main object of most chromatographic work it is now clear that the particular application to the complex mixtures of breakdown products obtained by splitting proteins, polysaccharides, nucleic acids and other large molecules has already given chromatography a prom- inent place among the tools of the structural chemistry of these extremely important substances. Among the newer applications it should also be noted that some quite promising attempts have been made to apply chromato- graphy to high polymers and to viruses and some other proteins. The adsorption analysis of gases and vapours also seems to have useful applica- tions.Many other examples will be given in the papers presented at this Discussion. No doubt chromatography can now be said to be of essential importance for the whole field of chemistry, and no other separation method has such a wide field of application. Despite this, one has still the impression that much work in this field is too empirical. That is natural enough, because if a separation is successful there is no immediate need for going deeper into the subject in that particular case. But if the separation is not successful there is every reason to do so. We are still far from the goal of placing chromatographic methods on a rational basis to the same extent as has been done, for example, with fractional distillation.The formal theory of the method has been worked out in detail, and we shall listen to some valuable contributions on this aspect in the first section of this Discussion. In my own laboratory we have found it useful to distinguish between three main types of chromatography, namely, frontal analysis, elution analysis and displacement analysis, as the conditions for the successful operation of a column are quite different in these three cases. The theoretical treatment of elution in partition chromatography, as worked out by Martin and Synge, is particularly illuminating since it describes the action of a partition column as a counter-current extraction arrangement with a very large number of theoretical plates. The fundamental charac- teristics of a column are of course the affinity between the solute and the particles of the column in the medium used for the separation, that is, the adsorption or the partition isotherm, or, with a common denominator, the " distribution " isotherm, and that naturally will enter into any theoretical treatment of the operations as the chief unknown variable. One of the great advantages of partition chromatography is that this function can be predicted reasonably well on the basis of known solubilities in the two phases.The situation is not quite as favourable in adsorption chromato- graphy. It appears to be very difficult to prepare reproducible adsorbents if they are to be sufficiently powerful to be used in chromatographic columns. The influence of small impurities on such large active surfaces is naturally very great.In my opinion, however, one should not be too pessimistic about the possibilities here, because one has the impression that on the whole the rdative magnitudes of adsorption affinity in a series of substances will be fairly constant, even if the capacity may vary from batch to batch. That is certainly true for a number of active charcoals, as shown, for example, by Claesson in his work on fatty acids where Traube's rule determines the variation of adsorption with molecular size in an homologous series. Similar regularities have been found with amino acids and sugars on charcoal. The change of affinity by changing the composition of the solvent is, of course, of the greatest importance in the " development '' of chromatograms, and here the change from non-polar to polar solvents (or the reverse) has been a standard practice in chromatography for many years.Such effects can be predicted qualitatively, at least with a reasonable degree of certainty, and have been studied in detail, for example, by Trappe in his work on the " elu-ARNE TISELIUS 9 tropic series” in the elution of lipids. The ion-exchange resins, as, for example, Amberlite, Wofatite and Dowex, are really to a certain extent “ adsorbents made to order” and have already proved to be extremely useful both for separation of inorganic ions, of amino acids and many other substances. On the other hand, these adsorbents show a specificity which is not only dependent upon electrochemical properties. It would be highly desirable to study such phenomena in detail, and to investigate the influence of the size of pores in these materials and the possibility of preparing very porous resins for use with larger molecules.Speaking of adsorbents which can be made to have a certain specificity, F. H. Dickey has recently published a paper in which silica gel was precipi- tated together with, for example, methyl orange. The resulting adsorbent, after removing the indicator, had a very marked specific affinity for this substance. If this principle would prove to be of general applicability, it would, of course, mean a great step forward. Mixed solvents, obtained, for example, by the addition of a polar solvent to a non-polar, working with a polar adsorbent, are generally used for elution or for the development of chromatograms, as has already been mentioned.This is probably due to a displacement effect, as was already realized by Tswett in his first communication. The reverse effect also must exist, although it has not been studied in great detail. Thus addition of a second component to the medium must, under certain circumstances, promote the adsorption of the solute. This component may be adsorbed in a polymolecular layer, which may have an affinity for the solute, which the original adsorbent did not have. We are here on the borderline between adsorption and partition chromatography-in the latter case the layer has extended so much that one can speak of a separate phase. In discussing the various forms of partition chromatography and the eventual influence of adsorption in these, I believe it is well to keep in mind that such phenomena are likely to occur.There is also the possibility that the substance added to the medium will not appreciably change the properties of the adsorbent but will change the thermodynamic potential of the solute by change in electrolytic dissociation, by complex formation or by other effects which express themselves as a change in solubility of the solute. This is made use of in certain cases of elution by modification in pH, by addition of citrate or other complex- forming agents. But here also the reverse effect has been observed. Thus any agent which will decrease the solubility of a substance should increase its adsorption, provided that the adsorbent is not influenced.Some dye- stuffs which are normally not adsorbed on filter paper will do so on addition of salting-out agents, for example, ammonium sulphate, and good chromato- grams can be obtained by elution with water. Some proteins show this phenomenon too, especially on Celite and silica gel, and it has been used for the chromatographic separation of some viruses. We thus have in our hands varying methods of modifying the adsorption affinity and specificity of adsorbents by addition of suitable substances. With the enormous number of alternative combinations possible it is natural that this field has as yet been very little explored, but it is well to remember that it is no longer necessary to feel oneself limited to the adsorption charac- teristics of the commercially available adsorbents as they are obtained from the manufacturer.However, for the application in chromatography it is essential not only to have adsorbents of satisfactory affinity and speci- ficity, they must also have a reasonably high capacity and be capable of acting reversibly at a fairly fast rate. That often limits the choice quite considerably. For the elution procedure it is very essential to avoid the Proc. Nut. Acad. Sci. (Washington), 1949, 35, 227. a *I0 GENERAL INTRODUCTION unpleasant lagging behind and broadening of the zones which is usually called “ tailing.” The tailing may be due to a too slow establishment of equilibrium, but just as often it is the consequence of the adsorption or partition isotherm being curved which makes the higher concentrations of a zone travel faster than the lower.A great advantage of partition chromato- graphy is the fact that the isotherm is generally almost linear. With adsorp- tion this is not so often the case, at least not for strongly adsorbed substances. A t low concentrations, however, many adsorption isotherms are linear, as required by the Langmuir theory. This is, of course, ultimately a question of the homogeneity of the surface with respect to affinities, but also a question of the space available and when saturation is approached the isotherm is bound to bend and become concave, both in adsorption and partition processes. In such cases it is evident that elution cannot separate two components A and B (A more strongly bound than B) if the lower concen- trations of I3 would show a higher relative adsorption or partition than the higher concentrations of A.A great advantage of displacement and frontal analysis is that the stationary concentration of the zones is constant even in such a case and thus a separation can be realized, and very large columns may be used without any spreading out of the zones. In the development of chromatographic technique during recent years there has been a tendency to use the so-called “ liquid” chromatogram method for following the separation. This method has great advantages as it makes one independent of optical methods for observing the zones directly on the column. The new automatic sample-collectors of Moore and Stein have proved extremely useful for this purpose. The other alternative of continuous observation of some convenient physicochemical property related to the concentration of the percolate as it leaves the column has been studied particularly in my own laboratory and also in a number of other places.These methods are particularly useful in the study of dis- placement and frontal analysis when the number of observations must be very large to get the concentration-volume curve accurately reproduced. Some combination of both methods would seem to be the most ideal proce- dure, but it is highly desirable to increase the sensitivity of the optical (or other) methods applied to be able to deal with the very low concentrations used in most work in partition chromatography. This is also essential for the proper interpretation of frontal analysis curves, where one wants to use low concentrations in order to avoid displacement effects as far as possible.The direct observation of zones on the column has many advantages, and several interesting suggestions for new procedures have been made for the observation of colourless substances. Ultra-violet absorption observation of columns in quartz tubes offers many interesting possibilities, but one is, of course, then always limited in the choice of the adsorbent. The filter-paper chromatography offers some problems of this kind-there it now seems particularly important to develop the method further in the quantitative direct ion. Chromatography has so far been based only upon adsorption and partition phenomena, but it is obvious that any phenomenon which would-in a specific manner-influence the rate at which a zone of a substance travels through a column might be utilized for separation purposes.Electrophoresis and ionophoresis are related processes and have been utilized in a way analogous to chromatographic separation, using the column chiefly as a stabilizing medium to avoid gravitational or thermal disturbances. A combination of chromatography and electrophoresis has been tried by Strain. I t seems as if electrophoretic separation in columns of ion-exchange resins would offer interesting possibilities, as it must be expected that the specific affinityARNE TISELIUS IS of the substances to the resin will strongly influence their migration. There is another possibility, which I have several times discussed with Dr. Martin and Dr. Synge, namely, to make use of differences in the frictional resistance in a gel. Such separations would mainly depend upon differences in molecular size, and would thus be very valuable in many cases, but offers great technical difficulties. Some of the separations observed in ionophoresis in gels may be due to such effects, in part at least. I am afraid that in this Introduction I have dealt more with methods than with applications, but it is perhaps not necessary to this audience to exemplify further the possibilities of a method of such an enormously wide scope as chromatography. The various titles of the many papers to be presented at this Discussion and the presence here of representatives of all fields of chemistry and from many countries is a proof as good as any of this point. I shall be very happy if the remarks I have made here would stimulate the Discussion, to which we are all looking forward with the greatest interest. Biokemiskn Institutionest, U$$sala Universitet, U@#saLa, S-zeden.

ISSN:0366-9033    

DOI:10.1039/DF9490700007

出版商:RSC    

年代:1949

数据来源: RSC

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I1 I . PHYSICOCHEMICAL PRINCIPLES AND THEIR UTILIZATION INTRODUCTORY PAPER BY STIG CLAESSON Received 2nd September, 1949 The importance of chromatography is still rapidly increasing as can be clearly seen from the great number of very interesting contributions to this Discussion. Papers have been presented to this Section which have deepened our understanding of the chromatographic method and important advances have also been made in the experimental application of these ideas. It is certainly true that well-designed experimental arrangements are of the utmost importance when difficult separations are attempted. Remarkable results can be achieved in this way which is evident from many contributions to this Discussion. A typical example of this kind of work is also Moore and Stein’s experiments on the separation of amino acids, some of the most carefully planned and beautiful experiments in chromatography ever published.However, if a problem can be solved in different ways, the simplest way is always the best. A simple fraction collector is therefore often quite as usnful as a more intricate apparatus for the continuous recording of the concentra- tion of the effluent, and in many cases the inspection of the column with an ultra-violet lamp is quite satisfactory, particularly when the quenching of the fluorescence of the adsorbent is observed.2 The most important factor in all chromatographic work is, of course, the properties of the adsorbent in the column. Nothing can therefore ever 1 Moore and Stein, J . Uiol. Chewt., 1948, 176, 337, 367 ; 1949, 178, 53.79 2 Sease, J . Amer. Chenz. SOL, 1947, 69, 2242. 3 Brockmann and Volpus, Uer., 1947, 80, 77.12 THEORY OF CHROMATOGRAPHY compete in importance with the introduction of new and powerful adsorbent columns and it is also well known to everybody that the greatest progress in chromatography in recent years is due to the introduction of two new types of adsorbent, the partition column and the ion-exchange column. It is therefore always of the greatest interest to follow the introduction of new principles for controlling the adsorption process either by changing the adsorbents or solvents used. Several very interesting papers dealing with such problems are found here. In this connection a paper published by Dickey should be mentioned.* Following some ideas of Pauling he was able to show that silica gel pre- cipitated in the presence of, e.g., propyl orange after washing had a greater adsorption affinity for that compound than for methyl, ethyl or butyl orange.If such results can be improved and extended they will be of such value that their importance can hardly be overestimated. There are, however, two factors about which so little is known that they almost prevent progress in certain branches of chromatography. One is the problem of the connection between adsorption and chemical structure. Some progress has certainly been made in this field, but much still remains to be done before we can choose the adsorbents in a scientific way instead of using our intuition which is almost the best we can do to-day. The other factor which prevents the progress of chromatography is the lack of reproducible adsorbents of constant quality standardized in suitable ways. This difficulty could certainly be overcome by the chemical industry of to-day if the importance of this factor was made sufficiently clear to manufacturers of chemical products by a suitable group of scientists, e.g., those present at this Discussion. Fysikalisk-Kemiska Institutionen, Uppsaln, Sweden. Dickey, €'roc. Nat. Acad. Sci., 1949, 35, 227.

ISSN:0366-9033    

DOI:10.1039/DF9490700011

出版商:RSC    

年代:1949

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THEORY OF CHROMATOGRAPHY VII. The General Theory of Two Solutes following Non-linear Isotherms BY E. GLUECICAUF Received 2nd J u n e , 1949 The movement of solutes in an ideal adsorbing column during a chromatographic separation depends on the adsorption isotherm, i.e., the amounts fl and fz adsorbed in equilibrium with the concentrations c1 and cZ of the two solutes. This results from the conservation of solute mass in an adsorbing column under the normal chromato- graphic conditions, and it requires that coexistent c1 and c2 are functions of each other only. The solution of the equation leads to a family of curves (characteristics) in a system with the co-ordinates c1 and cZ. Co-existing concentrations of the two solutes occurring in the same chromatographic boundary lie on one such curve and this curve is uniquely determined by the composition of the original solution and by the solution used for developing the chromatogram. The sequence of concentrations in the column (i.e., the path followed in the diagram of characteristics) as well as any new con- centration plateaux developing spontaneously, and the occurrence of sharp or diffuse boundaries, can be predicted by the application of a few simple rules, which are deduced.In this way a unified method is provided which can predict the chromatographic development of any binary mixture of which the binary adsorption isotherms are known. Calculation leads to the fundamental relationship, df,/dc, = d f2/dc2. This applies both to solvent development and displacement development.E. GLUECKAUF 13 General Theory.-If a small amount of solvent dv containing an adsorbable solute of concentration c passes through a narrow section of the column weighing dx g., then conservation of mass requires that the amount of solute lost from the solvent during passage of this section must have increased the solute content of this section (f(c)dx), see Fig.I. I I 4 I I I FIG. I. d X -Balance of solute mass Conhnf chmyes from 1 d x t c j 15 dx (&-I f % dv) dV I I I I I I ! for a solution flowing through a column of sorbent. This, as has been shown by Wilson,l must be true for every individual solute, and leads to the differential equation * (11 In the further development of this function it must be taken into con- sideration that fi is usually a function of the concentrations of all the solutes present in the solution : fi = fi (Cl, c2, . .. . Ci) . . ' (4 For a sharp boundary, the conservation of mass requires that dv . A c ~ = dx . Afi , . (3) and as this applies to all the solutes present, the movement of such a sharp boundary can be expressed in terms of any of the solutes : For a diffuse boundary, eqn. (I) can be transformed as was done by de Vault : and For further discussion we confine ourselves to only two solutes and we assume that the isotherms fi andf, are such that (I) for c1 = o , f i = 0, for c2 = 0, fz = o that 3f2/3c, has the same sign as 3fl/3c2. and The first assumption is obvious and the Gibbs-Duhem equation requires that the last assumption is always satisfied. (11) 1 Wilson, J . Amer.Chew. SOC., 1940, 62, 1583. De Vault, ibid., 1943, 65, 532.14 THEORY OF CHROMATOGRAPHY Two situations are possible : (i) either there is a functional relationship between c1 and c2 alone, not involving ZI and x . or (ii) there is no such relationship. A mathematical analysis of the properties of the system ( 5 4 b) shows that case (i) holds only for the very simplest boundary conditions, i.e., if a solution of constant concentration is fed into an adsorption column which itself contains the solutes in another constant concentration. Fortunately this is the rule in normal chromatographic procedure. One starts with the boundary conditions c1 = 0, c2 = o (empty column), then adds the solution to be separated into its constituents with the constant composition (cl.= cl0, c2 = cZ0), and then develops this band of constant concentrations w t h another solution where again c1 = o and c2 = 0.As soon as we introduce special boundary conditions, e.g., an initial distribution of solutes in the column of a type which does not arise from feeding a solution of constant composition, we have case (ii). In these cases it will not be possible to find a simple dependence between the variations of x and ZI due to the fact that there is a varying c1 - c2 combination which does not permit the function df,/dci to be expressed in terms not containing x and ZI themselves. For all practical purposes case (ii) will only arise when, in the rare case of isotherms convex against the c-axis (when a diffuse front is produced), the band is developed with a different solvent.I t is not believed that these cases are sufficiently import ant to warrant an exceedingly involved discussion. In the event that there is a functional relationship (6) between c1 and c2 (case (i)), eqn. (9, b) become Then we have a relationship of the type, c1 = '2 (c2) * (6) and eqn. (8) makes it possible to derive this relationship between coexistent values of c1 and c2 (eqn. 6). Partial differentiation of both sides of (8) leads to which form was first used by Offord and we is^.^ For any given c1 and c2, the quadratic eqn. (9) leads to two alternative values for dcl/dc2. In view of assumption I1 with respect to fi(c, c,) and f2(c1 c,), the free term in this equation is negative. Consequently the two roots of the quadratic are both real, one being positive, the other negative. Eqn.(9) makes it possible to construct the curves of c1 as function of c2 for any given isotherm and for any given starting point cl0 czo, the next points on the curve being given by c,' = c20 - Ac, and cl' = cl0 - [%I . a,,,\ c10 c z o Offord and Weiss, Nature, 1945, 155, 725.E. GLUECKAUF 1.5 and so on. In this manner we can determine the two curves passing through the point cIo c,’, with positive slope, by using first the positive values of dc,/dc,, and with negative slopes, by using the negative values. Starting with all possible points cl0 c20 in the quadrant c1 > 0, c, > 0, we shall obtain two whole families of curves, covering the entire quadrant. These families of curves are known as the “ families of characteristics ” of the differential eqn. (8) (Fig.2). Only such combinations of c1 and c2 can occur in the same chromatographic boundary which lie on the same characteristic. (Instead of constructing the cl-c2 characteristics, it is sometimes more convenient t o use the fi-f2 characteristics. The choice depends on the equation for the adsorption isotherm. All the general conclusions drawn for the c characteristics apply equally to the f characteristics, though the equations are obviously modified.) FIG 2.--c1-c2 diagram of characteristics for the adsorption isotherms a1 c1= fl( fl + f 2 ) for 22 = 2. a2 c:! = fi (fl + 1 2 ) a, Only sets of concentrations lying on the same characteristic can coexist chromatographic boundary. in the same The fact that the entire quadrant c1 > 0, c2 > o is covered by these curves is of importance, because it implies that the “envelope” of these curves, which itself represents a solution to the differential equation, lies outside this quadrant and therefore does not include cases where both concentrations have values above zero, which alone are of practical signifi- cance.Thus no ambiguity is possible. It is also important to note that a particular characteristic is defined by only one concentration set c1 c2, so that, if a column is filled with a solution cIo cg0 and then developed with a solution -El C,, we cannot as a rule16 THEORY OF CHROMATOGRAPHY expect that these two solutions lie on the same characteristic. Each solution will possess its own pair of characteristics, which is of some importance for the form of chromatographic bands of two solutes (Fig.3). Thus, when a solution is poured on a column which has previously been treated with another solution containing the same solutes at different con- centrations, they will, as a rule, produce two separate boundaries, separated by a band of constant concentrations cl* c2*, which correspond to the point of intersection of two characteristics passing through the points clo c20 and Cl Cz, respectively. But before discussing this in detail, it is necessary to return once more to eqn. (7) which concerns the v-x relationship, i.e., the movement of a point of given concentrations c1 c2. We can write +P 1 - 3 - 2 - 1 0 f 2 3 4 5 6 7 parabolic envelope of the linear characteristics. FIG.3.-p,-p, diagram of characteristics for any Langmuir isotherm showing the If we take the case of a binary isotherm where 3fl/3c2 is negative (which is almost invariably the case), then it follows that dv/dx for a given point c1 c2 is always smaller, and consequently the rate of movement dx/dv of a point c1 c, is always larger for positive values of dc2/dcl than for negative ones. A concentration plateau cl* c2* in a chromatogram is only stable and growing when its forward point moves faster than its rear. It follows from this that a concentration plateau cl* c2* i s produced spontaneously in a boundary, if the boundary section in front of it travels a l o q the positive characteristic of cl* c2* (i.e., towards elo c20 in Fig. 2) while the rear travels along a characteristic where dcl/dc2 i s negative (or zero) (RULE I ) .This eliminates any ambiguity as regards the route which a boundary can take between two points, as the alternative concentration plateau at the point P (Fig. 2) cannot form spontaneously, if the solution Cl C2 follows the solution clo cZ0. (If, however, El Cz is in front of cIo c20, then for the same reason the boundaxy contains a concentration plateau near P.)E. GLUECKAUF =7 When changes occur in a column along the characteristics of the equation then c2 is always a function of c1 defined by the original solution (cl0 c,’) of the particular case. We can therefore write where F , is a new function of c1 only, which fully corresponds to the adsorption isothermf(c) in the case of a single solute, with the only exception that the function F, depends also on the initial conditions cl0 c2’.Counting ZI from the change-over to the new solution Cl C2 we can integrate to and Once the transformation to the single solute equation has been made, the sequence of chromatographic band movements follows in a straightforward way, as has been described in previous publication^.^ The question where a diffuse or a sharp boundary arises is again very similar to the case of a single solute as neither dcl/dx nor dc2/dx must become positive in a frontal boundary or negative in a rear boundary. I t thus depends on F,” and on F,” = [ 31, which type of boundary occurs. But, unlike the case of a single solute, it requires some knowledge of the characteristics to see under what conditions F,” and F,“ are positive or negative.Langmuir 1sotherms.-The general case can be best understood by illustrating it with the Langmuir isotherm, which offers the advantage that eqn. (8) can be directly solved by integration, so that general expressions can be obtained for all functions, and for the movements of all points in the chromatographic band. Writing the Langmuir isotherm in the form and where ra,>a, and p are all some positive constants. introduce (As shown by Kemball, Rideal and Guggenheim the use of different values of p1 and p2 offends the Gibbs-Duhem law.) I t is convenient to Fal = b, and pa2 = b, and a,/a, = b2/b, = K. For these isotherms (8) reduces to h , ( ~ + b2cJdcldc2 + b1 b2c2(dcJ2 - b1 b,c1(dcJ2 - b2(1+ b 1 ~ 1 ) dcldc, . (14) “lueckauf, Proc. Roy.SOC. A , 1946, 186, 35. 6 Glueckauf, J . Chew. SOC., 1947, 1321. 6 Coates and Glueckauf, J . Chem. SOC., 1947, 1309. Kemball, Rideal, and Guggenheim, Trans. Faraday SOC., 1948, 44, 952.I8 THEORY OF CHROMATOGRAPHY Putting for brevity c 1 = p 1 , - bl b 2 b2 - bl .~ bl b2 b 2 - bl and c 2 = p 2 J and dividing by (dp2)2, we obtain P 2 ( g ) 2 - ( I + p l - p 2 ) . g - p 1 = 0 . . Differentiating again with respect to p , [. P 2 *d dP - (I + P 1 -,,,I. $:;. = 0. . Hence, either 9 , is a linear function of 9, or Eliminating dfi,/dp, in (16) by means of (IS), we obtain This function is a parabola which merely touches the quadrant (1 + Pi - $2)' -k 4pd'2 = 0 - 6 2 - 6 c2 > o at the point c1 = 0, c2 = ', lying otherwise entirely outside this quadrant.I t does not, therefore, apply to any real concentrations and need not concern us. The only solutions of (16) where 9 , and 9, are both positive arise therefore from bl b2 where 9, is a linear function of p,. This leads to or, returning to the concentrations, p , = A d , - A/(I +A), . (zra) Here A is an integration constant, depending on the given starting concen- trations. In a system of c1 and c2 co-ordinates, eqn. ( z ~ a or b) represents straight lines which envelop the parabola of eqn. (19), and this makes it easy to construct the lines ( Z I ~ or b) geometrically (Fig. 3 ) . If we are given some point c1 c2 we can draw through this point the two tangents to the curve (19) and these tangents give us the two characteristics passing through the point c, c2, which contain all the coexistent concentration sets.From the fact that the lines (21) are tangents to a convex curve, it follows that no two different lines possess the same value A. Consequently A, which is the slope of the lines ( z r a ) , can serve as a parameter to distinguish one characteristic from another. As two tangents pass through every point c1 > 0, c2 > 0, we can introduce the two A values as parameters characterizing every mixture c1 c2. We take the positive As as the parameter p, and the negative ones as the parameter v. The positive-slope family of characteristics has values of p 2 o while the negative-slope family has values of v between o and -I.E. GLUECKAUF If we are given a point c1 c,, then p and v can be determined (21b).for A in , I ' I9 by solving . (2IC) and identifying the positive root of the quadratic with p and its negative root with v. The values of p as function of C J C , and (b, - bl)/b, b, c2 are shown i i i Fig. 2.4 From the equations of the two characteristics (see (21b))' and we can solve for 9 , and p 2 , or c1 and c2. P l = pP2 - p/(I + P)' P l = vp, - V/(I + v), -p. v ' (22) b, - bl c1 = b,b, - (I + p)(rf) ' which gives us the values of c1 and c2, in terms of the parameters p and v. Both expressions are positive, as v is negative and > (-I). We can also express f l ( c l c2) and f2(c2 c,) as functions of the parameters p and v : we then obtain the movement of a point of concentration c1 c2 : along the positive characteristic, and along the negative characteristic, by partially differentiating (23) and (25) with respect to v or p.Thus we obtain the simple functions : (See Fig. 4 (a), A+B or D-tC.) (See Fig. 4 (a), D+A or C-tB.) It is easy to see from this that the movement along the positive characteristic proceeds faster than along the negative one, as was already predicted generally for isotherms where afl/3c, and 3f2/3cl are negative. The condition that we have a sharp boundary requires that dcldx, deduced from (28) or (zg), is positive at a falling concentration (e.g., when forming20 THEORY OF CHROMATOGRAPHY the front of a band) and negative at a rising concentration (e.g., as rear of a band). Using and as all variations must occur along one of the characteristics, we obtain by differentiating (zz), (23), (28) and (29) for constant v : where D = z$ (p + K)(v + K ) .v / K ; p2] - - (v + D -~ I) = always positive . - (32) IrP For a single solute isotherm of the Langmuir type [dc/dx], is always positive ; we can, therefore, conclude from the above information (30-33) that the more strongly adsorbed solute I1 behaves in the presence of solute I, as far as type of boundary is concerned, in the same way as if solute I were not there at all. This means that where the more strongly adsorbed solute c2 increases along the column we have a difuse boundary, while where it changes to a lower concentration we have a sharp boundary (RULE 2). Fig. 4 illustrates the practical aspect of this rule for the properties of various boundaries. The rate of movement of these sharp boundaries (s.b.) is given by and replacing the c and f values by p and v from eqn.(32-35) leads to (See Fig. 4 ( a ) , B+A or C+D.) (See Fig. 4 (a), A-D or B+C.) The subscripts for p and v refer to the two concentration sets which are connected by the sharp boundary (e.g., the points B and C in Fig. 4 (a)). The knowledge of the characteristics p and v thus enables us to derive, with the help of eqn. (28), (29), (34) and (35), the form and rate of movement of all boundaries in the column or in the eluate.E. GLUECKAUF 21 I decreasing I1 increasing I11 ~ decreasing IV 1 increasing -_- _ _ _ _ _ _ ~ ~ _ _ _ _ . Returning to the families of characteristics in the c1 - cg diagram, we see that the four cases mentioned in eqn. (28), (29), (34) and (35) do not describe all the possibilities which can occur if one solution is eluted by another.The two solutions following each other in these cases were excep- tional in so far as both concentration sets lie on one characteristic. This, however, is not usually the case. If the two solutions following each other have no characteristic in common, and this is the rule, a new concentration plateau develops corresponding to that point of intersection of two charac- decreasing increasing increasing decreasing D -+ B (see Fig. 4 ( b ) ) B -+ D (see Fig. 4 (c)) C -+ A (see Fig. 4 ( d ) ) A -+ C (see Fig. 4 (e)) - - ~-~ - ~ ~- ~ ~~ - - 41 b B 1 x/v d I L I x/v e x/" / \ a I 2 3 4 +P 1 FIG. 4.-(a) Diagram of characteristics connecting two binary solutions, e.g., A and C, or B and D. p and v are the values of the gradients [dcJdc,] of the characteristics.Concentration changes of the two solutes (solute I - , solute I1 - --- ) in a column produced when (b) Solution D elutes a solution B. (4 ,, B I , 1, 9 ) D. (4 ,, c ,, ,, J , A. (.) ,, A ,> ,, ,, c. teristics passing through the two points which is stable according to Rule I, which states that the positive characteristic always moves in front of a negative one. This means that we can have the following four cases, which can be classified only according to the variation of p and v along the column : Case 1 Er, i v 1 Reference for Fig. 4 (a)22 THEORY OF CHROMATOGRAPHY The resulting boundaries of the column chromatograms with their spon- taneously developing concentration plateaux are shown in Fig.4 (b)-(e). The rate of movement, i.e., relative position of the boundaries, has been obtained from the p- and v-values by means of eqn. (28), (29), (34) and (35). It should be pointed out that, while normally the sharp boundaries do not follow exactly the curves of the characteristics, they do so in the case of Langmuir isotherms, because here, due to the linearity of the characteristics, dc,/dc, is identical with AcJAc,. Formation and Development of a Two Solute Band.-During the formation of the original band, produced by feeding a solution cIo c20 into an empty column (c, = 0, c2 = o), we have clearly a case of type I1 (Fig. 4 (c)) with increasing p and v. An example is shown in Fig. 5 (a), the stable boundary course being represented S -+ Q + 0. A new concentration plateau is created at Q, representing the pure frontal band of solute I (Fig.5 (b)). 3 S 2 Yz a P 0 FIG. 5.-Formation and solvent-development of a chromatographic band of the original concentrations c,' czo. (a) As shown in the diagram of g,-g, characteristics. ( b ) Original band. (c) and (a) Stages in the development of the band. Distribution of solute in a column for the above case. If, after the formation of the original band, we proceed to " develop " the chromatogram by the addition of further amounts ZI of pure solvent (cl = 0, c2 = o), we form a rear boundary of the type I (Fig. 4 (b)) with decreasing p and v, which in Fig. 5 (a) is represented by 0 -+ P + S. This rear boundary forms a new concentration plateau, characterized by the point P in the fig2 (or g,-g,) diagram, which is the point at which fi becomes zero (Fig.5 (a) and (c)). The actual movements of the various points with continuing development (v) follow directly from the application of the equations given in Tables I to V.6E. GLUECKAUF 23 Fig. 5 (a) gives the fl-f2 characteristics, which for a Langmuir isotherm are identical in form with the p,-p, characteristics (eqn. (zra)), if one chooses the parameters, * (37) Pb b 2 - b l and g, = . ,f2. The equations of motion as function of the parameters 'p = ["I for the positive and = [s] for the negative characteristics are similar to the J f i + 3fi q3 eqn* (")J ('91, (34) and (35)' We obtain : (38) (39) For the construction of bands in the column, these fig2 characteristics are somewhat more convenient, as the areas under the x-f curves (or x-g curves) represent the original masses of the solutes which must be constant.(When dealing with elution (c-v) curves the advantage is with the c1-c2 characteristics, for the same reason.) Returning to our Rule I, we expect that the plateaux of P and Q (Fig. 5) should be stable and growing, as here positive precedes negative charac- teristic. However, at the point S this situation is reversed and this plateau, which had been experimentally introduced, prior to the development with pure solvent, is therefore unstable and constantly diminishing. Eventually the point S will disappear altogether, thereby making Q also unstable, so that both plateaux S and Q will eventually disappear. S slowly recedes on the line SP towards P ; so we arrive at the diagram shown in Fig.5 (a) by the sequence 0 - t P +S' -+ Q' - + R +O, which results in the chromato- gram shown in Fig. 5 (d), which type persists until the constantly receding point S' has merged with P, when Q' coincides with 0, i.e., when separation is complete. Displacement Development .-The use of the characteristics does not only apply to ordinary solvent development, but also to development with a solution containing a third solute. A particularly simple case arises when * Due to the fact that p(fi + fi) < I , it follows that > -4.24 THEORY OF CHROMATOGRAPHY the third solute is most strongly adsorbed, which leads to the so-called displacement development, In this case we get no ternary mixtures and we can, therefore, describe the situation by two adjoining maps of charac- teristics shown in Fig.6. The original concentrations (cIo czo) are represented by the point E, the concentration of the developing solute (c30) by point A. The connection between A and E, and E and 0 under Rule I, goes via the points B, D and F, all of which form stable plateaux in the chromatogram, while the original plateau E, where a negative characteristic precedes the positive one, is unstable and diminishes in length (Fig. 7 (a) and (b)). Eventually E will disappear completely and henceforward the path from D to 0 goes via the point G (Fig. 7 (c)). This change makes the previously formed plateaux D and F unstable, and they eventually disappear. The end of the chromatographic separation is then reached, leaving the plateaux A, B and G stable, all of which represent pure species of solute (Fig.7 (d)). FIG. 6.-Development of a binary band with a solution containing a third solute, as shown in the diagram of c,-c2 and c,-c, characteristics. Example : b , = 4, b , = I, b3 = 2. We can immediately see from Fig. 6 what would happen if the developing concentration c c , (A), is not high enough. In this case a diffuse boundary arises between the plateaux and E (see also Fig. 6) which disappears eventually due to the instability of plateau and the separation ends with the plateaux A, B, G. Furthermore, if c; is smaller than indicated by the point x where the enveloping parabola (not shown) touches the abscissa, the only way from to E leads via 0, which means that the third solute fails to make contact with the other solutes and then there is no longer any displacement development. Actually, all these cases have already been discussed indi~idually,~ but the use of the " characteristics " makes it possible to give a unified picture of all possible chromatographic processes involving the separation of two solutes. - - -E. GLUECKAUF 25 Separation of Three Solutes.-With a certain amount of algebraical discomfort, the theory can be extended to ternary solutions. We obtain then a three-dimensional system of characteristics, and changes caused by solvent development proceed by variation of one characteristic, while keeping the other two constant. However, as no new chromatographic features are derived from this extension, its discussion may well be postponed. v o - 2 v = o A I FIG, 7 . 4 n c e n t r a t i o n diagram in the column, deduced from Fig. 6, line A-B-D-E-F-0, etc. (a) Original two solute band (solute I -, solute I1 - - - -). (b) (c) Stages in the development with a third solute. (d) Complete separation. My thanks are due to the Director, Atomic Energy Research Establishment, for permission to publish this paper, and especially to Dr. B. Davison, A.E.R.E., for valuable mathematical advice. Atomic Energy Research Establishment, Harwell, Didcot, Berks.

ISSN:0366-9033    

DOI:10.1039/DF9490700012

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

CHROMATOGRAPHY WITH SEVERAL SOLUTES BY A. C. QFFORD AND JOSEPH WEISS Received 14th July 1949 The theory of chromatography of a single solute has been given previously by de Vault and by Weiss. The present investigation has for its object a theory of chromato- graphy for two or more solutes. The treatment is based on the differential equations connecting the concentrations c and c with the amounts adsorbed per unit length q and q respectively under equilibrium conditions where v is the volume and x the distance along the column. We first investigate under what initial conditions these equations will have a solution which is a function of v/x and using this result determine solutions in a few cases. We find that two very different phenomena may result. (i) In one case definite bands are formed in which the concentrations are constant.These constant values depend on the original bands and they can be determined in terms of the initial concentrations. (ii) In other cases bands of varying concentrations are formed. A typical example of the first type occurs when a uniform band of one or more solutes is developed with a solution of a different solute which is more strongly adsorbed than any of the solutes present initially (as illustrated for instance] by Tiselius’ displacement method). The second type of chromatogram may occur when a band of two solutes at constant concentrations is developed with pure solvent or when a band consisting of a single solute is developed with a solution of a second solute which is much less strongly adsorbed than the first solute.Several examples are discussed and various criteria for distinguishing these cases are given. The theory of chromatography of a single solute has been given by de Vault and by Weiss. 2 Subsequently a treatment for two or more solutes was briefly indicated by the present writers3 and in some special cases was given by a number of authors.4 The object of this paper is to give a general theory for the chromatography of two or more solutes. We begin with a general discussion and then give examples for two and three solutes. I. General Discussion.-The amounts adsorbed per unit cross-section at the level x of the band after the passing of a volume v are denoted by q ( x v) and q ( x v) . It is assumed that q and q depend only on the concen- trations c and c of the solutes (I) and (2) and that the adsorption equi- librium is practically always established.If q l q 2 c 1 and c z are continuous functions with continuous partial derivatives considerations of conservation of mass imply that and 1 De Vault J . Amer. Chem. SOL 1943 65 532 ; also Wilson ibid. 1940 62 1583. Weiss J . Chem. SOG. 1943 297. a Offord and Weiss Nature 1945 155 725. Cf. Glueckauf Proc. Roy. Soc. A 1946 186 3 5 ; J . Chem. SOL 1947 1308. 1321. 26 A. C. OFFORD AND JOSEPH WEISS 27 However we shall not assume that c1 and c 2 are continuous functions of x and v. This is based on results obtained by one of us and by others in the discussion of the chromatography of a single solute.At first sight it may seem somewhat artificial to admit discontinuous functions but it must be borne in mind that c 1 and c 2 represent certain mathematical functions which satisfy eqn. (1.1) and (1.2). These functions will in general represent the concentrations only approximately because they were derived under the assumption of equilibrium conditions and with the neglect of any diffusion and convection effects. Thus though the concentrations may be continuous it may well be that they can be best approximated by discontinuous functions. Experience in the chromatography of a single solute shows that this is actually the case. Let us suppose now that there is a certain front of discontinuity AB. Let ql(I) and q 2(1) be the amounts adsorbed per unit length in the band imme- diately above the level AB and q 1(11) and q 2(11) be the corresponding quantities in the band immediately below AB.We shall suppose that the concentra- tions remain constant as the level AB moves down the tube. Let ~ ~ ( ~ 1 c 2(1) c 1(11) and c 2(11) be the corresponding concentrations. Suppose that as the result of the addition of a volume 6v the level of discontinuity moves down the tube a distance 8% to a level A'B'. (see Fig. I). Then the decrease in the amounts of the substances (I) and ( 2 ) in the section (AB)(R'A') will be and these quantities will be equal to the increases in the amounts of solute in the small volume 6v which are { c l(II)-C l(1) } 6u and { c2(11)-Cz(I) respectively and hence 4- 42(rn-42(1) } h- { Cz(rl)-Cz(I) 7 Since 6v and 6x are not zero and since these equations must hold simul- taneously we get 6v - qi(II)-qi(I) 6% c l(II)-C 10) Allowing Bx and 6v to tend to zero we deduce that 1 J6v=0.- . (2.2) * = q2(11)-42(1) c 2(II)-C 2(1) * (2.31 This is a differential equation giving the rate at which a level of discontinuity AB moves down the tube in terms of the ratio of the discontinuities in the amounts adsorbed and the concentrations. We shall now make some general assumptions concerning q 1 and q 2 . We shall suppose that they depend only on c and c and that they are regular functions of c and c ; that is to say that they can be expanded in a Taylor series in c 1 and c 2 . Thus m=o m=o n=o n=o CHROMATOGRAPHY WITH SEVERAL SOLUTES 28 Since obviously q is zero if c is zero and q is zero if c is zero we must have a0,? = bm,o = o for all m and n.This implies that Thus the eqn. (I) become A A' FIG. I.-Front of discontinuity AB separating the regions (I) and (11). constant initial values + FIG. 2.-(a) z Solutes initial band. (b) z Solutes after partial development. vl'+(cl' One could solve any problem involving two solutes and without dis- continuities if one knew a solution of these equations under sufficiently general conditions. However no such solution is known. The classical X FIG. 4.-Originally uniform mix- ture of two substances (2) and (3) partly developed with a solution of a third substance (I) which is more strongly adsorbed.(1) c i 0 -+ constant (1) + (2) x1 (initial) value CIt c2' 1 c ; -1- constant initial value (1 cons tan t initial value 0 +variable -I"' FIG. 3.-Uniformly distributed solute (2) partly developed with a solution of a more strongly adsorbed sub- stance (I). c2=0) +constant initial values A. C. OFFORD AND JOSEPH WEISS gets In general the solution of these equations will be 3q - - 0. . * dc dc - 0. d y = d y - The necessary and sufficient conditions for them to have a solution other than this one is __ 3q 2 J C l 29 solution * assumes that c and c are regular functions of x and z and this hypothesis is difficult to apply since the initial conditions as usually stated in nearly every problem in chromatography require c and c to be discon- tinuous for x=o and v=o.There is however an important class of problems where it is possible to obtain solutions of these equations. It can be shown that under certain very special initial conditions the solutions of these equations will be approxi- mately functions of (z,/x). The initial conditions for this are that when x=o c and c are both constant for all z and that when z,=o the distributions q 1 and q 2 of solute in the tube are constant for all x . Now it happens that usually under the experimental conditions in chromatography these conditions are in fact often satisfied to a first approximation. For instance they are fulfilled when an (infinite) band containing adsorbed material from a solution of two solutes of constant concentrations is developed with pure solvent or with a solution of a third substance.In practice where the columns are of finite length it is reasonable to expect that the subsequent development would be the same since it is determined by the state of affairs at the top of the chromatogram. It would therefore seem that a solution of eqn. (5) under such initial conditions would furnish all that a practical chromatographer would desire. However another difficulty now arises. It appears that under such initial conditions eqn. (5) need not have a unique solution and so no mathematical argument can be fully conclusive and any results obtained will have to be tested by experiment. Let us proceed with the discussion of eqn.( 5 ) on these lines. Substituting y=(v/x) and assuming that c l and c 2 depend on y as defined above one (7.1) (7.2) . (2 - 3') This corresponds to the quadratic equation ac + 5~ JC 3q2 ac - 3c2 3c '12 = 0 (:%; 1 342) y 2 - which has the roots * See for example Goursat LeFons sur l'inthgratio9z des iquations nztx derive's partielles du premier ordre (1891)~ p. 2. CHROMATOGRAPHY WITH SEVERAL SOLUTES ’ * dc 1 according to which solution of (7) we take. In either case 30 and thus eqn. (7) may be written Whenever (7) is satisfied (6) gives that is or and similarly (Y-4 (Y- P) = 0 dc 1ldY -___- dc 2ldY = (Y - 3q l / > C 1) 3q 1/3c 2 dc - (P-3q1/3cJ (3q 1/3c 2) - cc = dql/dcl and p = dql/dcl a = dq2/dc2 so that in a variable band we have dq,/dc = dq 2/dC2 * I1 Examples (i) Chromatogram formed from two solutes and developed with pure solvent.-As we have already remarked our special conditions are satisfied in this case and we may apply the general theory developed above.Let us first consider the problem that arises when a solution of two solutes is poured on to a column of adsorbent material which is initially free of solutes. The first question to be decided is can a variable band be found in these circumstances? If this does occur then eqn. (10.1) and (10.2) hold throughout the band. Moreover it is clear that dc,/dcl will be positive since c and c will either increase together or decrease together. Further it is to be expected that the presence of one solute will hinder the adsorption of the others so that 3ql/3c and aq2/3cl will be negative and their product (3ql/dc,) (3q,/3cl) positive.Hence from the eqn. (8) one obtains for the discriminant and so it follows from eqn. (8) that (a - 3ql/3c,) is positive and ( p - 3ql/3c,) is negative. Consequently of the two eqn. (IO) (10.1) gives a negative value for dc 2 / d ~ and (10.2) a positive value. Therefore throughout this band we must have or But then (9) implies that or y = v/x = p V/X = dq Jdc . If now the level x is fixed and the volume v increased c and c will increase. But with normal isotherms dq ,/dc is a decreasing function of G (Len with all ordinary adsorption isotherms as c1 increases dqJdc decreases) and so (9) (10.1) .(10.2) (11) (12) - * * A. C. OFFORD AND JOSEPH WEISS 31 consequently the above equation can never be satisfied. Hence in this case there can be no variable band. - Eqn. (22) determines the concentration cl’ of the solute (I) at the level at which c 2 in zero and the level xl’ at which this occurs is given by (ii) Development of a column consisting of a uniform mixture of two solutes (1) and (2) with pure solvent.-We shall assume that when development occurs it will be a continuous phenomenon and so throughout the developed bands of the two solutes eqn. (7) holds. Let us now suppose that This corresponds to treating the solute (I) as the more strongly adsorbed one. Then from eqn. (4) and (8) one has (c1 0) = ( ~ q l / ~ ~ l ) c = o and P(c19 0) = (322/3c2)c,=0 and so (7) reduces to But (16) implies that for small c 1 * * (3q 1/3c I)c,,o > 2/3c 2)c*=0 9 * (v/x> = ( ~ 4 1 / ~ c l ) c = o - (18) and so since y= (711%) is a decreasing function of x (i.e.large for x small) we must have (19) On the other hand when c > 0 clearly dc 2/dc is positive and so as above and this implies that ’ - (21) or in other words that for c > o it is the second factor in (7) which vanishes while for c2=o it is the first factor. But by our hypothesis the development is a continuous phenomenon and this means that eqn. (7) must hold for all c2. If now we imagine c 2 tends to zero in (7) then since for c 2 > o the second factor is zero and for c2=o the first factor is zero it follows that at the point where c 2 just vanishes both factors must be zero and so at this level (22) v/x= P (32 1/3c l ) c z ~ o = (%7 2/3c 2)cp=0 4% 1’ = (32 1/3c l)c,=cl’ cz=O = (32 2/3c 2)cl,c,’ cl=o (23) Let us examine eqn.( 2 2 ) . The two functions z1=(~q1/3cl)c,=o and z %= (34 2)cp=o are both usually decreasing functions of the same variable cl. The first function x denotes the rate of change of the amount adsorbed of solute (I) with concentration of this solute in the absence of solute (2). This quantity for all ordinary isotherms decreases as the concentration c1 increases. The second function z represents likewise the rate of change of,adsorbed amount of solute ( 2 ) with concentration c 2 for very low values of c 2 and in a solution which also contains the solute (I) at a certain con- centration cl.Thus the curves for x 1 and z 2 are essentially different in character z will normally decrease as c increases but the decrease may be expected to be small compared with the corresponding decrease in 2,. This is the case if the “ strength ” at which solute (2) is adsorbed at very 32 CHROMATOGRAPHY WITH SEVERAL SOLUTES low concentrations is not greatly affected by the presence of solute (I) in the solution. If x 1 and x 2 decrease but x 1 decreases more rapidly than z, there will be a point at which xl=x2 (since initially x 1 is the larger). That is a cl' for which eqn. (22) is satisfied. Hence in this case when the column consisting of solutes (I) and ( 2 ) corresponding to constant concentrations clo and c20 (Fig.2 (a)) is developed with a volume 'u of solvent at the top of the tube there will be a clear band (from o to x, Fig. 2 (6)). If * (24) a band of solute (I) will occur the concentration at any level x (in the region from x to x,' ; Fig. 2 ( 6 ) ) being given by (25) . (26.1) (3q 1/3c l)c1=c2'o > (%I 2/3c 2)cl=c,=a 9 then at a level x I' given by V/X1'= (341/~cl)c,=c,=o 9 Xl 4% = (aqla/cl)c2_o 3 the concentration c1 of solute (I) within this band increases until it reaches a critical value cl' given by eqn. (22). After this there follows a mixed band (x ,' to x 2) where the concentrations c1 and c 2 are connected by the differential eqn. (12) subject to the conditions that when cl=cl' c,=o.This development continues to a certain level where a sharp change in the concentrations occurs and we arrive at the undeveloped point of the band with the concentrations at their original values of c and c 2 0 (Fig. 2(b)). Various other solutions may be possible in this case depending on the characters of the isotherms but the above solution appears to be of par- ticular interest. There is one further point of some importance. The chromatogram we have described is that obtained when an infinite band is developed. In practice the band will be of finite length of mixed solutes (I) and (2) followed usually by a band of the less strongly adsorbed solute (2). On developing with (pure) solvent the nature of the development at the top of the tube is clearly independent of the length of the band.The solution we have given then enables us to calculate the volume of solvent required to obtain com- ple te separation. Writing Q(C1) = 4 1 h 0) ? then the amount of solute (I) in the band between the levels x and x (see Fig. 2(6)) is 0 and on integration by parts this becomes 0 X c ' [Q (C 1)xI bearing in mind that Q(o) = 0. Now throughout this band? = Q' (cl) where v the volume added is of course a constant and so the above expression becomes (26.2) A. C. OFFORD AND JOSEPH WEISS 33 Xow suppose that the total amount of solute (I) present was ml. Then the tube being of unit cross-section the volume vs required to obtain total separation will be where cl’ is given by (22).(iii) An infinite column of uniformly distributed solute is developed with a solution of a more strongly adsorbed substance.-Our simple equations are valid in this case. We shall suppose as usual that solute (I) is the more strongly adsorbed. In any mixed band dc2/dcl is negative and this implies that If is small relative to the remaining terms this implies that Suppose a volume v of a solution of solute (I) is added then as we go down the tube (34 ,/3c increases as c decreases. Also for c 2=0 3q ,/3c is zero but not zero for c > 0. Hence the second term also increases (the product (?q ,/3c 2) (34 2/3c 1) being positive). Therefore both terms on the right-hand side increase which is impossible since with increasing x the left-hand side decreases.Hence we are led to a contradiction and must conclude that in this case there can be no variable band. Therefore the only solution possible is a discontinuous one. To satisfy eqn. (3) there must be two discontinuities separated by a mixed band. If cl’ and c2’ denote here the concentrations of the solutes in the mixed band then and are the two simultaneous equations for c and c 2’. If x denotes the level of the top of the mixed band and x 2 the level of the bottom (Fig. 3) after a volume ZJ has been added then by eqn. ( 3 ) since x=o when V=O (iv) Band consisting of a uniform mixture of two substances (2) and (3) which is developed with a solution of a third substance ( 1 ) . -In this case we have in general a cubic equation to solve and so a detailed analysis would be somewhat complicated.We shall consider only the case when the substance (I) is more strongly adsorbed than the other two substances and even then we shall only instance two cases which might arise. B FRONTAL ANALYSIS E * 34 (a) If the adsorption isotherms of the three substances are represented by q 1 q 2 and q 3 and if then there is no interaction between (I) and the mixture of the components (2) and (3). A band of solute (I) is formed at the top and the mixture (2) and (3) is developed as in Example (ii). then at the top of the tube we shall have a chromatogram corresponding to Example (iii). Altogether there may be fzve bands. In the to$ ban,d the substance (I) at concentration c l 0 .In the second band solutes (I) and (2) corresponding to concentrations cl’ and c2’. In the third b a d (2) at con- centration c2”. The fourth band may be a mixed band as in Example (i) and in thefzfth band (2) and (3) are present at their original concentrations c20 and c30 (see Fig. 4). The values of c l l c2’ and call are given by the following equations The levels x and x corresponding to the top and bottom of the first mixed band are then given by University of London Birkbeck College _ - - q2(‘1‘J ‘21 O) * 1 c 2‘ University of Durham King’s College ( 3 5 4 Newcastle-upon- Tyne I. London E.C.4. CHROMATOGRAPHY WITH SEVERAL SOLUTES BY A. C. QFFORD AND JOSEPH WEISS Received 14th July 1949 The theory of chromatography of a single solute has been given previously by de Vault and by Weiss.The present investigation has for its object a theory of chromato-graphy for two or more solutes. The treatment is based on the differential equations: connecting the concentrations c and c with the amounts adsorbed per unit length q and q respectively under equilibrium conditions where v is the volume and x the distance along the column. We first investigate under what initial conditions these equations will have a solution which is a function of v/x and using this result determine solutions in a few cases. We find that two very different phenomena may result. (i) In one case definite bands are formed in which the concentrations are constant. These constant values depend on the original bands and they can be determined in terms of the initial concentrations.(ii) In other cases bands of varying concentrations are formed. A typical example of the first type occurs when a uniform band of one or more solutes is developed with a solution of a different solute which is more strongly adsorbed than any of the solutes present initially (as illustrated for instance] by Tiselius’ displacement method). The second type of chromatogram may occur when a band of two solutes at constant concentrations is developed with pure solvent or when a band consisting of a single solute is developed with a solution of a second solute which is much less strongly adsorbed than the first solute. Several examples are discussed and various criteria for distinguishing these cases are given.The theory of chromatography of a single solute has been given by de Vault and by Weiss. 2 Subsequently a treatment for two or more solutes was briefly indicated by the present writers3 and in some special cases was given by a number of authors.4 The object of this paper is to give a general theory for the chromatography of two or more solutes. We begin with a general discussion and then give examples for two and three solutes. I. General Discussion.-The amounts adsorbed per unit cross-section at the level x of the band after the passing of a volume v are denoted by q ( x v) and q ( x v) . It is assumed that q and q depend only on the concen-trations c and c of the solutes (I) and (2) and that the adsorption equi-librium is practically always established.If q l q 2 c 1 and c z are continuous functions with continuous partial derivatives considerations of conservation of mass imply that : and 1 De Vault J . Amer. Chem. SOL 1943 65 532 ; also Wilson ibid. 1940 62 1583. a Offord and Weiss Nature 1945 155 725. Weiss J . Chem. SOG. 1943 297. Cf. Glueckauf Proc. Roy. Soc. A 1946 186 3 5 ; J . Chem. SOL 1947 1308. 1321. 2 A. C. OFFORD AND JOSEPH WEISS 27 However we shall not assume that c1 and c 2 are continuous functions of x and v. This is based on results obtained by one of us and by others in the discussion of the chromatography of a single solute. At first sight it may seem somewhat artificial to admit discontinuous functions but it must be borne in mind that c 1 and c 2 represent certain mathematical functions which satisfy eqn.(1.1) and (1.2). These functions will in general represent the concentrations only approximately because they were derived under the assumption of equilibrium conditions and with the neglect of any diffusion and convection effects. Thus though the concentrations may be continuous, it may well be that they can be best approximated by discontinuous functions. Experience in the chromatography of a single solute shows that this is actually the case. Let us suppose now that there is a certain front of discontinuity AB. Let ql(I) and q 2(1) be the amounts adsorbed per unit length in the band imme-diately above the level AB and q 1(11) and q 2(11) be the corresponding quantities in the band immediately below AB. We shall suppose that the concentra-tions remain constant as the level AB moves down the tube.Let ~ ~ ( ~ 1 , c 2(1) c 1(11) and c 2(11) be the corresponding concentrations. Suppose that as the result of the addition of a volume 6v the level of discontinuity moves down the tube a distance 8% to a level A'B'. (see Fig. I). Then the decrease in the amounts of the substances (I) and ( 2 ) in the section (AB)(R'A') will be and these quantities will be equal to the increases in the amounts of solute in the small volume 6v which are { c l(II)-C l(1) } 6u and { c2(11)-Cz(I) respectively and hence . (2.2) 7 J6v=0. - 4- 42(rn-42(1) } h- { Cz(rl)-Cz(I) 1 Since 6v and 6x are not zero and since these equations must hold simul-6v - qi(II)-qi(I) = q2(11)-42(1) taneously we get * (2.31 6% c l(II)-C 10) c 2(II)-C 2(1) * Allowing Bx and 6v to tend to zero we deduce that This is a differential equation giving the rate at which a level of discontinuity AB moves down the tube in terms of the ratio of the discontinuities in the amounts adsorbed and the concentrations.We shall now make some general assumptions concerning q 1 and q 2 . We shall suppose that they depend only on c and c and that they are regular functions of c and c ; that is to say that they can be expanded in a Taylor series in c 1 and c 2 . Thus, m=o n=o m=o n= 28 CHROMATOGRAPHY WITH SEVERAL SOLUTES Since obviously q is zero if c is zero and q is zero if c is zero we must have a0,? = bm,o = o for all m and n. This implies that Thus the eqn. (I) become A A' FIG. I.-Front of discontinuity AB separating the regions (I) and (11).constant initial values + (1) c i 0 -+ constant (1) + (2) x1 (initial) value CIt c2' 1 -I"' c ; -1- constant initial value FIG. 3.-Uniformly distributed solute (2) partly developed with a solution of a more strongly adsorbed sub-stance (I). 0 (1 +variable vl'+(cl' c2=0) +constant initial values cons tan t initial value X FIG. 2.-(a) z Solutes initial band. FIG. 4.-Originally uniform mix-ture of two substances (2) and (3) partly developed with a solution of a third substance (I) which is more strongly adsorbed. (b) z Solutes after partial development. One could solve any problem involving two solutes and without dis-continuities if one knew a solution of these equations under sufficiently general conditions.However no such solution is known. The classica A. C. OFFORD AND JOSEPH WEISS 29 solution * assumes that c and c are regular functions of x and z and this hypothesis is difficult to apply since the initial conditions as usually stated in nearly every problem in chromatography require c and c to be discon-tinuous for x=o and v=o. There is however an important class of problems where it is possible to obtain solutions of these equations. It can be shown that under certain very special initial conditions the solutions of these equations will be approxi-mately functions of (z,/x). The initial conditions for this are that when x=o c and c are both constant for all z and that when z,=o the distributions q 1 and q 2 of solute in the tube are constant for all x .Now it happens that usually under the experimental conditions in chromatography these conditions are in fact often satisfied to a first approximation. For instance they are fulfilled when an (infinite) band containing adsorbed material from a solution of two solutes of constant concentrations is developed with pure solvent or with a solution of a third substance. In practice where the columns are of finite length it is reasonable to expect that the subsequent development would be the same since it is determined by the state of affairs at the top of the chromatogram. It would therefore seem that a solution of eqn. (5) under such initial conditions would furnish all that a practical chromatographer would desire. However another difficulty now arises.It appears that under such initial conditions eqn. (5) need not have a unique solution and so no mathematical argument can be fully conclusive and any results obtained will have to be tested by experiment. Let us proceed with the discussion of eqn. ( 5 ) on these lines. Substituting y=(v/x) and assuming that c l and c 2 depend on y as defined above one gets In general the solution of these equations will be The necessary and than this one is dc dc, - 0. d y = d y -sufficient conditions for them to have a solution other 3q, - - 0. . * (7.1) __ 3q 2 J C l (2 - 3') This corresponds to the quadratic equation y 2 - (:%; 1 342) ac + 5~ JC 3q2 ac - 3c2 3c '12 = 0, . (7.2) which has the roots * See for example Goursat LeFons sur l'inthgratio9z des iquations nztx derive's partielles du premier ordre (1891)~ p.2 30 CHROMATOGRAPHY WITH SEVERAL SOLUTES and thus eqn. (7) may be written Whenever (7) is satisfied (6) gives (Y-4 (Y- P) = 0 - (9) dc 2ldY = (Y - 3q l / > C 1) dc 1ldY 3q 1/3c 2 ’ -___-that is, or (10.1) . (10.2) cc = dql/dcl and p = dql/dcl * (11) dc - (P-3q1/3cJ dc 1 (3q 1/3c 2) * -according to which solution of (7) we take. In either case, and similarly so that in a variable band we have I1 Examples (i) Chromatogram formed from two solutes and developed with pure solvent.-As we have already remarked our special conditions are satisfied in this case and we may apply the general theory developed above. Let us first consider the problem that arises when a solution of two solutes is poured on to a column of adsorbent material which is initially free of solutes.The first question to be decided is can a variable band be found in these circumstances? If this does occur then eqn. (10.1) and (10.2) hold throughout the band. Moreover it is clear that dc,/dcl will be positive since c and c will either increase together or decrease together. Further, it is to be expected that the presence of one solute will hinder the adsorption of the others so that 3ql/3c and aq2/3cl will be negative and their product (3ql/dc,) (3q,/3cl) positive. Hence from the eqn. (8) one obtains for the discriminant : a = dq2/dc2 , dq,/dc = dq 2/dC2 * * (12) and so it follows from eqn. (8) that (a - 3ql/3c,) is positive and ( p - 3ql/3c,) is negative.Consequently of the two eqn. (IO) (10.1) gives a negative value for dc 2 / d ~ and (10.2) a positive value. Therefore, throughout this band we must have or But then (9) implies that or y = v/x = p , V/X = dq Jdc . If now the level x is fixed and the volume v increased c and c will increase. But with normal isotherms dq ,/dc is a decreasing function of G (Len with all ordinary adsorption isotherms as c1 increases dqJdc decreases) and s A. C. OFFORD AND JOSEPH WEISS 31 consequently the above equation can never be satisfied. Hence in this case there can be no variable band. (ii) Development of a column consisting of a uniform mixture of two solutes (1) and (2) with pure solvent.-We shall assume that when development occurs it will be a continuous phenomenon and so throughout the developed bands of the two solutes eqn.(7) holds. Let us now suppose that This corresponds to treating the solute (I) as the more strongly adsorbed one. Then from eqn. (4) and (8) one has and so (7) reduces to (c1 0) = ( ~ q l / ~ ~ l ) c = o and P(c19 0) = (322/3c2)c,=0 But (16) implies that for small c 1, and so since y= (711%) is a decreasing function of x (i.e. large for x small) we must have (3q 1/3c I)c,,o > 2/3c 2)c*=0 9 * * (18) (v/x> = ( ~ 4 1 / ~ c l ) c = o - * (19) On the other hand when c > 0 clearly dc 2/dc is positive and so as above, and this implies that or in other words that for c > o it is the second factor in (7) which vanishes, while for c2=o it is the first factor. But by our hypothesis the development is a continuous phenomenon and this means that eqn.(7) must hold for all c2. If now we imagine c 2 tends to zero in (7) then since for c 2 > o the second factor is zero and for c2=o the first factor is zero it follows that at the point where c 2 just vanishes both factors must be zero and so at this level, Eqn. (22) determines the concentration cl’ of the solute (I) at the level at which c 2 in zero and the level xl’ at which this occurs is given by v/x= P - (21) (32 1/3c l ) c z ~ o = (%7 2/3c 2)cp=0 ’ - (22) 4% 1’ = (32 1/3c l)c,=cl’ cz=O = (32 2/3c 2)cl,c,’ cl=o (23) Let us examine eqn. ( 2 2 ) . The two functions z1=(~q1/3cl)c,=o and z %= (34 2)cp=o are both usually decreasing functions of the same variable cl. The first function x denotes the rate of change of the amount adsorbed of solute (I) with concentration of this solute in the absence of solute (2).This quantity for all ordinary isotherms decreases as the concentration c1 increases. The second function z represents likewise the rate of change of,adsorbed amount of solute ( 2 ) with concentration c 2 for very low values of c 2 and in a solution which also contains the solute (I) at a certain con-centration cl. Thus the curves for x 1 and z 2 are essentially different in character z will normally decrease as c increases but the decrease may be expected to be small compared with the corresponding decrease in 2,. This is the case if the “ strength ” at which solute (2) is adsorbed at ver 32 CHROMATOGRAPHY WITH SEVERAL SOLUTES low concentrations is not greatly affected by the presence of solute (I) in the solution.If x 1 and x 2 decrease but x 1 decreases more rapidly than z,, there will be a point at which xl=x2 (since initially x 1 is the larger). That is a cl' for which eqn. (22) is satisfied. Hence in this case when the column consisting of solutes (I) and ( 2 ) corresponding to constant concentrations clo and c20 (Fig. 2 (a)) is developed with a volume 'u of solvent at the top of the tube there will be a clear band (from o to x, Fig. 2 (6)). If then at a level x I' given by a band of solute (I) will occur the concentration at any level x (in the region from x to x,' ; Fig. 2 ( 6 ) ) being given by the concentration c1 of solute (I) within this band increases until it reaches a critical value cl' given by eqn.(22). After this there follows a mixed band (x ,' to x 2) where the concentrations c1 and c 2 are connected by the differential eqn. (12) subject to the conditions that when cl=cl' c,=o. This development continues to a certain level where a sharp change in the concentrations occurs and we arrive at the undeveloped point of the band with the concentrations at their original values of c and c 2 0 (Fig. 2(b)). Various other solutions may be possible in this case depending on the characters of the isotherms but the above solution appears to be of par-ticular interest. There is one further point of some importance. The chromatogram we have described is that obtained when an infinite band is developed. In practice the band will be of finite length of mixed solutes (I) and (2) followed usually by a band of the less strongly adsorbed solute (2).On developing with (pure) solvent the nature of the development at the top of the tube is clearly independent of the length of the band. The solution we have given then enables us to calculate the volume of solvent required to obtain com-ple te separation. Writing then the amount of solute (I) in the band between the levels x (see Fig. 2(6)) is (3q 1/3c l)c1=c2'o > (%I 2/3c 2)cl=c,=a 9 V/X1'= (341/~cl)c,=c,=o 9 * (24) 4% = (aqla/cl)c2_o 3 (25) Q(C1) = 4 1 h 0) ? and x and on integration by c ' [Q (C 1)xI bearing in mind that 0 Xl 0 parts this becomes Q(o) = 0. Now throughout this band? = Q' (cl) where v the volume X of course a constant and so the above expression becomes .(26.1) added is, (26.2 A. C. OFFORD AND JOSEPH WEISS 33 Xow suppose that the total amount of solute (I) present was ml. Then, the tube being of unit cross-section the volume vs required to obtain total separation will be where cl’ is given by (22). (iii) An infinite column of uniformly distributed solute is developed with a solution of a more strongly adsorbed substance.-Our simple equations are valid in this case. We shall suppose as usual that solute (I) is the more strongly adsorbed. In any mixed band dc2/dcl is negative and this implies that: If is small relative to the remaining terms this implies that Suppose a volume v of a solution of solute (I) is added then as we go down the tube (34 ,/3c increases as c decreases.Also for c 2=0 3q ,/3c is zero but not zero for c > 0. Hence the second term also increases (the product (?q ,/3c 2) (34 2/3c 1) being positive). Therefore both terms on the right-hand side increase which is impossible since with increasing x the left-hand side decreases. Hence we are led to a contradiction and must conclude that in this case there can be no variable band. Therefore the only solution possible is a discontinuous one. To satisfy eqn. (3) there must be two discontinuities separated by a mixed band. If cl’ and c2’ denote here the concentrations of the solutes in the mixed band then, and are the two simultaneous equations for c and c 2’. If x denotes the level of the top of the mixed band and x 2 the level of the bottom (Fig.3) after a volume ZJ has been added then by eqn. ( 3 ) since x=o when V=O, (iv) Band consisting of a uniform mixture of two substances (2) and (3) which is developed with a solution of a third substance ( 1 ) . -In this case we have in general a cubic equation to solve and so a detailed analysis would be somewhat complicated. We shall consider only the case when the substance (I) is more strongly adsorbed than the other two substances and even then we shall only instance two cases which might arise. 34 FRONTAL ANALYSIS (a) If the adsorption isotherms of the three substances are represented by q 1 q 2 and q 3 and if then there is no interaction between (I) and the mixture of the components (2) and (3). A band of solute (I) is formed at the top and the mixture (2) and (3) is developed as in Example (ii). then at the top of the tube we shall have a chromatogram corresponding to Example (iii). Altogether there may be fzve bands. In the to$ ban,d the substance (I) at concentration c l 0 . In the second band solutes (I) and (2) corresponding to concentrations cl’ and c2’. In the third b a d (2) at con-centration c2”. The fourth band may be a mixed band as in Example (i) and in thefzfth band (2) and (3) are present at their original concentrations c20 and c30 (see Fig. 4). The values of c l l c2’ and call are given by the following equations: The levels x and x corresponding to the top and bottom of the first mixed band are then given by * ( 3 5 4 _ - - q2(‘1‘J ‘21 O) E, * 1 c 2‘ University of London University of Durham, Birkbeck College King’s College, London E.C.4. Newcastle-upon- Tyne I

ISSN:0366-9033    

DOI:10.1039/DF9490700026

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

34 THEORY OF FRONTAL ANALYSIS AND DISPLACEMENT DEVELOPMENT BY STIG CLAESSON Received 29th August 1949 A simple theoretical model which is useful for practical applications of chromato- graphy is discussed for frontal analysis and displacement development. It can be said that the theory for chromatographic separation has been developed along two different lines. One group of workers has tried to use as simple theoretical models as possible in order to get simple but still useful results for the actual calculations when separating complex mixtures. Another STTG CLAESSON FIG. I. 35 group of wqrkers has tried to work out a very detailed theory for the very complex phenomena which take place in a column. Due to mathematical difficulties this type of work has mostly been carried out for systems with one or at most two solutes.Such results are usually of little use in practical applications of chromatography but they are of utmost importance when one is trying to get a deeper understanding of the chromatographic process in general or to reach the limit of resolving power as in isotope separation. In this paper the " simple " theory which is useful for practical applications of chromatography will be discussed for two rather important special cases namely front a1 analysis and displacement development. Frontal analysis introduced by Tiselius 1 in 1940 is the simplest possible arrangement for chromatography. The solution containing one or several solutes is forced through the column with adsorbent previously washed with pure solvent.The concentration of the effluent is determined and plotted as a function of the volume which has passed the filter. In this way characteristic curves are obtained showing one step for every solute. In Fig. I is seen a schematic curve for two solutes. The first step contains solute I the second step solute I and z and so on. The volume which has passed when a particular solute i breaks through is called the retention volume for component i. This volume includes the small volume necessary to replace the solvent in the column at the beginning of the experiment. After subtracting this small volume we get thc corrected retention volume ui. The corrected retention volume per gram adsorbent is called specific retention volume vg .Frontal analysis has the great advantage that the adsorbed substance does not have to be eluted again and this method can therefore be used even in the case of irreversible adsorption. That is of great value as it is often difficult to find adsorbents with sufficient selectivity which will give off the adsorbed substance quantitatively. The disadvantage is however that the heights of the steps are not proportional to the concentrations of the corresponding solutes in the original solution. The reason €or that is adsorption displacement which means that when solute z is passing down the column and adsorbed it will knock off some already adsorbed molecules of solute I from the adsorbent and consequently the concentration cI1 of solute T in step I is higher than in the original solution.We introduce the general notation ci,j for the concentration of solute i in step j . 1 Tiselius Arkiv. Kemi Min. Geol. B 1940 14 No. 22. FRONTAL ANALYSIS 36 For quantitative work it is therefore necessary to calculate the composition of the solution from the observed retention volumes and the heights of the steps. In order to get simple results of practical value for systems with many solutes it is necessary to make simplified assumptions. We assume instan- taneous equilibrium and perfectly plane fronts. The first treatment along these lines was given by de Vault and further discussed by the present authore3 In the following we will restrict the discussion to systems with only two solutes. Extension of the theory to an arbitrary number of solutes is quite simple 2-4 but will not bring forward anything new in principle.For a system with only one solute of concentration cll the amount adsorbed u; = fl(cll) per gram adsorbent in the column is evidently v4cll and we have zIo f l ( C l l > c11 1 - For two solutes we get in the same way . . 4 =f1 (c12 c22) = vx c12 - (VX - u;) c11 If we now introduce a suitable equation for the adsorption isotherm these expressions can be simplified considerably. The simplest results are obtained 4 = f 2 (c12 c22) = 0322. from Langmuir’s isotherm a = &22/(1 By dividing eqn. (2) and (3) we get - (2) - (3) * * (4) (5) 2 De Vault J . Amer. Chem. SOC. 1943 65 532. 4 = 4c12/(1 + 4c12 + J2c22) * + 4c12 + 4c22).We have thus obtained a very simple formula for the calculation of the correct concentration cI2 from the observed value cll. I t should be noted that ki is the retention volume for solute i at infinite dilution (from eqn. (I)). I n case of several solutes eqn. (6) will take the form I t is consequently possible to calculate all concentrations in this way provided that we know the k values which have to be determined from experiments with pure components or by other suitable methods. It must be borne in mind however that this displacement effect is not primarily dependent on the actual displacement process in the column. Changes of concentration of this type will occur in all systems where a separation is caused by differences in mobility between the different solutes and where these mobilities depend on the concentrations.These “ dis- placement ” effects which are a consequence of the conservation of the mass of the solute are also well known both in electrophoresis 5 and in 3 Claesson Arkiv. Kenzi Mitz. Geol. A 1946 23 NO. I . 4 Claesson Arkiv. Keuni Min. Geol. A 1946 24 No. 7 . 5 Dole J . Amer. Chew. SOC. 1946 42 769. 37 STIG CLAESSON ‘czz I G I 1% I proportional to the retention volumes. In fact as eqn. (9) only contains ratios of mobilities it will be sufficient to assume that ratios of mobilities are independent of the total concentration in order to get simple results. That is true both for electrophoresis and ult racentrifugation. It can be demonstrated in the same way that eqn.(7) will be unchanged even if more complicated isotherms than Langmuir’s are used as long as they have the same property I - - - - - - - - - - - - - - - - - -Fz Ogston and Johnston Trans. Faraduy Soc. 1946 42 769. Tiselius Arkiv. Kemi Min. Geol. A 1943 16 No. 18. CONSERVATION EQUATION OF CHROMATOGRAPHY 38 can apply eqn. (I) and as the specific retention volume then can be regarded as a measure of the rate we get wheref,(c,) is the adsorption isotherm for the developer. Consequently if the concentration of the developer is kept fixed in all experiments a particular solute i always has to be of the same concentration ci in order to fulfil eqn. (11). The height of a step is therefore independent of the amount of substance present and is a constant which can be used for identification of a substance.As the area of a step is proportional to the FIG. 3. amount of substance present and its height is constant it is obvious that the length of a step (b) is proportional to the amount of substance. We therefore get a qualitative analysis by measuring the heights of the steps and a quantitative by measuring their lengths. Displacement development is therefore a very elegant way of carrying out chromatographic separations as it can be used for preparative quali- tative and quantitative purposes at the same time. I t has however the drawback common to all methods using reversible adsorption that it may be difficult to find a developer which will displace the unknown mixture quantitatively.Fysikalis k- Kemiska Institutionen Uppsala. 34 THEORY OF FRONTAL ANALYSIS AND DISPLACEMENT DEVELOPMENT BY STIG CLAESSON Received 29th August 1949 A simple theoretical model which is useful for practical applications of chromato-graphy is discussed for frontal analysis and displacement development. It can be said that the theory for chromatographic separation has been developed along two different lines. One group of workers has tried to use as simple theoretical models as possible in order to get simple but still useful results for the actual calculations when separating complex mixtures. Anothe STTG CLAESSON 35 group of wqrkers has tried to work out a very detailed theory for the very complex phenomena which take place in a column.Due to mathematical difficulties this type of work has mostly been carried out for systems with one or at most two solutes. Such results are usually of little use in practical applications of chromatography but they are of utmost importance when one is trying to get a deeper understanding of the chromatographic process in general or to reach the limit of resolving power as in isotope separation. In this paper the " simple " theory which is useful for practical applications of chromatography will be discussed for two rather important special cases, namely front a1 analysis and displacement development. Frontal analysis introduced by Tiselius 1 in 1940 is the simplest possible arrangement for chromatography. The solution containing one or several solutes is forced through the column with adsorbent previously washed with pure solvent.The concentration of the effluent is determined and plotted as a function of the volume which has passed the filter. In this way characteristic curves are obtained showing one step for every solute. In Fig. I is seen a schematic curve for two solutes. The first step contains FIG. I. solute I the second step solute I and z and so on. The volume which has passed when a particular solute i breaks through is called the retention volume for component i. This volume includes the small volume necessary to replace the solvent in the column at the beginning of the experiment. After subtracting this small volume we get thc corrected retention volume ui. The corrected retention volume per gram adsorbent is called specific retention volume vg .Frontal analysis has the great advantage that the adsorbed substance does not have to be eluted again and this method can therefore be used even in the case of irreversible adsorption. That is of great value as it is often difficult to find adsorbents with sufficient selectivity which will give off the adsorbed substance quantitatively. The disadvantage is however that the heights of the steps are not proportional to the concentrations of the corresponding solutes in the original solution. The reason €or that is adsorption displacement which means that when solute z is passing down the column and adsorbed it will knock off some already adsorbed molecules of solute I from the adsorbent and consequently the concentration cI1 of solute T in step I is higher than in the original solution.We introduce the general notation ci,j for the concentration of solute i in step j . 1 Tiselius Arkiv. Kemi Min. Geol. B 1940 14 No. 22 36 FRONTAL ANALYSIS For quantitative work it is therefore necessary to calculate the composition of the solution from the observed retention volumes and the heights of the steps. In order to get simple results of practical value for systems with many solutes it is necessary to make simplified assumptions. We assume instan-taneous equilibrium and perfectly plane fronts. The first treatment along these lines was given by de Vault and further discussed by the present authore3 In the following we will restrict the discussion to systems with only two solutes.Extension of the theory to an arbitrary number of solutes is quite simple 2-4 but will not bring forward anything new in principle. For a system with only one solute of concentration cll the amount adsorbed u; = fl(cll) per gram adsorbent in the column is evidently v4cll and we have . . zIo f l ( C l l > 1 -c11 For two solutes we get in the same way : 4 =f1 (c12 c22) = vx c12 - (VX - u;) c11 - (2) 4 = f 2 (c12 c22) = 0322. - (3) If we now introduce a suitable equation for the adsorption isotherm these expressions can be simplified considerably. The simplest results are obtained from Langmuir’s isotherm 4 = 4c12/(1 + 4c12 + J2c22) * * (4) a = &22/(1 + 4c12 + 4c22). * (5) By dividing eqn. (2) and (3) we get We have thus obtained a very simple formula for the calculation of the correct concentration cI2 from the observed value cll.I t should be noted that ki is the retention volume for solute i at infinite dilution (from eqn. (I)). I n case of several solutes eqn. (6) will take the form I t is consequently possible to calculate all concentrations in this way provided that we know the k values which have to be determined from experiments with pure components or by other suitable methods. It must be borne in mind however that this displacement effect is not primarily dependent on the actual displacement process in the column. Changes of concentration of this type will occur in all systems where a separation is caused by differences in mobility between the different solutes and where these mobilities depend on the concentrations.These “ dis-placement ” effects which are a consequence of the conservation of the mass of the solute are also well known both in electrophoresis 5 and in 2 De Vault J . Amer. Chem. SOC. 1943 65 532. 3 Claesson Arkiv. Kenzi Mitz. Geol. A 1946 23 NO. I . 4 Claesson Arkiv. Keuni Min. Geol. A 1946 24 No. 7 . 5 Dole J . Amer. Chew. SOC. 1946 42 769 STIG CLAESSON 37 proportional to the retention volumes. mobilities it will be sufficient to assume that In fact as eqn. (9) only contains ratios of ratios of mobilities are independent of the I ‘czz 1% total concentration in order to get simple results. That is true both for electrophoresis and ult racentrifugation. in the same way that eqn. (7) will be unchanged even if more complicated isotherms than Langmuir’s are used as long as they have the same property It can be demonstrated - - - -Ogston and Johnston Trans.Faraduy Soc. 1946 42 769. Tiselius Arkiv. Kemi Min. Geol. A 1943 16 No. 18. I G I - - -Fz I - - - - - - - - - - 38 CONSERVATION EQUATION OF CHROMATOGRAPHY can apply eqn. (I) and as the specific retention volume then can be regarded as a measure of the rate we get wheref,(c,) is the adsorption isotherm for the developer. Consequently if the concentration of the developer is kept fixed in all experiments a particular solute i always has to be of the same concentration ci in order to fulfil eqn. (11). The height of a step is therefore independent of the amount of substance present and is a constant which can be used for identification of a substance. As the area of a step is proportional to the FIG. 3. amount of substance present and its height is constant it is obvious that the length of a step (b) is proportional to the amount of substance. We therefore get a qualitative analysis by measuring the heights of the steps and a quantitative by measuring their lengths. Displacement development is therefore a very elegant way of carrying out chromatographic separations as it can be used for preparative quali-tative and quantitative purposes at the same time. I t has however the drawback common to all methods using reversible adsorption that it may be difficult to find a developer which will displace the unknown mixture quantitatively. Fysikalis k- Kemiska Institutionen, Uppsala

ISSN:0366-9033    

DOI:10.1039/DF9490700034

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

THE CONSERVATION EQUATION OF CHROMATOGRAPHY BY W. M. SMIT Received 18th July 1949 The conservation equation of chromatography thus far used does not account for the volume occupied by the adsorbed phase. When the equation of conservation of matter is applied to the process of adsorptive percolation and this volume is accounted for a general equation is obtained which may be more correct. This is shown by applying this equation to the equilibrium theory. Several effects occurring in chromato- graphy which cannot be explained by the de Vault equation may now be interpreted a t least qualitatively. Moreover the equation obtained reveals the possibility of a new method of adsorptive percolation. W. M. SMTT 39 In the development of the mathematical description of the process of adsorptive percolation two different methods of approach may be distin- guished.One may start with the assumption of instantaneous equilibrium between solution and adsorbent (equilibrium theory) or one may dismiss the idea of complete equilibrium and treat the problem as a kinetic one (kinetic theory). However in both cases the law of conservation of matter supplies the initial equation which in both cases must be fundamentally the same. The equations obtained thus far give a ready explanation of the phenomena occurring during development. But when applying these equations in explanation of other processes of adsorptive percolation like the continuous introduction method (frontal analysis) or the elution method (displacement method) difficulties are encountered.Moreover the occurrence of so-called adsorption azeotropes,l especially with closely related compounds needs further interpretation. Some experience gained with the adsorptive percolation of hydrocarbons suggested that the failures mentioned might be caused by the use of an approximate conservation equation. It also represents the weight of are different functions. Symbols The following symbols are used in this paper :- a interstitial volume of an adsorption column per unit of length (ml./cm.). b index referring to a sharp boundary. C concentration of solute expressed as units of weight of solute per unit of volume of solution (g./ml.). CA CB concentrations of substance A and substance B respectively. d density (g./ml.).f f l ( C ~ ) adsorption isotherm of component N. solute adsorbed per unit of weight of adsorbent a t the concentration CN (g./g.). &(CN) first derivative of the adsorption isotherm (dfn(CN) /dCN) m weight of adsorbent contained in the column per unit of length (g./cm.). f ( q ) volume occupied by the adsorbed phase (solute + solvent) per unit of q weight of solute adsorbed per unit of length of a column (g./cm.). length if a quantity q of solute is adsorbed (ml./cm.). S solvent. V volume of liquid forced into the column (ml.). v interstitial volume of dry adsorbent present in the column per unit of weight (ml*/g.) * x distance from the top of the column (cm.). 0 I (when combined with C) indices indicating zero and unit concentration respectively .I 2 3 (when combined with f) indices indicating that the functions fl fi and fs tion equation was given by Wilson 1. The Conservation Equations used thus far.-The first conserva- where C is the concentration of the solute x the distance from the top of the column q the weight of solute adsorbed per unit of length and V the volume of liquid poured into the adsorption column. This equation has been corrected by de Vault 3 for the interstitial volume a per unit of length of the adsorption column. De Vault arrives at the equation 1 Hirschler and Amon. Ind. Eng. Chenz 1947 39 1565. 2 Wilson J . Amer. Chem. SOC. 1940 62 1583. 3 De Vault J . Amer. Chem. SOC. 1943 65 532. CONSERVATION EQUATION OF CHROMATOGRAPHY 40 The same equation has been used implicitly or explicitly by other authors whether they adopted the equilibrium or the kinetic standpoint for the equilibrium or the kinetic conditions are introduced subsequently when the relation between C and q is inserted.Eqn. (2) is based on the assumption that the interstitial volume a is constant. This is the point where intro- duction of a correction might be useful. The adsorbed phase occupies a certain fraction of the original interstitial volume thus causing the liquid to flow at a greater linear rate than when the adsorbed phase is not present. As the quantity of material adsorbed is a function of the concentration the free interstitial volume that is the volume available to the jlowing liquid must be a function of the concentration.2. Derivation of the Corrected Conservation Equation.-To intro- duce the correction for the volume of the adsorbed phase the conservation equation will be derived once more for a solution containing a single solute. The following assumptions which do not differ from those usually applied have been made. Within an adsorption band the concentration is a con- tinuous function of x. Within a cross-section of infinitesimal length & (3C/3x)v is considered constant. The same applies to (3q/3V) when an infinitesimal volume of liquid dV is introduced into the column. When an infinitesimal volume dV is introduced into the column an equal volume has to pass through a cross-section at x where x is a point within the adsorption band. As the concentration of the solution entering the cross-section at x and the concentration of the solution leaving the cross- section at (x + dx) differ by (3Cj3X)V h the amount of solute present in this section of the column is increased by (ac/3x)V dx.dv.This amount is distributed between the liquid phase and the adsorbed phase present in the section. The volume of the liquid phase present in the section is equal to (a -f(d)dx wheref(q) means the volume of the adsorbed phase per unit of length within the section considered. Thus the amount of solute present in the liquid phase is equal t o (a - f(4))Cdx. The amount of solute present in the adsorbed phase is equal to qdx where 4. = mfn(C,). Thus the total amount of solute present in the section is equal t o If an infinitesimal volume dV is forced through this section the increase of solute may therefore also be represented by (a - f W + 4 ) h .According to the law of conservation of matter the amount of solute intro- duced by the volume dV should be equal to the increase of the amount of solute present in the section. Thus 'Martin and Synge Biocheun. J. 1941 35 1941 Weiss J . Chew. sot. 1943 '45. 297. Thomas Ann. N.Y. Acad. Sci. 1948 49 161. W. M. SMIT 41 This is the corrected conservation equation of chromatography in its general form for a solution containing a single solute. As to.f(q) it should be borne in mind that it is not correct to put by f l ( C A ) . At equilibrium f ( d = 4 M where d means the density of the adsorbed solute.For besides the solute the solvent is adsorbed too. As t o this point it might be useful to stress that adsorption isotherms of pure gases and so-called isotherms of solutions containing a single solute usually have a different meaning. The adsorption isotherm of a pure gas shows the relation between the concentration !(pressure) of the gas and the weight adsorbed. However the so-called adsorption isotherm of a solute shows the relation between the concentration and the weight of solute adsorbed preferentially. ‘This relation might be better called a preferential adsorption isotherm. ‘To avoid confusion it may be stated here that in this paper the adsorption isotherm of a solute means the relation between the total amount of solute .adsorbed per unit of weight of adsorbent and the concentration of the :solution with which it is in equilibrium.3. Application of Eqn. (3) to the Equilibrium Theory.-Attempts will now be made to derive a fundamental equation of chromatography ,on the basis of the equilibrium theory for a liquid mixture of two substances A and B. Which of the two is considered as solvent does not matter. For the sake of convenience it is assumed that A and B have equal densities and no volume change occurs on adsorption or mixing. If the densities of A and B are represented by d and the concentration is expressed as g./ml. we may put C.4 + CB = d . Suppose the adsorption column considered contains m g. adsorbent per unit of length. The adsorption isotherm of A when diluted by B is represented y = mfi (CAI (for non-equilibrium another relation between q and C should be introduced but this will not be discussed).If f2(C4 represents the adsorption isotherm of B when diluted by A we may put and or f ( 4 ) = {fi(cA) + f d C B > Imld* If ZI is the interstitial volume of the dry adsorbent per unit of weight Substitution in eqn. (3) gives a = mu. (3cA/3x)V + m[(v -fi(cA)/a -f2(cB)/d>(acA/av)z - { (afl(cA)/3v)x + (3f2(CB)/3v)z)CA/d + (af,(cA)/av)%l = 0. On further substitution viz. ( ~ C A / ~ X ) V = - (3V/ax)cA (~CA/~V)X (afi (C A) /a v) = ( J f l ( CA) / ~ C A ) (JCA/J v) = f i c ~ ( ~CA/J V)z (3f,cB/3v)x = (afz(cB)/acB) (acB/av) = - fkcB(aCA/aV)~ the following equation is obtained - (dlr/Jx)~,(aC~/’~)% + mCv - f l ( c ~ ) / d - j 2 ( C ~ ) / d - {fi(c~) - fS(c~) ) C A / ~ fi(CA)I(acA/av)~ = 0 - PviWc + [vd - f l ( c ~ ) - f z ( C ~ ) + CB~~(CA) + C,fh(C~)]m/d = 0.B* 42 CONSERVATION EQUATION OF CHROMATOGRAPHY Thus (bX/bV)C = - _ _ _ ~ m[vd - f i ( c A ) -fi(cB) -k CBf;(CA) + c A f L ( c B ) l * a T concentrations thus if b( (&v/bV)CA}/aCA is positive it means that the highest concentration of A possible will move at the greatest rate. Consequently it will overtake all lower concentrations of A which might be in front of it. Thus a sharp front boundary will be finally formed and the lowest concentration possible will be trailing behind forming a diffuse boundary. If however b{ (bX/bV)C,}/bcA is negative the inverse will be obtained.A further calculation which will not be extended here shows that if adsorption occurs according to the Langmuir theory of adsorption a leading front boundary of A will be formed if ~ I ( ~ A > / ~ A > f2(cB>/cB holds good at all concentrations. This simple condition only means that the concentration of A in the liquid phase decreases if adsorbent is added to any mixture of A and B. If the adsorption isotherms of A and B are of the Freundlich type and fl(c~>/c~ > fi(c~)/C~ a leading front boundary will also be formed at least at low or high con- centrations. At medium concentrations no general conclusion can be drawn. Thus it may be assumed if f i ( c A > / c A > f2(cB)/cB a sharp front boundary of A will generally be formed.As has already been stated eqn. (4) only holds good within an adsorption band. When a sharp boundary occurs whether in front or in the rear another conservation equation has to be applied to the boundary. For- either in front or at the rear of such a boundary no trace of the substance considered should be present. Therefore the conservation equation of a sharp boundary is CdV = (a - f ( 4 ) ) C + q l h where C applies to the concentration in the boundary. If eqn. (5) is applied to the mixture of A and B and equilibrium is instantaneously established the following equation holds good for a sharp boundary where the subscript b indicates that the concentration refers to that in the sharp boundary. (4) De Vault,3 whose paper has been very useful to the author assumed the interstitial volume to be constant and obtained According to the initial assumptions eqn.(4) holds good only within the adsorption band. In the same way as de Vault did it can be shown that at a boundary of an adsorption band eqn. (4) may lead to physical impos- sibilities thus indicating the occurrence of a discontinuity (sharp boundary). If 'v is increased at a constant rate (3~/bV)c may be called the rate of transport of the concentration C A . If (bx/bV)cA increases with increasing (5) -~ W. M. SMIT and are obtained. 43 For the diffuse boundary (zero concentration) the following equation is obtained where the subscripts o and I indicate zero and unit concentration d respec- tively.Applying de Vault’s theory 4. Discussion.-Eqn. (4) represents the rate of transport of a certain concentration of A within the adsorption band. But the corresponding concentration of B should attain the same value. Thus Consequently (a@ Y) c = (a@ V> c, if CB = d - CF,. Eqn. (4) meets this condition whereas eqn. (4a) does not f:(CA) =fi(CB) unless which as a rule cannot be true. As has already been shown a sharp leading boundary of A will be formed if f i ( c A ) / c A > fi(CB)/CB- (ax/3VC* zvj I (eqn. (6)). This means that the rate of transport of any boundary is always smaller than the rate of transport of the liquid front which of course is equal to Both rates become equal if Cg = 0 i.e. CA = d. If however r/mvd. fi(cA)/cA < f i ( C B ) / c B the front boundary will be diffuse and a sharp boundary will be present at the rear.In this case the rate of transport of the boundaries is always greater than the rate of transport of the liquid front unless C A = d. Thus it may be stated if f i ( C d / C ~ > f & d / C ~ substance A will always lose on the liquid front whereas B has a tendency to overtake the liquid front until its concentration d has become equal to unity. This may serve as a suitable interpretation of the phenomena which obtain when a solution containing a single solute is introduced into the dry column (continuous introduction method). If a mixture of A and B is introduced into a dry column A will move at a slower rate than the liquid front. Therefore pure B will appear in front of the adsorption band and its quantity will increase if introduction is continued.Since at the top of the column the concentration of A is equal to the initial concentration and this is the highest Concentration of A possible under these conditions it will overtake all lower concentrations of A which might be in front of it. Thus within the adsorption band the concentration of A is equal to the initial concentration and the adsorption band will move at a constant rate. Elution (displacement) may now be explained too. Suppose two sub- stances A and B are present in an adsorption column and a new solvent S is added at the top of the column. Further .f3(cS>/cS >fi(cA)/c.I > f i ( c B > / c B . CONSERVATION EQUATION OF CHROMATOGRAPHY 44 According to the above deductions any A or B which happens to become diluted by S will move at a greater rate than the liquid front whereas the rate of S should be smaller than (or when undiluted equal to) the rate of the liquid front.Therefore S will remain undiluted and act like a piston forcing A and B in front of it. The same applies to A as compared to B. Therefore the final result of elution will be three adjacent adsorption bands containing S A and B at unit concentrations As to the developer method little further explanation seems to be necessary. If more than one solute is present and separation of the solutes has been obtained already the rates of transport of the different solutes are established by eqn. (6) and (7) and thus the distance between the adsorption bands will usually increase on further development and at the same time the bands will be broadened.The mere process of separation of a mixture of solutes is more com- plicated because of mutual alteration of the adsorption isotherms. Never- theless it may be accepted that the separation itself occurs qualitatively along the same lines. As a rule when the developer method is applied a sharp front boundary occurs. Experimentally asymmetric adsorption bands are usually found the highest concentration being near the front. Eqn. (6) and (7) reveal another possibility which may not as yet have been recognized. An adsorption band of A when present will move at a greater rate than the liquid front But if the whole column has previously been wetted with B the substance A cannot reach the liquid front before leaving the column.If a mixture of A and B is introduced into a column previously wetted with B the lowest concentration of A will move at the greatest rate. Thus an adsorption band with a diffuse front and a sharp rear boundary will be formed immediately. This band is transported through the column if there- upon pure B is added to the column at the top. The band is broadened during this procedure but its rate is conditioned by eqn. (6) and (7). If still another solute say D is present which is also adsorbed less strongly than B it will behave in the same way but it may be that both limits of the rate of transport of the adsorption band of D are smaller than those of B.Thus this method which is supposed to be called development with an eluent may cause separation. According to former conceptions this method would be impossible because B the eluent should not allow A or D to become adsorbed. However a very simple experiment showed the present conception to be correct. A mixture of cetane and cetene could be partly separated by silica gel previously wetted with benzene using benzene as developing liquid. I t is well known that benzene may act as an eluent (displacer) for cetane and cetene when adsorbed on silica gel. The advantage of this method is that the total volume of " developer " necessary to collect all the solutes at the bottom of the column is smaller than the interstitial volume of the column.This means a saving of time. However the separating efficiency of this method may be in many cases low. The occurrence of rates of transport greater than the rate of the liquid front is also demonstrated in an earlier paper,5 though not mentioned explicitly. Rates of transport up to about 1-3 times the rate of the liquid front have been found with chloroform and dodecylbenzene. A few words may be added as to the occurrence of so-called adsorption azeotropes as described by Hirschler and Am0n.l When an adsorption azeotrope occurs fi (c A ) / c A - 5 Smit Anal. chim. A d a 1948. 2 671. w. M. SMIT At the azeotropic concentration ~ ~ ( C A ) / C A -~~(CB)/CB = 0. 45 changes sign at a certain concentration of say A. At concentrations lower than the azeotropic concentration the sign may be positive and when the mixture is introduced into the column pure B appears in front of the adsorption band containing A.Beyond the azeotropic concentration pure A will appear in front and B is contained within the adsorption band. The rate of transport of A and B both become equal to the rate of the liquid front and no separation occurs. So the conclusion derived from eqn. (6) and (7) agrees with experiment. At the same time it is clear that the occurrence of adsorption azeotropes is limited to substances having comparable adsorption affinities. No attempt has been made to solve the differential equations nor to develop formulze for more than two substances. As de Vault already stated this becomes very complicated.Moreover as real adsorption isotherms are used in our equation the quantitative solution is of no use as the real adsorption isotherms are not available. The main purpose of this paper has been to arrive at a rather simple formula which permits a qualitative explanation of the different methods of percolation but on the other hand it might show that the quantitative deductions made thus far have to be handled with care. Acknowledgment is due to the Management of the Bataafsche Petroleum Maatschappij for their permission to publish this paper. KoninklijkelShell-Laboratorium Amsterdam. THE CONSERVATION EQUATION OF CHROMATOGRAPHY BY W. M. SMIT Received 18th July 1949 The conservation equation of chromatography thus far used does not account for the volume occupied by the adsorbed phase.When the equation of conservation of matter is applied to the process of adsorptive percolation and this volume is accounted for a general equation is obtained which may be more correct. This is shown by applying this equation to the equilibrium theory. Several effects occurring in chromato-graphy which cannot be explained by the de Vault equation may now be interpreted, a t least qualitatively. Moreover the equation obtained reveals the possibility of a new method of adsorptive percolation W. M. SMTT 39 In the development of the mathematical description of the process of adsorptive percolation two different methods of approach may be distin-guished. One may start with the assumption of instantaneous equilibrium between solution and adsorbent (equilibrium theory) or one may dismiss the idea of complete equilibrium and treat the problem as a kinetic one (kinetic theory).However in both cases the law of conservation of matter supplies the initial equation which in both cases must be fundamentally the same. The equations obtained thus far give a ready explanation of the phenomena occurring during development. But when applying these equations in explanation of other processes of adsorptive percolation like the continuous introduction method (frontal analysis) or the elution method (displacement method) difficulties are encountered. Moreover the occurrence of so-called adsorption azeotropes,l especially with closely related compounds needs further interpretation.Some experience gained with the adsorptive percolation of hydrocarbons suggested that the failures mentioned might be caused by the use of an approximate conservation equation. Symbols The following symbols are used in this paper :-a interstitial volume of an adsorption column per unit of length (ml./cm.). b index referring to a sharp boundary. C concentration of solute expressed as units of weight of solute per unit of CA CB concentrations of substance A and substance B respectively. d density (g./ml.). f f l ( C ~ ) adsorption isotherm of component N. &(CN) first derivative of the adsorption isotherm (dfn(CN) /dCN), m weight of adsorbent contained in the column per unit of length (g./cm.). q weight of solute adsorbed per unit of length of a column (g./cm.).f ( q ) volume occupied by the adsorbed phase (solute + solvent) per unit of S solvent. V volume of liquid forced into the column (ml.). v interstitial volume of dry adsorbent present in the column per unit of weight x distance from the top of the column (cm.). 0 I (when combined with C) indices indicating zero and unit concentration I 2 3 (when combined with f) indices indicating that the functions fl fi and fs volume of solution (g./ml.). It also represents the weight of solute adsorbed per unit of weight of adsorbent a t the concentration CN (g./g.). length if a quantity q of solute is adsorbed (ml./cm.). (ml*/g.) * respectively . are different functions. 1. The Conservation Equations used thus far.-The first conserva-tion equation was given by Wilson : where C is the concentration of the solute x the distance from the top of the column q the weight of solute adsorbed per unit of length and V the volume of liquid poured into the adsorption column.This equation has been corrected by de Vault 3 for the interstitial volume a per unit of length of the adsorption column. De Vault arrives at the equation : 1 Hirschler and Amon. Ind. Eng. Chenz 1947 39 1565. 2 Wilson J . Amer. Chem. SOC. 1940 62 1583. 3 De Vault J . Amer. Chem. SOC. 1943 65 532 40 CONSERVATION EQUATION OF CHROMATOGRAPHY The same equation has been used implicitly or explicitly by other authors whether they adopted the equilibrium or the kinetic standpoint for the equilibrium or the kinetic conditions are introduced subsequently when the relation between C and q is inserted.Eqn. (2) is based on the assumption that the interstitial volume a is constant. This is the point where intro-duction of a correction might be useful. The adsorbed phase occupies a certain fraction of the original interstitial volume thus causing the liquid to flow at a greater linear rate than when the adsorbed phase is not present. As the quantity of material adsorbed is a function of the concentration the free interstitial volume that is the volume available to the jlowing liquid must be a function of the concentration. 2. Derivation of the Corrected Conservation Equation.-To intro-duce the correction for the volume of the adsorbed phase the conservation equation will be derived once more for a solution containing a single solute.The following assumptions which do not differ from those usually applied, have been made. Within an adsorption band the concentration is a con-tinuous function of x. Within a cross-section of infinitesimal length &, (3C/3x)v is considered constant. The same applies to (3q/3V) when an infinitesimal volume of liquid dV is introduced into the column. When an infinitesimal volume dV is introduced into the column an equal volume has to pass through a cross-section at x where x is a point within the adsorption band. As the concentration of the solution entering the cross-section at x and the concentration of the solution leaving the cross-section at (x + dx) differ by the amount of solute present in this section of the column is increased by This amount is distributed between the liquid phase and the adsorbed phase present in the section.The volume of the liquid phase present in the section is equal to wheref(q) means the volume of the adsorbed phase per unit of length within the section considered. Thus the amount of solute present in the liquid phase is equal t o The amount of solute present in the adsorbed phase is equal to qdx where 4. = mfn(C,). Thus the total amount of solute present in the section is equal t o If an infinitesimal volume dV is forced through this section the increase of solute may therefore also be represented by (3Cj3X)V h, (ac/3x)V dx.dv. (a -f(d)dx, (a - f(4))Cdx. (a - f W + 4 ) h . According to the law of conservation of matter the amount of solute intro-duced by the volume dV should be equal to the increase of the amount of solute present in the section.Thus 'Martin and Synge Biocheun. J. 1941 35 1941 Weiss J . Chew. sot. 1943 '45. 297. Thomas Ann. N.Y. Acad. Sci. 1948 49 161 W. M. SMIT 41 This is the corrected conservation equation of chromatography in its As to.f(q) it should be borne in mind that it is not correct to put general form for a solution containing a single solute. f ( d = 4 M where d means the density of the adsorbed solute. For besides the solute the solvent is adsorbed too. As t o this point it might be useful to stress that adsorption isotherms of pure gases and so-called isotherms of solutions containing a single solute usually have a different meaning. The adsorption isotherm of a pure gas shows the relation between the concentration !(pressure) of the gas and the weight adsorbed.However the so-called adsorption isotherm of a solute shows the relation between the concentration and the weight of solute adsorbed preferentially. ‘This relation might be better called a preferential adsorption isotherm. ‘To avoid confusion it may be stated here that in this paper the adsorption isotherm of a solute means the relation between the total amount of solute .adsorbed per unit of weight of adsorbent and the concentration of the :solution with which it is in equilibrium. 3. Application of Eqn. (3) to the Equilibrium Theory.-Attempts will now be made to derive a fundamental equation of chromatography ,on the basis of the equilibrium theory for a liquid mixture of two substances A and B.Which of the two is considered as solvent does not matter. For the sake of convenience it is assumed that A and B have equal densities and no volume change occurs on adsorption or mixing. If the densities of A and B are represented by d and the concentration is expressed as g./ml., we may put Suppose the adsorption column considered contains m g. adsorbent per unit of length. The adsorption isotherm of A when diluted by B is represented by f l ( C A ) . At equilibrium, (for non-equilibrium another relation between q and C should be introduced but this will not be discussed). If f2(C4 represents the adsorption isotherm of B when diluted by A we may put C.4 + CB = d . y = mfi (CAI f ( 4 ) = {fi(cA) + f d C B > Imld* If ZI is the interstitial volume of the dry adsorbent per unit of weight, a = mu.Substitution in eqn. (3) gives (3cA/3x)V + m[(v -fi(cA)/a -f2(cB)/d>(acA/av)z - { (afl(cA)/3v)x + (3f2(CB)/3v)z)CA/d + (af,(cA)/av)%l = 0. On further substitution viz., ( ~ C A / ~ X ) V = - (3V/ax)cA (~CA/~V)X, (afi (C A) /a v) = ( J f l ( CA) / ~ C A ) (JCA/J v) = f i c ~ ( ~CA/J V)z and (3f,cB/3v)x = (afz(cB)/acB) (acB/av) = - fkcB(aCA/aV)~, the following equation is obtained : - (dlr/Jx)~,(aC~/’~)% + mCv - f l ( c ~ ) / d - j 2 ( C ~ ) / d - {fi(c~) - fS(c~) ) C A / ~ or fi(CA)I(acA/av)~ = 0, - PviWc + [vd - f l ( c ~ ) - f z ( C ~ ) + CB~~(CA) + C,fh(C~)]m/d = 0. B 42 CONSERVATION EQUATION OF CHROMATOGRAPHY * (4) Thus a (bX/bV)C = - _ _ _ ~ m[vd - f i ( c A ) -fi(cB) -k CBf;(CA) + c A f L ( c B ) l De Vault,3 whose paper has been very useful to the author assumed the interstitial volume to be constant and obtained T According to the initial assumptions eqn.(4) holds good only within the adsorption band. In the same way as de Vault did it can be shown that at a boundary of an adsorption band eqn. (4) may lead to physical impos-sibilities thus indicating the occurrence of a discontinuity (sharp boundary). If 'v is increased at a constant rate (3~/bV)c may be called the rate of transport of the concentration C A . If (bx/bV)cA increases with increasing concentrations thus if b( (&v/bV)CA}/aCA is positive, it means that the highest concentration of A possible will move at the greatest rate. Consequently it will overtake all lower concentrations of A which might be in front of it.Thus a sharp front boundary will be finally formed and the lowest concentration possible will be trailing behind, forming a diffuse boundary. b{ (bX/bV)C,}/bcA is negative, the inverse will be obtained. A further calculation which will not be extended here shows that if adsorption occurs according to the Langmuir theory of adsorption a leading, front boundary of A will be formed if ~ I ( ~ A > / ~ A > f2(cB>/cB holds good at all concentrations. This simple condition only means that the concentration of A in the liquid phase decreases if adsorbent is added to any mixture of A and B. If the adsorption isotherms of A and B are of the Freundlich type and fl(c~>/c~ > fi(c~)/C~, a leading front boundary will also be formed at least at low or high con-centrations.At medium concentrations no general conclusion can be drawn. Thus it may be assumed if a sharp front boundary of A will generally be formed. As has already been stated eqn. (4) only holds good within an adsorption band. When a sharp boundary occurs whether in front or in the rear, another conservation equation has to be applied to the boundary. For-either in front or at the rear of such a boundary no trace of the substance considered should be present. If however, f i ( c A > / c A > f2(cB)/cB, Therefore the conservation equation of a sharp boundary is CdV = (a - f ( 4 ) ) C + q l h (5) where C applies to the concentration in the boundary. If eqn. (5) is applied to the mixture of A and B and equilibrium is instantaneously established the following equation holds good for a sharp boundary : -~ where the subscript b indicates that the concentration refers to that in the sharp boundary W.M. SMIT 43 For the diffuse boundary (zero concentration) the following equation is obtained : where the subscripts o and I indicate zero and unit concentration d respec-tively. Applying de Vault’s theory, and are obtained. 4. Discussion.-Eqn. (4) represents the rate of transport of a certain concentration of A within the adsorption band. But the corresponding concentration of B should attain the same value. Thus if CB = d - CF,. Eqn. (4) meets this condition whereas eqn. (4a) does not, which as a rule cannot be true. (a@ Y) c = (a@ V> c,, unless f:(CA) =fi(CB), As has already been shown a sharp leading boundary of A will be formed if f i ( c A ) / c A > fi(CB)/CB-Consequently I (ax/3VC* zvj (eqn.(6)). This means that the rate of transport of any boundary is always smaller than the rate of transport of the liquid front which of course is equal to r/mvd. If however, fi(cA)/cA < f i ( C B ) / c B , the front boundary will be diffuse and a sharp boundary will be present at the rear. In this case the rate of transport of the boundaries is always greater than the rate of transport of the liquid front unless C A = d. Thus f i ( C d / C ~ > f & d / C ~ it may be stated if substance A will always lose on the liquid front whereas B has a tendency to overtake the liquid front until its concentration d has become equal to unity.This may serve as a suitable interpretation of the phenomena which obtain when a solution containing a single solute is introduced into the dry column (continuous introduction method). If a mixture of A and B is introduced into a dry column A will move at a slower rate than the liquid front. Therefore pure B will appear in front of the adsorption band and its quantity will increase if introduction is continued. Since at the top of the column the concentration of A is equal to the initial concentration and this is the highest Concentration of A possible under these conditions it will overtake all lower concentrations of A which might be in front of it. Thus within the adsorption band the concentration of A is equal to the initial concentration and the adsorption band will move at a constant rate.Suppose two sub-stances A and B are present in an adsorption column and a new solvent S is added at the top of the column. Both rates become equal if Cg = 0 i.e. CA = d. Elution (displacement) may now be explained too. Further .f3(cS>/cS >fi(cA)/c.I > f i ( c B > / c B 44 CONSERVATION EQUATION OF CHROMATOGRAPHY According to the above deductions any A or B which happens to become diluted by S will move at a greater rate than the liquid front whereas the rate of S should be smaller than (or when undiluted equal to) the rate of the liquid front. Therefore S will remain undiluted and act like a piston forcing A and B in front of it. The same applies to A as compared to B.Therefore the final result of elution will be three adjacent adsorption bands containing S A and B at unit concentrations As to the developer method, little further explanation seems to be necessary. If more than one solute is present and separation of the solutes has been obtained already the rates of transport of the different solutes are established by eqn. (6) and (7) and thus the distance between the adsorption bands will usually increase on further development and at the same time the bands will be broadened. The mere process of separation of a mixture of solutes is more com-plicated because of mutual alteration of the adsorption isotherms. Never-theless it may be accepted that the separation itself occurs qualitatively along the same lines.As a rule when the developer method is applied a sharp front boundary occurs. Experimentally asymmetric adsorption bands are usually found the highest concentration being near the front. Eqn. (6) and (7) reveal another possibility which may not as yet have been recognized. An adsorption band of A when present will move at a greater rate than the liquid front But if the whole column has previously been wetted with B, the substance A cannot reach the liquid front before leaving the column. If a mixture of A and B is introduced into a column previously wetted with B the lowest concentration of A will move at the greatest rate. Thus an adsorption band with a diffuse front and a sharp rear boundary will be formed immediately. This band is transported through the column if there-upon pure B is added to the column at the top.The band is broadened during this procedure but its rate is conditioned by eqn. (6) and (7). If still another solute say D is present which is also adsorbed less strongly than B it will behave in the same way but it may be that both limits of the rate of transport of the adsorption band of D are smaller than those of B. Thus this method which is supposed to be called development with an eluent may cause separation. According to former conceptions this method would be impossible because B the eluent should not allow A or D to become adsorbed. However, a very simple experiment showed the present conception to be correct. A mixture of cetane and cetene could be partly separated by silica gel previously wetted with benzene using benzene as developing liquid.I t is well known that benzene may act as an eluent (displacer) for cetane and cetene when adsorbed on silica gel. The advantage of this method is that the total volume of " developer " necessary to collect all the solutes at the bottom of the column is smaller than the interstitial volume of the column. This means a saving of time. However the separating efficiency of this method may be in many cases low. The occurrence of rates of transport greater than the rate of the liquid front is also demonstrated in an earlier paper,5 though not mentioned explicitly. Rates of transport up to about 1-3 times the rate of the liquid front have been found with chloroform and dodecylbenzene. A few words may be added as to the occurrence of so-called adsorption azeotropes as described by Hirschler and Am0n.l When an adsorption azeotrope occurs fi (c A ) / c A -5 Smit Anal.chim. A d a 1948. 2 671 w. M. SMIT 45 changes sign at a certain concentration of say A. At concentrations lower than the azeotropic concentration the sign may be positive and when the mixture is introduced into the column pure B appears in front of the adsorption band containing A. Beyond the azeotropic concentration pure A will appear in front and B is contained within the adsorption band. At the azeotropic concentration ~ ~ ( C A ) / C A -~~(CB)/CB = 0. The rate of transport of A and B both become equal to the rate of the liquid front and no separation occurs. So the conclusion derived from eqn. (6) and (7) agrees with experiment. At the same time it is clear that the occurrence of adsorption azeotropes is limited to substances having comparable adsorption affinities. No attempt has been made to solve the differential equations nor to develop formulze for more than two substances. As de Vault already stated this becomes very complicated. Moreover as real adsorption isotherms are used in our equation the quantitative solution is of no use as the real adsorption isotherms are not available. The main purpose of this paper has been to arrive at a rather simple formula which permits a qualitative explanation of the different methods of percolation but on the other hand it might show that the quantitative deductions made thus far have to be handled with care. Acknowledgment is due to the Management of the Bataafsche Petroleum Maatschappij for their permission to publish this paper. KoninklijkelShell-Laboratorium, Amsterdam

ISSN:0366-9033    

DOI:10.1039/DF9490700038

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

45 w. M. SMIT (1) in which * ( 3 ) (4) * On four preceding papers. This derivation presupposes that none of cj's is constant and P is not constant either. But assuming all ails to be different one can easily show that P = const. is irreconcilable with eqn. ( I ) and can occur only in the regions where all concentrations are constant ; consequently the eqn. ( I ) is not applicable. If one of the concentrations is constant one can similarly show that either P is constant or the concentration in GENERAL DISCUSSION * Dr. B. Davison (A.E.R.E. Haywell) (communicated) I want to point out that for the case of Langmuir isotherms the method described by Dr. Glueckauf in his paper can be easily generalized to any number of solutes simultaneously present in solution.For simplicity I shall speak of three solutes only. The basic equations are Putting and re-writing (I) in terms of pj's and PI differentiating again using P as independent variable and comparing with (3) we get either Pj=bjcj; P = P 1 +P2 + P a dquestion is zero and we have one solute less. GENERAL DISCUSSION * 46 or and consequently p, p and p are linear functions of P. Hence it can also be shown that ( 5 ) (8) (14 3G. 3S1 = mj P i- nj in which mj and yj are some constants. To satisfy (4) it is necessary that some of the concentrations are negative which is impossible ; thus (5) is the only solution of (I) having a physical signi- ficance. To determine the line (5) passing through the given initial concentrations (clo c,' c,") one may proceed as follows.Expressing ni's in terms of mi's and the initial concentrations and then substituting (5) into (I) gives in which fi" are the values of fi for initial concentrations and according to (3) m + w 2 + m3 = I. Denoting the common value of the expressions (6) by S expressing mj's in terms of S and substituting into m + m2 + m = I we have This is a cubic equation for which it can be shown that all roots are real and initid concentration. If a < a2 < a3 and the three roots of (7) are S < S < S different. Thus there are three characteristics of type (5) passing through each o < S < a < S < a < S < a,. . If S, S2 and S are known the corresponding c, c and G are given by If we use Sj's as the unknown functions rather than the concentrations (to facilitate manipulation) de Vault's partial differential equations become Each characteristic (5) is obtained by varying the corresponding Sj keeping the other two constant.From (10) one can easily derive that the speed of propagation of any particular combination of concentrations in the region of variable S i s given by From (8) and (11) we see that the regions with S variable propagate a t the fastest and the regions with S variable propagate a t the slowest rate. It also follows from (I I) that if the value of Sj at the rear of the region with variable Sj is greater than its value in the front part of this region (SjR > SiF) then the rear part propagates slower than the front part and we have a diffuse boundary.Similarly for SIR < SjF a self-sharpening boundary is obtained. From formulze (9) we can also derive which together with (8) predicts which concentrations increase and which decrease in each particular region. For instance since all 2 (i = I 2 3) have the same sign either all three concentrations increase or all three decrease in the most forward region ( S variable)-this generalizes Glueckauf's Rule I. Also since all % ( j = I 2 3) have the same sign the concentration of the most 3Sj adsorbed solute always increases along the column boundary while for a sharp boundary it decreases. This generalizes Glueckauf's Rule 2 . The conclusions are illustrated on the diagrams. Fig. IU (continuous line) gives the S diagram for the case when one set of three concentrations cl0 cZo c30 is developed by another set of three con_centrations cl c2 c3 of the same solutes and these cjo Gj are such that all Sj" > Sj (all three 47 - - - c3 'I1;fl R=O boundaries are diffuse).Fig. ~b gives schematically the corresponding concentra- tion diagram. The case of three diffuse boundaries will arise for instance if a set of three concentrations is developed by clear solvent. In this latter case the concentration diagram reduces to that represented in Fig. z (rear part). The dotted line in Fig. IU gives the S diagram for the case when on the contrary cl c2 c3 are developed by cl0 c20 c30. The corresponding concentrations diagram for GI = G2 = C3 = o is given in the front part of Fig.2 . B FIG. ra b and 2. ulcl = ql(ql + q2)@1-1) etc.1 n2 > ?a1 > 1. - - - Dr. F. G. Angel1 (Stockton-upon-Tees) said From the standpoint of a practical chromatographer it is worth while exploring or if necessary experimentally determining the adsorption isotherms of two substances on a particular adsorbent before proceeding with any chromatographic separation. According to Dr. Glueckauf's paper conditions for sharp boundaries in adsorption chromatography have been uniquely defined and provided that equilibrium is established under experimental conditions it would appear that much time and labour could be spared and more chances of success accrue by such a procedure than by the more customary empirical method of trial and error with various adsorbents.Dr. E. Glueckauf (A.E.R.E. Hurwell) said Dr. Smit mentions the need for further knowledge about chromatography in the case of adsorption azeotropes. Actually we can predict fairly well what will happen in such cases provided that the binary isotherms are a t least approximately known. Let us assume as an example that these isotherms are of the exponential type where a t low concentrations solute z is the more strongly adsorbed and where Such a system gives adsorption azeotropes whenever the total adsorption density GENERAL DISCUSSION 2 X f R' A' B' f'=O A I \ See Glueckauf J . Chem. SOC. 1947 1302. GENERAL DISCUSSION The qrq2 characteristics (see preceding paper by Glueckauf) of co-existing concentrations have approximately the form shown in Fig.I (only one positive characteristic being shown). The essential difference from the normal type of characteristic is that the positive characteristic is convex against the point 7 following complications will occur at first solute z will separate both in rear and front with enrichment of solute I in the centre of the mixed kand. Due to the spreading of the band during solvent development the maximum concentra- tion of the mixed band moves along the line BZA. After Z has been passed the formerly sharp front of the mixed kand becomes a diffuse one ; solute I enters the frontal band of solute 2 and eventually overtakes it. After a somewhat complex state of affairs (which can in detail be deduced from the diagram of charac- teristics) we arrive at the shape of the normal band distribution with a frontal band of pure solute I.This procedure wastes a good deal of time and adsorbent in the process ; which could be saved by working from the start at lower adsorp- tion densities i.e. with more dilute q1 = 0 q 2 = 0. Applying the rules given in the paper earlier in this Discussion we can see that nothing abnormal will happen during solvent development if the original start- ing point q I o q 2 O lies below the azeotropic line DZE. If lying above this line the FIG. I. solutes. One can compare this process with the distillation of binary azeotropes after the addition of a third inert solvent. Dr. W. M. Smit (Amsterdam) (communicated) In stating that there is a need of further knowledge on the behaviour of adsorption azeotropes it has not been my intention to say that i t is not yet possible to give a qualitative prediction of what will happen in an adsorption column if an adsorption azeotrope occurs.When “more strongly adsorbed ” stands for “ moving a t a lower rate ” a qualitative prediction of the phenomena which will occur in the case of adsorption azeotropes is even possible without using any formula. By simple reasoning we may come to the same conclusions as arrived a t by Dr. Glueckauf. However the adsorption azeotropes have been mentioned with a different purpose viz. to demonstrate the need of correcting the conservation equation. If a solution of azeotropic concentration is poured into an adsorption column no separation occurs and the adsorption front will move a t the same rate as the liquid front.The equations for the rate of transport of the front of an adsorption band derived from the conservation equations of Wilson or of de Vault (see paper on the conservation equation) lead to conclusions which do not tally with this simple fact. De Vault’s equation only gives the right value for the rate of transport of the adsorption front if the amount adsorbed at the azeotropic concentration is zero which cannot be true. The same obtains for the rate of transport of the adsorp- tion front of a pure solvent containing no solutes. Another point is that Dr. Glueckauf refers to ternary systems (a solvent with two solutes) whereas the literature quoted by me deals with binary systems (a solvent with one solute).Finally an interesting feature is that in the case of a binary system de Vault’s conservation equation will lead to the same results as the corrected equation if f(C,) in de Vault’s equation is replaced by another function F(C,) where F(C,) stands for the amount of substance A adsorbed preferentially. But this GENERAL DISCUSSION 49 implies that F(C,) cannot be represented by an adsorption isotherm of the Langmuir type the Freundlich type or the exponential type as given by Dr. Glueckauf. For F(C,) should a t least meet the following condition F(C,) = 0 for C = o and C = I . It may be a point for furtherinvestigation to determine how the equations for solutions containing more than one solute are affected by using the corrected conservation equation.Dr. B. Davison (A.E.R.E. Harwell) said The treatment of the problem given by Prof. A. C. Offord and Dr. J. Weiss in their paper is in many respects inadequate. In particular when confronted with a situation when at the first sight there are several alternatives they invariably choose the alternative which a more detailed analysis shows to be inacceptable. This can be most clearly seen if we re-examine the examples considered in their paper. Let me first draw attention to formula (23). It will be noticed that the value of c,’ given by this formula depends only upon the form of q1 and q2 as functions of c and c, and not upon the concentrations which were initially present.For instance for the Langmuir isotherms a1 c1 “ = I + blcl + b2c2 q 2 = I+ b,c a2 c2 + bx ’ the formula (23) leads to a t the other c,’ = - I ____ a - a . b l a2 It is rather difficult to visualize how this is possible in particular if the initiaI concentration cl0 of the first solute is very much smaller than the value of cl’ given by (23). Indeed if a < a i.e. if solute z is little adsorbed impossibly high values of 6,’ would be postulated by eqn. (23). Eqn. (23) also leads to the interesting value (26.2) for the content of solute I in the pure rear band which can exceed the total amount of solute I introduced into the column. This suggests that we should re-examine the derivation of (23). The formula (23) was derived from the assumption that (7) should hold for all values of y = vjx.But (7) was derived as the condition for the eqn. (6) to have other solutions than dc,/dy = dc,/dy = 0. Thus all we can say is that in the vicinity of any particular value of v / x either (7) should hold or both concentrations should be constant. Let us see what will happen if we introduce a constant concentration band between the regions of c 2 > o and c = 0 G variable. Then a t one boundary of this region and the impossible eqn. (23) (or a t any rate the equation with which one finds a great difficulty to reconcile oneself) disappears. This can be taken as a proof that a t least for Langmuir isotherms there will necessarily be a constant concentration band separating the regions c2 > o and c2 = 0 c variable.In the light of these remarks formulz (26) must be modified (and perhaps also some other formulze). I also want to draw attention to formulae (30) which postulates two discon- tinuities separated by a mixed band. For the sake of clarity I shall apply them to the case of Langmuir isotherms where their inadequacy becomes very ovbious. According to the assumption that solute I is more strongly adsorbed a > a,. Now if the postulated mixed band is to be formed spontaneously between two discontinuities the boundary between the mixed band and the original solute should move faster than the boundary between the mixed band and the pure solution of the developing solute. In the notation of formula (31) i t follows that ’2 ’1 J ’/’ > < v/’l GENERAL DISCUSSION and according to (31) C1‘ 41 (Cl’ 62’) < 4 2 (Cl’ G 2 3 For Langmuir isotherms this becomes ca’ i.e.a < a contrary to the assumption that solute I is more strongly adsorbed. To resolve this contradiction let us re-examine the derivation of (30). Formulae (30) is an application of formula (2-3) * (4 41m - 41m = 42W) - %(I) . Cl(II)?- h ( 1 ) (a) cl’ = 0 C2(II) - C2(1) derived from the conservation of material of both solutes. It is essential for the derivation of this formula that both solutes be present a t the boundary. If for instance the solute I is not present on either side of the boundary then the eqn. (2.1) disappears and we can no longer use equation (a). If it is known beforehand that on one side of the boundary both solutes are present then of course the formula (2.3) is necessarily satisfied.But if on one side of the boundary only one solute say solute 2 is present and we want to determine the concentrations on the other side of the boundary then there are two possibilities. Either these concentrations will be determined by eqn. (a) or only the solute 2 is present and el’ = 0. In the example under consideration we have thus three possibilities for the concentrations between the two dis- continuities namely (6) cl’ and c2’ (b) c2’ = o determined by the eqn. (30). We have already seen that the possibility (6)-the one chosen by Offord and Weiss-leads at least in the case of Langmuir isotherms to contradiction. isotherms to the same contradiction ( x = x2 ( I + b cl) and hence x > x,).The possibility ( b ) i.e. that of c,’ = 0 also leads at least in the case of Langmuir But the possibility ( a ) i.e. that of el’ = 0 leads to x2 = x (I + b2e,) and hence x > x as it ought to be. Thus the possibility ( a ) is the only acceptable one. I should also point out that if one eliminates from Offord and Weiss’s paper all the inacceptable choices of alternatives and uses the logically acceptable ones one arrives at the same results as have been already published by Glueckauf. Dr. E. Glueckauf (A.E.R.E. Harzuell) said I wish to call Prof. Offord’s attention to eqn. (17) which is a simple quadratic for y . In view of eqn. (IZ) 41 and q2 are both functions of c and c2 only and (3ql/J~,)C*=o and (342/3~2)c,= have thus values independent of y .I ask Prof. Offord why he postulates (see eqn. 23) that the two roots of the quadratic (17) should be equal when this is absolutely impossible. This answer is essential because the rest of paragraph I1 is based on eqn. (23). In order to avoid confusion in future research I wish to remove any impression that there are two alternative theories or even two alternative ways of approach in this matter. Actually the fundamental approach is very similar as can be seen from the identity of a number of differential equations and logical treatment of these equations would lead anybody to the same results as were obtained in my researches during the last five years. However the meaning which these equations take on after integration and the physical limits imposed by the fact that we are dealing with physical realities have sometimes escaped the authors and of several mathematically Possible solutions they have repeatedly chosen the physically impossible ones.Thus we are confronted with concentration bands which beginning with a zero-existence have a negative growth-rate ; with other bands which can contain more solute than has been put into the column. GENERAL DISCUSSION In paragraph I the treatment of a sharp boundary between two binary solu- tions (i.e. eqn. ( 2 3 ) (3) and Fig. I) is inadequate and misleading. In para- graph 11 all equations after (21) are erroneous including Fig. 2. In paragraph I11 all conclusions after eqn. (29) are wrong including Fig.3. In paragraph IV all conclusions are wrong including Fig. 4. I must apologize to Prof. Offord and Dr. Weiss for stating this so bluntly the reason being that the Fig. 2 3 4 look so extremely plausible to the chemist who uses chromatography as a technical tool. The disturbing effects of using a finite grain-size and of non-equilibrium phenomena which have not been taken into account in their papers result in exactly those phenomena which Offord and Weiss claim to have found for an “ideal” column by means of their wrong interpretations. These disturbing phenomena cause a smearing- out of the flat region missing at X,’ in their Fig. 2 and they cause variable mixed boundaries under the conditions of Fig. 3 and 4. But these phenomena though shown in the Figures are not represented by the equations of Offord and Weiss.Prof. A. C. Offord (London) (fiarlly contributed) In spite of the remarks of Dr. Glueckauf I must emphasize that our approach to the problem is funda- mentally different from his and any points of resemblance in the two theories are largely a matter of accident. One of the main problems treated in both contributions is the development of a chromatogram consisting of a band of two substances by pure solvent. Dr. Glueckauf starts with the simultaneous partial differential equations where i = I 2 andf and f are functions of c1 and c only. These equations are subject to the initial conditions that G = G = o when x = 0 and that c1 = elo and c = c20 when z = 0. He then gives what is claimed to be a solution of these equations which is broadly of the following form.The first quadrant is divided into five sectors by four straight lines (i) (ii) (iii) and (iv) issuing from the origin and the functions c1 and c are given by various expressions different for each sector. This solution has however no mathematical validity for the partial derivatives of the functions c1 and c2 do not exist on the lines (i) (ii) (iii) and (iv). In fact these differential equations have no solutions under the boundary conditions stated. Our contribution was a continuation of the work of Weiss on the problem for a single solute and i t has much in common with this earlier investigation. The mathematical treatment for the single solute (see the Appendix 2 hinges on the fact that we are not dealing with a sharp discontinuity at the top of the band but with a continuous tailing-off.Indeed an explicit form was assumed for this tail off (see eqn. (A 13)). The conclusion arrived a t was however that in the case of isotherms with pronounced curvature the major term giving the concentration in the rear part of the elute was in fact independent of the initial shape assumed for the top of the band. Hence in spite of the fact that in its final form the result appears to depend on the displacement relations only i t was only by making use of the notion of a diffused initial boundary (created by the presence of diffusion) that the problem could be given a tangible mathematical form . Unfortunately the same treatment cannot be applied in the case of two solutes but none the less similar considerations apply.The main point remains that the boundary conditions c = c = o when x = 0 and el = clO c = c,’ when li‘ = 0 have neither mathematical nor physical validity and i t is essential not only for physical but also for mathematical reasons to treat this boundary as a diffused boundary which it most certainly is. If we fail to do this we do not arrive a t an “ ideal ” problem as asserted by Dr. Glueckauf but at one which has no meaning. I t should have been explained that our treatment only applies when the isotherms have a pronounced curvature. The results we give are not true for 2 J . Chem. SOC. 1943 297. GENERAL DISCUSSION 52 linear isotherms where the phenomenon of diffusion plays a very much more marked role.Unlike Dr. Glueckauf we can lay no claim to a definitive mathe- matical theory valid in all cases and in our communication we considered only a few problems which we regarded as of particular interest. In the case of the special problem under consideration we remarked that various other solutions were possible depending on the particular isotherms. This applies to the parti- cular example selected by Dr. Davison where the more strongly adsorbed substance is present only in weak concentration. Our solution does not apply in this case. If one substance is present only as a trace then another mathematical treatment is possible and we hope to return to this problem later. Dr. Glueckauf's main objection arises from our treatment of eqn.(7) in our paper leading to eqn. (23) which is vital for all further consideration. This treatment is however necessary if the eqn. (I) are to have a solution (with continuous partial derivatives) valid over the part of the chromatogram where the faster moving solute first appears. Eqn. ( 2 2 ) is not incompatible with the differ- ential eqn. (IZ) because in our treatment there is no initial condition associated with (12) and so this equation does not determine G in terms of c,. Dr. J. Weiss (Newcastle) said As we have pointed out in our paper the problem is in certain respects not fully defined and i t is essential to take into account the experimental fact that-on account of diffusion and convection -there is neveY a sharp boundary a t the top of the band.By paying attention to this fact diffusion has been taken into account implicitly in our theory. This. is not the case in Glueckauf's treatment and i t is an important point in our theory . As we have stated we have assumed that the development of a chromatogram of two solutes with the solvent is in general a continuous process. We have assumed this in view of the experimental results of various workers as there is. no evidence to show that one obtains under these conditions a band of constant concentration as postulated by Glueckauf. This is in fact demonstrated clearly even by the experimental work of Coates and Glueckauf.* Fig. 2 and 4 of this paper are very instructive and I would like to recommend anyone interested in this problem to consult these Figures and to see for themselves that the " constant band " predicted by this theory is a figment of imagination.These authors have also presented there some theoretical curves. However in order to show even a rough agreement between theory and experiment they had to choose the arbitrary constants in the theoretical equations in such a way that the " constant band "-which is the main point in Glueckauf's theory-is practically eliminated i.e. is reduced to such small proportions that it falls well within the region of the experimental error. I shall be only too pleased to accept Glueckauf's theory if he can produce any experimental evidence in favour of i t that he obviously cannot and he explains by " disturbing effects " while our theory can a t least give an account of the main experimental features.We find i t hard to understand also what Dr. Glueckauf says in the last para- graph of his remark. Diffusion phenomena might be expected to have the effect of spreading out the band and so making the flat region which he predicts more pronounced. Dr. J. F. Duncan (A.E.R.E. Harwell) (commzwicated) In order to make an experimental test of the theory developed by Dr. E. Glueckauf the case shown in Fig. 4 ( d ) of his paper was simulated with the following solutions Solution I 0.025 N NaNO, 0.1 N KNO, 0.125 N HNO,. Solution z 0.15 N NaNO, 0.06 N KNO, 0.04 N HN03. Assuming the mass action law to obtain we have 3 This Discussion. J .Chem. SOC. 1947 1308. GENERAL DISCUSSION from which bi = (KNa+ - I)iCS 9 and similarly for potassium 53 where the terms have the same meanings as in our paper. Eqn. (I) has the form of a Langmuir equation. Hence b = ( K K + - I)/CS . Assuming K N ~ + = 1-52 and putting KK+ = 2.00 (a reasonable value estimated from the results of other workers) we get P N a + = 5cNas and P K + = 5cKs. pNa+ = 0.125 p,+ = 0.5 Hence for solution I and for solution 2 P N ~ + = 0.75 p,+ = 0.03. This represents the case when a solution A in the column is followed by a solution C the characteristic diagram being similar to that shown in Fig. 4 (a). The result of the experiment showed quite clearly that an intermediate concen- tration plateau forms with both concentrations higher than A in the manner shown in Fig.4 ( d ) . Dr. J. Weiss (Newcastle-upon-Tyne) (conzmztnicated) Dr. Davison has confined his assertions to a special form of the Langmuir isotherm for two solutes viz. a G1 41 = I + blGl + b,c etc. and furthermore he makes a somewhat arbitrary choice of the constants. I t is obvious that our theory holds as stated only when q1 and q are functions of c and c through the whole course of the development (when both these solutes are present). However in the Langmuir equations 4 and 4 are functions of c and c only if b c and b G are both of the order of I. I t is well known that if this is not the case the equations go over into either (a) the linear isotherms if b c and b c < I or (b) q1 and q2 become constant (independent of c1 and c2) if b,c and b c > I.I t is clear that this imposes considerable restric- tions on the application of this form of the Langmuir isotherm. Thus it clearly cannot be applied to the displacement development (Tiselius’ case) because when the more strongly adsorbed component is present in excess according to this simple form of the Langmuir isotherm the amount adsorbed would become independent of the concentration of this solute. These facts have not been taken into account by Dr. Davison and thus his remarks are hardly of any significance. Dr . E . Glueckauf ( A . E. R. E. Harwell) (communicated) Two matters seem clear from the paper and the contributions of Dr. Offord and Dr. Weiss (i) that the former has little experience in the application of boundary conditions to hyperbolic differential equations which govern the process concerned (which by the way is in close analogy to the propagation of waves in channels) ; (ii) that the latter tries unsuccessfully to sidetrack the issue by counter attacks which being without substance can be easily answered.No reply has been made concerning Dr. Davison’s explicit criticism of eqn. (30) which invalidates the whole of their para. 3 and 4 and it must be assumed that Offord and Weiss concur. Dr. Offord’s explanation “ that their treatment only applies when the isotherms have a pronounced curvature ” and when the more strongly adsorbed substance is present at higher concentrations is not given expression in any of the boundary conditions used in their paper and it is difficult to escape the impression that it has been an afterthought.In particular Dr. Offord omits to say that when ti Duncan and Lister Chern. and Ind. 1949 24. CONSTITUTIONAL AND STERIC PROPERTIES 54 he limits application to isotherms of pronounced curvature and excludes cases where the more adsorbed material (I) is in the minority (which in terms of the Langmuir isotherm is equivalent to the conditions b,c + b,c > I and cl0 > c,”) his result approximates more and more to that of my theory. This must be so because his mathematically correct solution is the point (see my Fig. 3 ) where the parabolical envelope touches the ordinate a t p = I . It required exceptionally special assumptions (which the authors do not state in their paper and which do not frequently occur under experimental conditions) before Offord and Weiss’s solution would apply even approximately.Dr. Weiss emphasizes that “ in theiv paper the problem is in certain respects mathematically not fully defined.’’ The reason for this is that they have omitted to introduce an essential boundary condition namely the concentration of the original solution cIo and G,’. This omission is caused by the fact that they impose the condition of continuity of the partial derivatives-a condition which is quite legitimate and in fact necessary in the elliptic differential equations but has no justification in equations of the hyperbolic type. And having imposed such a condition they are no longer able to satisfy the initial conditions given by cl0 c,”.That chromatographic conditions (in particular the value of c a t C = 0) depend on the original concentrations cl0 cZo has been shown experimentally and very clearly indeed in the Fig. (4)s which Dr. Weiss has examined in SO careful and unbiased a manner. The constant band ” which is by no means the main point of my theory but merely follows from i t for certain isotherms is not and should not be predominant in this case where the exchange constants a and a differ as little as they do. I also wish to draw attention to the communi- cation by Dr. Duncan concerning experimental data. Dr. Weiss who has izevev published any chromatographic experiment has stated ‘ ( that he will be only too pleased to accept Glueckauf’s theory if he can produce any experimental evidence in favour of it.” I therefore hope that Dr.Weiss will be pleased to do so. Coates and Glueckauf J . Chew. SOC. 1947 1313. w. M. SMIT 45 GENERAL DISCUSSION * Dr. B. Davison (A.E.R.E. Haywell) (communicated) I want to point out that for the case of Langmuir isotherms the method described by Dr. Glueckauf in his paper can be easily generalized to any number of solutes simultaneously present in solution. For simplicity I shall speak of three solutes only. The basic equations are (1) in which Putting Pj=bjcj; P = P 1 +P2 + P a * ( 3 ) and re-writing (I) in terms of pj's and PI differentiating again using P as independent variable and comparing with (3) we get either (4) * On four preceding papers.This derivation presupposes that none of cj's is constant and P is not constant either. But assuming all ails to be different one can easily show that P = const. is irreconcilable with eqn. ( I ) and can occur only in the regions where all concentrations are constant ; consequently the eqn. ( I ) is not applicable. If one of the concentrations is constant one can similarly show that either P is constant or the concentration in dquestion is zero and we have one solute less 46 GENERAL DISCUSSION or and consequently p, p and p are linear functions of P. Hence in which mj and yj are some constants. To satisfy (4) it is necessary that some of the concentrations are negative, which is impossible ; thus (5) is the only solution of (I) having a physical signi-ficance.To determine the line (5) passing through the given initial concentrations (clo c,' c,") one may proceed as follows. Expressing ni's in terms of mi's and the initial concentrations and then substituting (5) into (I) gives = mj P i- nj * ( 5 ) in which fi" are the values of fi for initial concentrations and according to (3), m + w 2 + m3 = I. Denoting the common value of the expressions (6) by S , expressing mj's in terms of S and substituting into m + m2 + m = I we have This is a cubic equation for which it can be shown that all roots are real and different. Thus there are three characteristics of type (5) passing through each initid concentration. If a < a2 < a3 and the three roots of (7) are S < S < S , it can also be shown that If S, S2 and S are known the corresponding c, c and G are given by o < S < a < S < a < S < a,.. (8) If we use Sj's as the unknown functions rather than the concentrations (to facilitate manipulation) de Vault's partial differential equations become Each characteristic (5) is obtained by varying the corresponding Sj keeping the other two constant. From (10) one can easily derive that the speed of propagation of any particular combination of concentrations in the region of variable S i s given by From (8) and (11) we see that the regions with S variable propagate a t the fastest and the regions with S variable propagate a t the slowest rate. It also follows from (I I) that if the value of Sj at the rear of the region with variable Sj is greater than its value in the front part of this region (SjR > SiF) then the rear part propagates slower than the front part and we have a diffuse boundary.Similarly for SIR < SjF a self-sharpening boundary is obtained. From formulze (9) we can also derive (14 which together with (8) predicts which concentrations increase and which decrease in each particular region. For instance since all 2 (i = I 2 3) have the same sign either all three concentrations increase or all three decrease in the most forward region ( S variable)-this generalizes Glueckauf's Rule I. Also since all % ( j = I 2 3) have the same sign the concentration of the most adsorbed solute always increases along the column 3G. 3S1 3S GENERAL DISCUSSION 47 'I1;c3 X 2 R=O A B f R' A' B' f'=O boundary while for a sharp boundary it decreases.This generalizes Glueckauf's Rule 2 . Fig. IU (continuous line) gives the S diagram for the case when one set of three concentrations cl0 cZo c30 is developed by another set of three con_centrations cl c2 c3 of the same solutes and these cjo Gj are such that all Sj" > Sj (all three The conclusions are illustrated on the diagrams. - - -FIG. ra b and 2. boundaries are diffuse). Fig. ~b gives schematically the corresponding concentra-tion diagram. The case of three diffuse boundaries will arise for instance if a set of three concentrations is developed by clear solvent. In this latter case the concentration diagram reduces to that represented in Fig. z (rear part). The dotted line in Fig.IU gives the S diagram for the case when on the contrary, cl c2 c3 are developed by cl0 c20 c30. The corresponding concentrations diagram for GI = G2 = C3 = o is given in the front part of Fig. 2 . Dr. F. G. Angel1 (Stockton-upon-Tees) said From the standpoint of a practical chromatographer it is worth while exploring or if necessary experimentally determining the adsorption isotherms of two substances on a particular adsorbent before proceeding with any chromatographic separation. According to Dr. Glueckauf's paper conditions for sharp boundaries in adsorption chromatography have been uniquely defined and provided that equilibrium is established under experimental conditions it would appear that much time and labour could be spared and more chances of success accrue by such a procedure than by the more customary empirical method of trial and error with various adsorbents.Dr. E. Glueckauf (A.E.R.E. Hurwell) said Dr. Smit mentions the need for further knowledge about chromatography in the case of adsorption azeotropes. Actually we can predict fairly well what will happen in such cases provided that the binary isotherms are a t least approximately known. Let us assume as an example that these isotherms are of the exponential type : ulcl = ql(ql + q2)@1-1) etc.1 where a t low concentrations solute z is the more strongly adsorbed and where Such a system gives adsorption azeotropes whenever the total adsorption - - -n2 > ?a1 > 1. density I \ See Glueckauf J . Chem. SOC. 1947 1302 GENERAL DISCUSSION The qrq2 characteristics (see preceding paper by Glueckauf) of co-existing concentrations have approximately the form shown in Fig.I (only one positive characteristic being shown). The essential difference from the normal type of characteristic is that the positive characteristic is convex against the point q1 = 0 q 2 = 0. Applying the rules given in the paper earlier in this Discussion we can see that nothing abnormal will happen during solvent development if the original start-ing point q I o q 2 O lies below the azeotropic line DZE. If lying above this line the following complications will occur at first solute z will separate both in rear and front with enrichment of solute I in the centre of the mixed kand. Due to the spreading of the band during solvent development the maximum concentra-tion of the mixed band moves along the line BZA.After Z has been passed the formerly sharp front of the mixed kand becomes a diffuse one ; solute I enters the frontal band of solute 2 and eventually overtakes it. After a somewhat complex state of affairs (which can in detail be deduced from the diagram of charac-teristics) we arrive at the shape of the normal band distribution with a frontal band of pure solute I. This procedure wastes a good deal of time and adsorbent in the process ; which could be saved by working from the start at lower adsorp-tion densities i.e. with more dilute solutes. One can compare this process with the distillation of binary azeotropes after the addition of a third inert solvent. 7, FIG.I. Dr. W. M. Smit (Amsterdam) (communicated) In stating that there is a need of further knowledge on the behaviour of adsorption azeotropes it has not been my intention to say that i t is not yet possible to give a qualitative prediction of what will happen in an adsorption column if an adsorption azeotrope occurs. When “more strongly adsorbed ” stands for “ moving a t a lower rate ” a qualitative prediction of the phenomena which will occur in the case of adsorption azeotropes is even possible without using any formula. By simple reasoning we may come to the same conclusions as arrived a t by Dr. Glueckauf. However the adsorption azeotropes have been mentioned with a different purpose viz. to demonstrate the need of correcting the conservation equation.If a solution of azeotropic concentration is poured into an adsorption column no separation occurs and the adsorption front will move a t the same rate as the liquid front. The equations for the rate of transport of the front of an adsorption band derived from the conservation equations of Wilson or of de Vault (see paper on the conservation equation) lead to conclusions which do not tally with this simple fact. De Vault’s equation only gives the right value for the rate of transport of the adsorption front if the amount adsorbed at the azeotropic concentration is zero, which cannot be true. The same obtains for the rate of transport of the adsorp-tion front of a pure solvent containing no solutes. Another point is that Dr. Glueckauf refers to ternary systems (a solvent with two solutes) whereas the literature quoted by me deals with binary systems (a solvent with one solute).Finally an interesting feature is that in the case of a binary system de Vault’s conservation equation will lead to the same results as the corrected equation if f(C,) in de Vault’s equation is replaced by another function F(C,) where F(C,) stands for the amount of substance A adsorbed preferentially. But thi GENERAL DISCUSSION 49 implies that F(C,) cannot be represented by an adsorption isotherm of the Langmuir type the Freundlich type or the exponential type as given by Dr. Glueckauf. For F(C,) should a t least meet the following condition: F(C,) = 0 for C = o and C = I . It may be a point for furtherinvestigation to determine how the equations for solutions containing more than one solute are affected by using the corrected conservation equation.Dr. B. Davison (A.E.R.E. Harwell) said The treatment of the problem given by Prof. A. C. Offord and Dr. J. Weiss in their paper is in many respects, inadequate. In particular when confronted with a situation when at the first sight there are several alternatives they invariably choose the alternative which a more detailed analysis shows to be inacceptable. This can be most clearly seen if we re-examine the examples considered in their paper. It will be noticed that the value of c,’ given by this formula depends only upon the form of q1 and q2 as functions of c and c, and not upon the concentrations which were initially present. Let me first draw attention to formula (23).For instance for the Langmuir isotherms a1 c1 a2 c2 “ = I + blcl + b2c2 q 2 = I+ b,c + bx ’ the formula (23) leads to I a - a c,’ = - ____ . b l a2 It is rather difficult to visualize how this is possible in particular if the initiaI concentration cl0 of the first solute is very much smaller than the value of cl’ given by (23). Indeed if a < a i.e. if solute z is little adsorbed impossibly high values of 6,’ would be postulated by eqn. (23). Eqn. (23) also leads to the interesting value (26.2) for the content of solute I in the pure rear band which can exceed the total amount of solute I introduced into the column. This suggests that we should re-examine the derivation of (23). The formula (23) was derived from the assumption that (7) should hold for all values of y = vjx.But (7) was derived as the condition for the eqn. (6) to have other solutions than dc,/dy = dc,/dy = 0. Thus all we can say is that in the vicinity of any particular value of v / x either (7) should hold or both concentrations should be constant. Let us see what will happen if we introduce a constant concentration band between the regions of c 2 > o and c = 0 G variable. Then a t one boundary of this region a t the other and the impossible eqn. (23) (or a t any rate the equation with which one finds a great difficulty to reconcile oneself) disappears. This can be taken as a proof that a t least for Langmuir isotherms there will necessarily be a constant concentration band separating the regions c2 > o and c2 = 0 c variable.In the light of these remarks formulz (26) must be modified (and perhaps also some other formulze). I also want to draw attention to formulae (30) which postulates two discon-tinuities separated by a mixed band. For the sake of clarity I shall apply them to the case of Langmuir isotherms where their inadequacy becomes very ovbious. According to the assumption that solute I is more strongly adsorbed a > a,. Now if the postulated mixed band is to be formed spontaneously between two discontinuities the boundary between the mixed band and the original solute should move faster than the boundary between the mixed band and the pure solution of the developing solute. In the notation of formula (31) i t follows that ’2 > ’1 J ’/’ < v/’ GENERAL DISCUSSION and according to (31), 41 (Cl’ 62’) < 4 2 (Cl’ G 2 3 C1‘ ca’ For Langmuir isotherms this becomes i.e.a < a, contrary to the assumption that solute I is more strongly adsorbed. (30) is an application of formula (2-3) To resolve this contradiction let us re-examine the derivation of (30). Formulae * (4 41m - 41m = 42W) - %(I) . Cl(II)?- h ( 1 ) C2(II) - C2(1) derived from the conservation of material of both solutes. It is essential for the derivation of this formula that both solutes be present a t the boundary. If for instance the solute I is not present on either side of the boundary then the eqn. (2.1) disappears and we can no longer use equation (a). If it is known beforehand that on one side of the boundary both solutes are present then of course the formula (2.3) is necessarily satisfied.But if on one side of the boundary only one solute say solute 2 is present and we want to determine the concentrations on the other side of the boundary then there are two possibilities. Either these concentrations will be determined by eqn. (a) or only the solute 2 is present and el’ = 0. In the example under consideration we have thus three possibilities for the concentrations between the two dis-continuities namely (a) cl’ = 0 (b) c2’ = o (6) cl’ and c2’ determined by the eqn. (30). We have already seen that the possibility (6)-the one chosen by Offord and Weiss-leads at least in the case of Langmuir isotherms to contradiction. The possibility ( b ) i.e. that of c,’ = 0 also leads at least in the case of Langmuir isotherms to the same contradiction ( x = x2 ( I + b cl) and hence x > x,).But the possibility ( a ) i.e. that of el’ = 0 leads to x2 = x (I + b2e,) and hence x > x as it ought to be. Thus the possibility ( a ) is the only acceptable one. I should also point out that if one eliminates from Offord and Weiss’s paper all the inacceptable choices of alternatives and uses the logically acceptable ones one arrives at the same results as have been already published by Glueckauf. Dr. E. Glueckauf (A.E.R.E. Harzuell) said I wish to call Prof. Offord’s attention to eqn. (17) which is a simple quadratic for y . In view of eqn. (IZ), 41 and q2 are both functions of c and c2 only and have thus values independent of y . I ask Prof. Offord why he postulates (see eqn.23) that the two roots of the quadratic (17) should be equal when this is absolutely impossible. This answer is essential because the rest of paragraph I1 is based on eqn. (23). In order to avoid confusion in future research I wish to remove any impression that there are two alternative theories or even two alternative ways of approach in this matter. Actually the fundamental approach is very similar as can be seen from the identity of a number of differential equations and logical treatment of these equations would lead anybody to the same results as were obtained in my researches during the last five years. However the meaning which these equations take on after integration and the physical limits imposed by the fact that we are dealing with physical realities, have sometimes escaped the authors and of several mathematically Possible solutions they have repeatedly chosen the physically impossible ones.Thus we are confronted with concentration bands which beginning with a zero-existence have a negative growth-rate ; with other bands which can contain more solute than has been put into the column. (3ql/J~,)C*=o and (342/3~2)c,= GENERAL DISCUSSION In paragraph I the treatment of a sharp boundary between two binary solu-tions (i.e. eqn. ( 2 3 ) (3) and Fig. I) is inadequate and misleading. In para-graph 11 all equations after (21) are erroneous including Fig. 2. In paragraph I11 all conclusions after eqn. (29) are wrong including Fig. 3. In paragraph IV all conclusions are wrong including Fig.4. I must apologize to Prof. Offord and Dr. Weiss for stating this so bluntly, the reason being that the Fig. 2 3 4 look so extremely plausible to the chemist who uses chromatography as a technical tool. The disturbing effects of using a finite grain-size and of non-equilibrium phenomena which have not been taken into account in their papers result in exactly those phenomena which Offord and Weiss claim to have found for an “ideal” column by means of their wrong interpretations. These disturbing phenomena cause a smearing-out of the flat region missing at X,’ in their Fig. 2 and they cause variable mixed boundaries under the conditions of Fig. 3 and 4. But these phenomena though shown in the Figures are not represented by the equations of Offord and Weiss. Prof.A. C. Offord (London) (fiarlly contributed) In spite of the remarks of Dr. Glueckauf I must emphasize that our approach to the problem is funda-mentally different from his and any points of resemblance in the two theories are largely a matter of accident. One of the main problems treated in both contributions is the development of a chromatogram consisting of a band of two substances by pure solvent. Dr. Glueckauf starts with the simultaneous partial differential equations where i = I 2 andf and f are functions of c1 and c only. These equations are subject to the initial conditions that G = G = o when x = 0 and that c1 = elo and c = c20 when z = 0. He then gives what is claimed to be a solution of these equations which is broadly of the following form. The first quadrant is divided into five sectors by four straight lines (i) (ii) (iii) and (iv) issuing from the origin, and the functions c1 and c are given by various expressions different for each sector.This solution has however no mathematical validity for the partial derivatives of the functions c1 and c2 do not exist on the lines (i) (ii) (iii) and (iv). In fact these differential equations have no solutions under the boundary conditions stated. on the problem for a single solute and i t has much in common with this earlier investigation. The mathematical treatment for the single solute (see the Appendix 2 hinges on the fact that we are not dealing with a sharp discontinuity at the top of the band but with a continuous tailing-off. Indeed an explicit form was assumed for this tail off (see eqn.(A 13)). The conclusion arrived a t was however that in the case of isotherms with pronounced curvature the major term giving the concentration in the rear part of the elute was in fact independent of the initial shape assumed for the top of the band. Hence in spite of the fact that in its final form the result appears to depend on the displacement relations only, i t was only by making use of the notion of a diffused initial boundary (created by the presence of diffusion) that the problem could be given a tangible mathematical form . Unfortunately the same treatment cannot be applied in the case of two solutes, but none the less similar considerations apply. The main point remains that the boundary conditions c = c = o when x = 0 and el = clO c = c,’ when li‘ = 0 have neither mathematical nor physical validity and i t is essential not only for physical but also for mathematical reasons to treat this boundary as a diffused boundary which it most certainly is.If we fail to do this we do not arrive a t an “ ideal ” problem as asserted by Dr. Glueckauf but at one which has no meaning. I t should have been explained that our treatment only applies when the isotherms have a pronounced curvature. The results we give are not true for Our contribution was a continuation of the work of Weiss 2 J . Chem. SOC. 1943 297 52 GENERAL DISCUSSION linear isotherms where the phenomenon of diffusion plays a very much more marked role. Unlike Dr. Glueckauf we can lay no claim to a definitive mathe-matical theory valid in all cases and in our communication we considered only a few problems which we regarded as of particular interest.In the case of the special problem under consideration we remarked that various other solutions were possible depending on the particular isotherms. This applies to the parti-cular example selected by Dr. Davison where the more strongly adsorbed substance is present only in weak concentration. Our solution does not apply in this case. If one substance is present only as a trace then another mathematical treatment is possible and we hope to return to this problem later. Dr. Glueckauf's main objection arises from our treatment of eqn. (7) in our paper leading to eqn. (23) which is vital for all further consideration. This treatment is however necessary if the eqn.(I) are to have a solution (with continuous partial derivatives) valid over the part of the chromatogram where the faster moving solute first appears. Eqn. ( 2 2 ) is not incompatible with the differ-ential eqn. (IZ) because in our treatment there is no initial condition associated with (12) and so this equation does not determine G in terms of c,. Dr. J. Weiss (Newcastle) said As we have pointed out in our paper the problem is in certain respects not fully defined and i t is essential to take into account the experimental fact that-on account of diffusion and convection -there is neveY a sharp boundary a t the top of the band. By paying attention to this fact diffusion has been taken into account implicitly in our theory. This.is not the case in Glueckauf's treatment and i t is an important point in our theory . As we have stated we have assumed that the development of a chromatogram of two solutes with the solvent is in general a continuous process. We have assumed this in view of the experimental results of various workers as there is. no evidence to show that one obtains under these conditions a band of constant concentration as postulated by Glueckauf. This is in fact demonstrated clearly even by the experimental work of Coates and Glueckauf.* Fig. 2 and 4 of this paper are very instructive and I would like to recommend anyone interested in this problem to consult these Figures and to see for themselves that the " constant band " predicted by this theory is a figment of imagination.These authors have also presented there some theoretical curves. However in order to show even a rough agreement between theory and experiment they had to choose the arbitrary constants in the theoretical equations in such a way that the " constant band "-which is the main point in Glueckauf's theory-is practically eliminated i.e. is reduced to such small proportions that it falls well within the region of the experimental error. I shall be only too pleased to accept Glueckauf's theory if he can produce any experimental evidence in favour of i t that he obviously cannot and he explains by " disturbing effects " while our theory can a t least give an account of the main experimental features. We find i t hard to understand also what Dr. Glueckauf says in the last para-graph of his remark.Diffusion phenomena might be expected to have the effect of spreading out the band and so making the flat region which he predicts more pronounced. Dr. J. F. Duncan (A.E.R.E. Harwell) (commzwicated) In order to make an experimental test of the theory developed by Dr. E. Glueckauf the case shown in Fig. 4 ( d ) of his paper was simulated with the following solutions : Solution I 0.025 N NaNO, 0.1 N KNO, 0.125 N HNO,. Solution z 0.15 N NaNO, 0.06 N KNO, 0.04 N HN03. Assuming the mass action law to obtain we have 3 This Discussion. J . Chem. SOC. 1947 1308 GENERAL DISCUSSION 53 from which where the terms have the same meanings as in our paper. form of a Langmuir equation. Hence and similarly for potassium Assuming from the results of other workers) we get Hence for solution I , This represents the case when a solution A in the column is followed by a solution C the characteristic diagram being similar to that shown in Fig.4 (a). The result of the experiment showed quite clearly that an intermediate concen-tration plateau forms with both concentrations higher than A in the manner shown in Fig. 4 ( d ) . Eqn. (I) has the bi = (KNa+ - I)iCS 9 b = ( K K + - I)/CS . K N ~ + = 1-52 and putting KK+ = 2.00 (a reasonable value estimated P N a + = 5cNas and P K + = 5cKs. pNa+ = 0.125 p,+ = 0.5, and for solution 2 P N ~ + = 0.75 p,+ = 0.03. Dr. J. Weiss (Newcastle-upon-Tyne) (conzmztnicated) : confined his assertions to a special form of the Langmuir solutes viz., a G1 etc., 41 = I + blGl + b,c, and furthermore he makes a somewhat arbitrary choice Dr.Davison has isotherm for two of the constants. I t is obvious that our theory holds as stated only when q1 and q are functions of c and c through the whole course of the development (when both these solutes are present). However in the Langmuir equations 4 and 4 are functions of c and c only if b c and b G are both of the order of I. I t is well known that if this is not the case the equations go over into either (a) the linear isotherms if b c and b c < I or (b) q1 and q2 become constant (independent of c1 and c2) if b,c and b c > I. I t is clear that this imposes considerable restric-tions on the application of this form of the Langmuir isotherm. Thus it clearly cannot be applied to the displacement development (Tiselius’ case) because when the more strongly adsorbed component is present in excess according to this simple form of the Langmuir isotherm the amount adsorbed would become independent of the concentration of this solute.These facts have not been taken into account by Dr. Davison and thus his remarks are hardly of any significance. Dr . E . Glueckauf ( A . E. R. E. Harwell) (communicated) Two matters seem clear from the paper and the contributions of Dr. Offord and Dr. Weiss (i) that the former has little experience in the application of boundary conditions to hyperbolic differential equations which govern the process concerned (which, by the way is in close analogy to the propagation of waves in channels) ; (ii) that the latter tries unsuccessfully to sidetrack the issue by counter attacks which, being without substance can be easily answered.No reply has been made concerning Dr. Davison’s explicit criticism of eqn. (30), which invalidates the whole of their para. 3 and 4 and it must be assumed that Offord and Weiss concur. Dr. Offord’s explanation “ that their treatment only applies when the isotherms have a pronounced curvature ” and when the more strongly adsorbed substance is present at higher concentrations is not given expression in any of the boundary conditions used in their paper and it is difficult to escape the impression that it has been an afterthought. In particular Dr. Offord omits to say that when ti Duncan and Lister Chern. and Ind. 1949 24 54 CONSTITUTIONAL AND STERIC PROPERTIES he limits application to isotherms of pronounced curvature and excludes cases where the more adsorbed material (I) is in the minority (which in terms of the Langmuir isotherm is equivalent to the conditions b,c + b,c > I and cl0 > c,”) his result approximates more and more to that of my theory.This must be so because his mathematically correct solution is the point (see my Fig. 3 ) where the parabolical envelope touches the ordinate a t p = I . It required exceptionally special assumptions (which the authors do not state in their paper and which do not frequently occur under experimental conditions) before Offord and Weiss’s solution would apply even approximately. Dr. Weiss emphasizes that “ in theiv paper the problem is in certain respects mathematically not fully defined.’’ The reason for this is that they have omitted to introduce an essential boundary condition namely the concentration of the original solution cIo and G,’. This omission is caused by the fact that they impose the condition of continuity of the partial derivatives-a condition which is quite legitimate and in fact necessary in the elliptic differential equations but has no justification in equations of the hyperbolic type. And having imposed such a condition they are no longer able to satisfy the initial conditions given by cl0 c,”. That chromatographic conditions (in particular the value of c a t C = 0) depend on the original concentrations cl0 cZo has been shown experimentally and very clearly indeed in the Fig. (4)s which Dr. Weiss has examined in SO careful and unbiased a manner. constant band ” which is by no means the main point of my theory but merely follows from i t for certain isotherms, is not and should not be predominant in this case where the exchange constants, a and a differ as little as they do. I also wish to draw attention to the communi-cation by Dr. Duncan concerning experimental data. Dr. Weiss who has izevev published any chromatographic experiment has stated ‘ ( that he will be only too pleased to accept Glueckauf’s theory if he can produce any experimental evidence in favour of it.” I therefore hope that Dr. Weiss will be pleased to do so. The Coates and Glueckauf J . Chew. SOC. 1947 1313

ISSN:0366-9033    

DOI:10.1039/DF9490700045

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

54 ADSORPTION AND SOME CONSTITUTIONAL AND STERIC PROPERTIES BY L. ZECHMEISTER discussed. Received 15th July 1949 The concept " anchoring group " is proposed and some pertinent experimental approaches are indicated. A report is submitted on the separation of cistrcrns stereo- isomers and the dependence of the adsorption affinity on molecular morphology is Although it is not assumed that a single correct picture can be given for the adsorption and geometrical orientation of organic molecules on a solid surface the following concept may serve as a basis of discussion. Two essential features can be postulated for the simple and stable fixation of an organic solute on an adsorbent first its molecules should fit into the cavities of a certain size located on the active surface; and second adequately oriented forces should be operative between the adsorbent and the adsorbed molecule.These two postulates are of course closely interrelated. Very little is known about the geometrical orientation of adsorbed mole- cules but one may speculate that the manner of orientation would influence among others the relative behaviour of closely related compounds. If the L. ZECHMEISTER 55 orientation be roughly parallel to the surface it could be perhaps expected that within a homologous series a maximum of the adsorption affinity would be reached at a certain chain length. On the other hand it may be considered that in the case of a roughly vertical orientation of the adsorbed molecules with reference to the surface a continuous increase or decrease in the adsorption affinities would be observed by passing stepwise from lower members of the series to some higher ones.Whatever the orientation may be in a given system it seems reasonable to assume that not each section of the adsorbed molecule will be equally responsible for the fixation process. On the contrary we propose that a decisive part will be played by certain atomic groups which are conveniently designated as “ anchoring groups.” This concept is similar to that in which it is presumed that the fixation of a drug to bacteria takes place by the intermediary of a haptene group (or groups). In favourable instances an anchoring group could be located experimentally by showing that a modi- fication of that particular section of the molecule causes an unusually sharp change in the adsorption affinity.Perhaps a single example will demonstrate the general lines along which we think that laboratory work may help in collecting material for a more satisfactory discussion of such problems. It was found that the introduction of a methyl group in the a-position to the sulphur atom of cc-terthienyl markedly increased the adsorption affinity; and that upon dimethylation this effect was about doubled. On the other hand Kofler reported that the adsorbability of some tocol derivatives decreased upon methylation near the phenolic OH group although this substitution had but little effect in a more remote position to the hydroxyl mentioned. In the case of thiophene rings the introduced methyl group seems to have been involved in the fixation process but the anchoring group of the tocol derivative was probably the phenolic hydroxyl whose function is sterically hindered by the presence of adjacent methyl groups.While manifold possibilities are open for experimentation in the direction just outlined our problem changes its character entirely when we pass from chemical to stereochemical considerations. Then in a way the situation appears to be simplified since in stereochemistry the dependence of the adsorption affinity on the overall shape of the molecules is to be considered and not the presence or absence of certain functional groups. The compounds which are suitable for such investigations should possess numerous stereo-isomeric forms whose respective morphological types should vary as widely as possible.This postulate is unequally fulfilled in various stereo-isomeric sets ; for example the epimerization of a sugar would not essentially alter the overall shape of its molecules. The conditions for studies of this kind are especially favourable in the field of the natural and synthetic polyenes which compounds possess a long conjugated carbon-carbon double-bond system in an open chain. Their structure is “ morphologically sensitive ’’ to spatial variations. Although the ordinary or all-tram form (Fig. I) which shows a rod-like general shape is not greatly modified by a single trans+& rotation which occurs near an end of the system it undergoes a radical change and is converted into a V-like pattern when such a spatial re-arrangement takes place at or near the centre (11).On the other hand in the course of continued tralzs+cis rotations after having passed through several bent forms a straightening-out takes place. Thus the molecules of the resulting poly-cis compound show a rod- like overall shape (111) which is morphologically similar to that of the all- a-carotene p-carotene y-carotene and lycopene can trans molecules (I). As is well known the respective all-trans forms of the polyene hydro- carbons (C ,H 5 CONSTITUTIONAL AND STERIC PROPERTIES 56 be easily converted into a mixture of stereo-isomers containing mainly compounds with mono-cis and di-cis configurations. Such a mixture is resolved chromatographically whereupon some insight into the respective configurations can be gained by means of appropriate spectroscopic methods.Whereas any manner of bending of an all-trans carotene molecule causes a displacement of the main extinction maxima towards shorter wavelengths in the visible spectral region the simultaneously occurring alteration in the adsorption affinity does not follow such a simple rule. It was found for each of the three carotenes for example that a stereo-isomer which very probably contains a single peripherally-located cis double bond is adsorbed above the corresponding all-tram compound in the Tswett column. In contrast those spatial forms of the carotenes which possess a centrally-located cis double bond (and in some instances another cis bond) show considerably weaker adsorbability than the all-trans isomer.Thus we have the following chromatographic sequence (Top) Peripheral mono-cis carotene } essentially straight types. All-trans carotene (Bottom) Central mono-cis carotene etc. essentially bent type. I. II. 111. Fig. I . Typical spatial forms of a long conjugated double bond system in an open chain I all-trans ; 11 central mono-cis; and 111 poly-cis (penta-cis). I t should be noted in this connection that the adsorption affinity of a monohydroxy-P-carotene viz. all-trans cryptoxanthin (C ,H 6.0H) undergoes the changes just described when stereo-isomerized in vitro ; hence in this instance the presence of a single hydroxyl group has no decisive influence on the relative adsorbabilities of the respective spatial forms.In contrast each cis isomer so far observed of the analogous dihydroxy compounds viz. zeaxanthin and lutein (H0.C 4oH 6,.0H) is located in chromatograms above the corresponding all-trans zone. Thus the outstanding features of this Irans+cis isomerization are weakening of the colour and an L. ZECHMEISTER 57 increase in the adsorption affinity. One could suppose that in these stereo- chemical sets both hydroxyl groups would participate in the anchoring process and that the decrease of the distance between them as caused by bending of the molecule would promote fixation. However it is also possible that such spatial forms of the dihydroxycarotenes do exist which show decreased adsorption affinities as compared with the all-trans form ; but for some reason they do not appear in appreciable quantities in the stereo- isomeric mixtures which can be obtained by current methods.Some representatives of a different type of stereo-isomeric hydrocarbons viz. poly-cis y-carotenes and poly-cis lycopenes which so far we were unable to prepare in the laboratory could be isolated from some plant materials. They contain four to seven of their double bonds in cis configura- tion. If we pass from an all-trans compound to either of its poly-cis forms then the remarkable weakening in the colour runs parallel with a substantial decrease in the adsorption affinity. A similar parallelism is also observed within the subclass of the poly-cis lycopenes whose chromatographic sequence and spectral sequence are identical.With reference to both of these physical characteristics the individual differences are much smaller within the class of the poly-cis isomers than between either poly-cis form and all-trans lycopene. Before closing these considerations we should stress that the described variations in the adsorption affinity which are a function of morphological changes are of the same order of magnitude as analogous effects caused by reasonably-chosen structural conversions. For example a solution containing several spatial forms of both a-carotene and p-carotene gave the following chromatographic sequence on calcium hydroxide when developed with petroleum ether (Top) Neo-P-carotene V Neo-a-carotene U All-trans p-carotene Neo-a-carotene V Neo-P-carotene B Neo-(3-carotene E Neo-a-carotene W Neo-P-carotene F All-trans a-carotene (Bottom) Neo-a-carotene B Evidently the weaker adsorption affinity of all-trans a-carotene (con- taining 10 conjugated double bonds) as compared with that of all-trans ?- carotene (11 such bonds) can be compensated and even overruled by a suitable adjustment of the molecular form.Furthermore an increase of the adsorption affinity several times stronger than that caused by the reaction a-carotene +p-carotene is obtained when we convert a poly-cis compound into the corresponding all-trans form. Finally the author wishes to express his appreciation to his colleagues (Mrs.) A. Chatterjee R. B. Escue F. Haxo R. M. Lemmon A.L. LeRosen W. H. McNeely J. H. Pinckard A. Polgh- A. Sandoval W. A. Schroeder J. W. Sease and B. Wille. California Institute of Technology Pasadena. 54 ADSORPTION AND SOME CONSTITUTIONAL AND STERIC PROPERTIES BY L. ZECHMEISTER Received 15th July 1949 The concept " anchoring group " is proposed and some pertinent experimental approaches are indicated. A report is submitted on the separation of cistrcrns stereo-isomers and the dependence of the adsorption affinity on molecular morphology is discussed. Although it is not assumed that a single correct picture can be given for the adsorption and geometrical orientation of organic molecules on a solid surface the following concept may serve as a basis of discussion. Two essential features can be postulated for the simple and stable fixation of an organic solute on an adsorbent first its molecules should fit into the cavities of a certain size located on the active surface; and second adequately oriented forces should be operative between the adsorbent and the adsorbed molecule.These two postulates are of course closely interrelated. Very little is known about the geometrical orientation of adsorbed mole-cules but one may speculate that the manner of orientation would influence, among others the relative behaviour of closely related compounds. If th L. ZECHMEISTER 55 orientation be roughly parallel to the surface it could be perhaps expected that within a homologous series a maximum of the adsorption affinity would be reached at a certain chain length.On the other hand it may be considered that in the case of a roughly vertical orientation of the adsorbed molecules with reference to the surface a continuous increase or decrease in the adsorption affinities would be observed by passing stepwise from lower members of the series to some higher ones. Whatever the orientation may be in a given system it seems reasonable to assume that not each section of the adsorbed molecule will be equally responsible for the fixation process. On the contrary we propose that a decisive part will be played by certain atomic groups which are conveniently designated as “ anchoring groups.” This concept is similar to that in which it is presumed that the fixation of a drug to bacteria takes place by the intermediary of a haptene group (or groups).In favourable instances an anchoring group could be located experimentally by showing that a modi-fication of that particular section of the molecule causes an unusually sharp change in the adsorption affinity. Perhaps a single example will demonstrate the general lines along which we think that laboratory work may help in collecting material for a more satisfactory discussion of such problems. It was found that the introduction of a methyl group in the a-position to the sulphur atom of cc-terthienyl markedly increased the adsorption affinity; and that upon dimethylation, this effect was about doubled. On the other hand Kofler reported that the adsorbability of some tocol derivatives decreased upon methylation near the phenolic OH group although this substitution had but little effect in a more remote position to the hydroxyl mentioned.In the case of thiophene rings the introduced methyl group seems to have been involved in the fixation process but the anchoring group of the tocol derivative was probably the phenolic hydroxyl whose function is sterically hindered by the presence of adjacent methyl groups. While manifold possibilities are open for experimentation in the direction just outlined our problem changes its character entirely when we pass from chemical to stereochemical considerations. Then in a way the situation appears to be simplified since in stereochemistry the dependence of the adsorption affinity on the overall shape of the molecules is to be considered and not the presence or absence of certain functional groups.The compounds which are suitable for such investigations should possess numerous stereo-isomeric forms whose respective morphological types should vary as widely as possible. This postulate is unequally fulfilled in various stereo-isomeric sets ; for example the epimerization of a sugar would not essentially alter the overall shape of its molecules. The conditions for studies of this kind are especially favourable in the field of the natural and synthetic polyenes which compounds possess a long, conjugated carbon-carbon double-bond system in an open chain. Their structure is “ morphologically sensitive ’’ to spatial variations. Although the ordinary or all-tram form (Fig. I) which shows a rod-like general shape is not greatly modified by a single trans+& rotation which occurs near an end of the system it undergoes a radical change and is converted into a V-like pattern when such a spatial re-arrangement takes place at or near the centre (11).On the other hand in the course of continued tralzs+cis rotations, after having passed through several bent forms a straightening-out takes place. Thus the molecules of the resulting poly-cis compound show a rod-like overall shape (111) which is morphologically similar to that of the all-trans molecules (I). As is well known the respective all-trans forms of the polyene hydro-carbons (C ,H 5 a-carotene p-carotene y-carotene and lycopene ca 56 CONSTITUTIONAL AND STERIC PROPERTIES be easily converted into a mixture of stereo-isomers containing mainly compounds with mono-cis and di-cis configurations.Such a mixture is resolved chromatographically whereupon some insight into the respective configurations can be gained by means of appropriate spectroscopic methods. Whereas any manner of bending of an all-trans carotene molecule causes a displacement of the main extinction maxima towards shorter wavelengths in the visible spectral region the simultaneously occurring alteration in the adsorption affinity does not follow such a simple rule. It was found for each of the three carotenes for example that a stereo-isomer which very probably contains a single peripherally-located cis double bond is adsorbed above the corresponding all-tram compound in the Tswett column. In contrast, those spatial forms of the carotenes which possess a centrally-located cis double bond (and in some instances another cis bond) show considerably weaker adsorbability than the all-trans isomer.Thus we have the following chromatographic sequence: (Top) Peripheral mono-cis carotene All-trans carotene (Bottom) Central mono-cis carotene etc. essentially bent type. } essentially straight types. I. II. 111. Fig. I . Typical spatial forms of a long conjugated double bond system in an open chain : I all-trans ; 11 central mono-cis; and 111 poly-cis (penta-cis). I t should be noted in this connection that the adsorption affinity of a monohydroxy-P-carotene viz. all-trans cryptoxanthin (C ,H 6.0H), undergoes the changes just described when stereo-isomerized in vitro ; hence, in this instance the presence of a single hydroxyl group has no decisive influence on the relative adsorbabilities of the respective spatial forms.In contrast each cis isomer so far observed of the analogous dihydroxy compounds viz. zeaxanthin and lutein (H0.C 4oH 6,.0H) is located in chromatograms above the corresponding all-trans zone. Thus the outstanding features of this Irans+cis isomerization are weakening of the colour and a L. ZECHMEISTER 57 increase in the adsorption affinity. One could suppose that in these stereo-chemical sets both hydroxyl groups would participate in the anchoring process and that the decrease of the distance between them as caused by bending of the molecule would promote fixation. However it is also possible that such spatial forms of the dihydroxycarotenes do exist which show decreased adsorption affinities as compared with the all-trans form ; but for some reason they do not appear in appreciable quantities in the stereo-isomeric mixtures which can be obtained by current methods.Some representatives of a different type of stereo-isomeric hydrocarbons, viz. poly-cis y-carotenes and poly-cis lycopenes which so far we were unable to prepare in the laboratory could be isolated from some plant materials. They contain four to seven of their double bonds in cis configura-tion. If we pass from an all-trans compound to either of its poly-cis forms, then the remarkable weakening in the colour runs parallel with a substantial decrease in the adsorption affinity. A similar parallelism is also observed within the subclass of the poly-cis lycopenes whose chromatographic sequence and spectral sequence are identical.With reference to both of these physical characteristics the individual differences are much smaller within the class of the poly-cis isomers than between either poly-cis form and all-trans lycopene. Before closing these considerations we should stress that the described variations in the adsorption affinity which are a function of morphological changes are of the same order of magnitude as analogous effects caused by reasonably-chosen structural conversions. For example a solution containing several spatial forms of both a-carotene and p-carotene gave the following chromatographic sequence on calcium hydroxide when developed with petroleum ether: (Top) Neo-P-carotene V Neo-a-carotene U All-trans p-carotene Neo-a-carotene V Neo-P-carotene B Neo-(3-carotene E Neo-a-carotene W Neo-P-carotene F All-trans a-carotene (Bottom) Neo-a-carotene B Evidently the weaker adsorption affinity of all-trans a-carotene (con-taining 10 conjugated double bonds) as compared with that of all-trans ?-carotene (11 such bonds) can be compensated and even overruled by a suitable adjustment of the molecular form. Furthermore an increase of the adsorption affinity several times stronger than that caused by the reaction a-carotene +p-carotene is obtained when we convert a poly-cis compound into the corresponding all-trans form. Finally the author wishes to express his appreciation to his colleagues, (Mrs.) A. Chatterjee R. B. Escue F. Haxo R. M. Lemmon A. L. LeRosen, W. H. McNeely J. H. Pinckard A. Polgh- A. Sandoval W. A. Schroeder, J. W. Sease and B. Wille. California Institute of Technology, Pasadena

ISSN:0366-9033    

DOI:10.1039/DF9490700054

出版商:RSC    

年代:1949

数据来源: RSC