84
ANALYTICAL CHEMISTRY
(117) Sideri, C. X., and Osol, d.,J . A m . P h a r n ~ Assoc., . Sci. Ed., 42, 586 (1953). (118) Sidgwick, K. V., “Electronic Theory of Valency,” Oxford, Oxford University Press, 1927. (119) Siegel, E. F., and Rloran, 11.K., J . A m . Chcni. Soc., 69, 1457 (1947). (120) Siggia, S..and Floramo, S . .I.,- ~ N A I , .CHEM.,25, 797 (1953). (121) Siggia,,S., and Hanna, J. G., Ibid.,23, 1717 (1951). (122) Siggia, S.,and Sepal, E., Ibid.,25,830 (1953). (123) Smith, T. L., and Elliott, J. H., J . A m . Cheni. Soc., 75, 3566 (1953). (124) Spandau, H., and Brunncck, E., 2. a m r g . u . a l l g e m . Chem., 270, 201 (1952). (128) Stewart, T. D., and Cook, V., J . Am. c I J ~ ( n i Soc., 50, 1973 (1928). (126) Tamres, ll.,Ibid.,74, 3375 (1952). ANAL.C H E i f . , (127) T e r r y , D. E., Kendrick, E. R., and Moe, 0. &I., 24, 313 (1952). (128) Thiers, R., Itidustricchim. E d g e , 17, 1143 (1952). (129) Tomieek, O., “Chemical Indicators,” London, Butterworths Scientific Publications, 1951. (130) Tomieek, O., and HeyrovskG, .I.,Collection Caechoslos. Chem. C m n m n s . , 15, 984 (1951).
(131j (132) (133) (134)
Ibzd., p. 997. TomiEek. O., and Suk, V., Chem. L i q t y , 46, 139 (1952). TomiCek, O., and ZukriegelovA, M , ,Ibzd., p. 263. Usanovich, hI., and Yatsimiiskii, K., Zhirr. ObshcRei A-h!/u., 11,
954 (1941). (135) Ibid.. P. 959. (136) Velbef, S., Eggerson, K , and Liiiliolt \, C., A c l a Chenc. Scririd , 6, 1066 (1952). F., Indonesian J . .’id. Sci.. 108. 19 (1952). (137) Vermast, F. -4. (138) Vespe, V., and Fritz, J. S., J . A m . Pharm. Assoc., Sci.Ed., 41, 197 (1952). (139) Wagner, C. D., Brown, R. H., a n d Peters, E. D., J . Am. Chrm. Soc., 69, 2611 (1947). (140) Wagner, W. F., and Kauffman. IT. R., A s ~ L .CHEM.,25, 538 (1953). 1141) Ward, L. F., Moore, R. T., and Hall, J. S., Ibid. 25, 1070 (1953). , 254 (1953). (142) Tashbrook, C. C., A m Z ~ s t78, (143) Wichtl, AI., Scientw Pharm., 21,20 (1953). 1144) Wynne-Jones, W.F. K., J . Chenc. Soc., 1931, 795. (145) Zezyulinskii, V. >I,, Dolzlady Aknd. S a i ~ kS.S.S.R., 81, 033 (1951).
Potentiometric Titrations N. HOWELL FURMAN Frick Chemical Laboratory, Princeton University, Princeton,
T
HIS paper attempts to summarize the chief trends in the development and application of titration methods that are essentially potentiometric. The period covered begins M ith the time of preparation of a prior review of the subject by the author (67). THEORETICAL DEVELOPMEWS
Polarization Curves and Electrotitration Methods. A much clearer idea of the fundamental principles and the interrelations of various electrotitration processes has resulted from R ork from a few active centers during the period 1951 to 1953. Clarification has resulted in the main through theoretical and experimental study of polarhation curves and a much cIearer understanding of the phenomena that are involved in indication processes. The most estensive theoretical treatment of polarization curves as related to electroanalytical methods is due to Gauguin and coworkers Charlot, Berth, Badoz-Lambing, and Coursier in France. Their papers deal with general relations (22, 7 5 ) , potentiometry a t a small constant current (1 to 5 pa.) (14, 71, 7 4 ) , interpretations of potential measurements (@), coulometry a t constant current ( 7 S ) , and amperometry at constant e.ni f. (8, 14, 7 6 ) . Polarization curves are considered in terms of “ideal electrolysis,” where current density, a t an anode, for example, is given by:
(i
=
k n ~ F(Red)e
PnF E ) , k RT
being a constant depending upon
choice of potential origin, (Red) the activity of the reductant, nT total number of electrons per molecule of reductant in the electrode process, and n the number of electrons in the slowest intermediate stage-i.e., n < nr. p is the transfer coefficient. With the aid of equations of this type for the various oxidants and reductants, and polarographic diffusion expressions, mathematical expressions are derived for anodic, cathodic, and mixed anodiccathodic polarization curves for reversible and irreversible oxidanbreductant systems or for mixtures of such systems, both reversible and irieversible. The position of the polarization curves along the applied voltage axis is important in determining the magnitude of the e.m.f. change that is produced a t the end point a t small constant applied current. Reilley, Cooke, and the writer (147) developed a three-di-
N. 1. inenqional model that is useful in understanding the ideal r e h tionships betFeen polarographic E M values and polarographic bvaves, current, and ratios of oxidant and reductant concentrations, in reversible systems. This model shows clearly the relation between potentiometric and amperometric titration? and polarography. By considering planes of currents slightly atlove and below zero-e.g., 1 to 2 pa.-the possibility of derivative relationships leading to potentiometric changes a t end points is easily seen. The relationships are more clearly visualized from polarization curves (146). Because data are needed with vnrious media and various sizes and shapes of electrodes both for medium alone and with oxidant and reductant systems, a cui-rent-scanning technique was developed by Adams, Reilley, and the writer ( 5 ) and found to be much more rapid and convenient with solid electrodes than making a voltage scan. Time maxima do not occur with current as the controlled variable. These curves in actual situations are of great value both for studying new titrimetric possibilities and for coulometric studies. Delahay ( 46). as \veil as Gauguin et aZ. (74) and Badoz-Lambing (8) also realized clearly the importance of polarimtion curves to the coulometric titration methods. With an efficient technique ( 5 ) it is posqible to find out rapidly R hether a coulometric reagent may be generated a t an adequate approach to 100% efficiency in themedium desired and with the particular size and shape of electrode and current density that one intends to use. Duyckaerts ( 5 3 ) has also considered the bearing of polarization curves upon methods involving constant current or constant voltage between electrodes. He developed ( 5 4 ) a theoretical treatment that has much in common with the work of Gauguin and associates and the author’s studies. His equations are developed from the standpoint of the rate theory of Eyring and associatesfor example, for an irreversible system with oxidant and conjugate reductant present:
-anF E i = k, (Ox) e RT
- kO(Red)e(1 RTa)nFE,
where k,,kz are rate constants and 01 is a coefficient of activation. His treatment for a reversible system reduces to the classical polarographic equation. He has stated clearly the relation between the various electrometric methods and also considers the bearing of constant resistance on electrometric methods.
V O L U M E 2 6 , NO. 1, J A N U A R Y 1 9 5 4
85
tion in behiivior of an indicator elect,rode that is polarized I)y a Kambara et ul. (100, 189) have exainiuetl the theory of the suitably chosen attackable electrode may be used for indication. diffusion current. a t the rotating platinum electrode and have conThis type of procedure is fairly familiar in the aniperomet ric: cluded that the current’-voltage equation or di:igrams are essentitration field. tially similar to those for a mercury electrode. Potentiometry at Constant Current Density. I ~ R I V A T I V E Theory of Equivalence-Point Potentials and Equilibria. Bishop ( 2 6 ) has pointed out that in cases where all the coeffic>ieiits POLAROGRAPHIC TITRATIOS. The subject of potentiometry a t bRed2 = cRedl tlOss, in a redox reaction are different, aOxl constant current density has been developed ( 7 4 ) both for systems calcu1:itions of equilibria are difficult or impossible unless n 6 consisting of a reference electrode and an inclicator electrode and = c (E. Ohlweiler ( 1 3 8 ) has discussed the theory of the for two similar indicator electrodes. The latter process is identicquivalence point and the completeness of t,he reaction in rcclox cal x i t h the method thst, the mthor’s group has called “deriva,tive reactions. po1arogr:iphic titratim” ( 1 4 6 ) . K i t h dissinii1:tr electrodes there Gr:in ( 8 0 )has at1vw:itetl the plotting of potentiometric titration is a strict limitation in the amount of current t.hat can he passed. data in linear form by using expressions such as ( V O 1’) This newer field of potentiometric titration will become a very 10“’- pH = K , ( V , - V ) ,Lvhere K1]K?:ire constants contxining useful supplement to preexisting methotis. activity coofficients and Vo,Ve,antl V : u p , respectivcly. initial Dead-Stop Method. liiUPEROMETRY .iT c O S B T . i N T POTESTIAL. volume, volume a t the equivalence point, and volumc of reagent The chief differences of opinion between the authors of recent added a t :in>- point.. -1similar equation wit,h appropriate rontopic appear to arise from lack of definition asstauts :rpplies he>-orid the end point. Similar expressious are mall” applied e.m.f. As originally proposed by given for other types of t,itrations. The process may he simplified Foulk anti Bawclen (1926), the ap~iliede.ni.t. was designated as by plotting points caorrcrted for volumc changes on 1og:criihmic 10 t o 1,5 niv. If this definition is adhered to, very many of the papcr. later applications that have been cllissecl as (lead-stop titrations APPARATUS deviate more or less markedly in the nature of the cn(l-iwint’indication a i d are very clearly of amperometrir titration iuiture, in Automatic Titration. Frediani (60) developed a special a[)the sense that a very large e.m.f. hetneen electrotles must be a11paratus now conunercially available (Becknian Co.) for the deadplied in order to cut the polarizat,ion curves from vitrious constop titration as applied in the determination of water by the centrations of an oxidant or :t reductant. Stonct and Srholten Karl Fischttr method. A time-delay snitch preadjusted for any ( 1 7 0 )have used applied volt,ages from 0. to 0.4 volt’. .A11 these time up to 1 minute prevents false end points due to the .slow t const ant pot eiit i d ’’ methods approach forms of “amperam yielding of water from suspended matter. Titration is (76). automatkally resumed if the electrodes drift, hack within the Delahay ( 4 5 ) interpreted dead-stop phenomena :it 10 mv.nppreset interval. plied e.m.f. iri terms of polarization curves of rcver.=iblcand irC:irson (3.5)developed an automatic apparatus for coulometric reversible systems, The latter type of system mag sllow anodic titmtion and also ( 3 6 ) a circuit that gives automatic titration tQ and cathodic overvoltages so great t,liat no electrolysis c a n proan end point with a portion of t,he solution reserved in a side tube. ceed a t the small applied potent,ial. If :Lreversible system apAt t-his point n piston operated magnetically forces the reserved pears at the end point, a n appreciable current can pass a t t’he solution into the main vessel and the titration is resumed autoe.m.f. applied. An increase in the a.pplied e.m.f. can eventually matically at a slow rate until the true end point is reached. Other cause an irreversible system to pass current. The electrolytic automatic titrators have been described by Juliard and van Caprocess at one electrode may not he the reverse of the process at kenberghe ( 9 9 ) ! Dunn et ul. (61)(dual assembly for aqueous and the other clectrotle. Stone and Scholten (170) found that nitrite nonaqueous media), Saeki (I;.$), Wise 11.96), and Fowler et nl. in diaxotation reactions a t 0.4 volt applied yields nitric oxide at, (5Q). The last is a combined robot titrator and dispenser capahlo the cathode and nitrogen dioside a t the anode. of 100 titrations per hour. Stock ( 1 6 9 ) lxts reviewed dead-stop applicatioIis. Rratlbury Several pH nirters and racuulu-tube voltmeters have becxn de(26) has given a thorough general discussion of the method in Champaygne sciilwd hy Tali:& et al. (174), Ishibashi et al. (94), terms of overpotential effects (activation overpotential, concen( S T ) , and Antlerson and Greenwood (6). DuBoff and Prat.t (4itrat,ionof the reduced solutions, allon-ing them to How int.0 ferric iron from a reduct.or for the win, and by selective reduc+on t,o vanadium(1V) for vanailium alone. Oelsen ~t al. (136) describes elrc*trolgt,ictitration for a d s , and oxidmts eucth as manganese( VI1 i. cahromium(VI), or vanadium
(V. Ceric Titrations. Takahashi rt al. ( 1 7 3 ) used oxidation by excess of standard ceric solution followed by back-titration with ferrous sulfate for determination of polyhydric alcohols or alginic arid. Wells (194) titrated micro amounts of iron (0.27 to 0.5 mg.) with O.OO5N nitratoceratr solution after reduction in a silver reductor. Bichromate Processes. Collier :tnd Fricker (40)used a modified dead-stop method for titration of ferrous ion with bichromate in approximately normal solutions. Gokhale ( 7 8 ) allowed cuprous chloride to react with ferric alum with air excluded and titrated the ferrous ion with bichromate. Oxides of Chromium and Manganese. Jatkar and Mainkar (g6)report a series of intermediat,e oxides formed by reaction of manganese(I1j with permanganate in acid and alkaline solutions, and a similar series for chromium (97). Periodate. Singh et al. (160) report determinations of mer('urous, stannous, arsenious, and ant,imonous compounds and of hydrazine and thiocyanate by reaction with potassium pwiodate. Mazor and Erdey (190) determine bivalent vanadium with periodate. Zinc is reported t,o be less effective than cadmium in reducing the vanadium. They report reduction of periodate to iodide in the process vanadium(I1) .-,vanadium(II1). Bromate Processes. The direct estimation of furfural by hromate titration in presence of bromide and niolybdate catalyst has been described by Domansky (48). Britton and Britton ( 2 7 ) have studied the titration of bromate ion by iodide ion. Vanadate as Oxidant. Rius and Diaz-Flores (151) titrate titanium(II1) or molybdenuni(II1) with vanadate which is reduced t,o vanadium( 111). By differential processes involving titrations with vanadate or with ferric solution mixtures of t'itanium, molybdenum, and vanadium can be analyzed. Hypochlorite. The indirect determination of bivalent metals that form double ammonium salts is possible by oxidation of the aninionia derived from the salt with hypochlorite or with hypobromite, according to Simon et al. (165). Bromine. Kiess (106) investigated the dead-stop method in the titration of iodide with bromine. Iodate Methods. McBridc et al. (116) applied potentiometric titration to iodine chloride or iodine end points in the t h a t i o n of hydrazine or of its organic derivatives ( 1 1 6 ) . A similar study was reported earlier by Miller and Furman ( 1 2 3 ) on one or two derivatives. Britton d a / . (29) investigated the titration of iodic acid with iodide.
88 Mixtures of Copper and Oxidants. Nievas and Berges (133) studied mixtures of copper and iodate, bromate, or permanganate. Iodate is determined after complexing copper with citrate and adding iodide. Similar methods are applied to the other oxidants. Ferricyanide. Adams et al. (4)used the derivative method (146) for the titration of glucose. Matsuno and Sano (119) used both the ferricyanide and the copper methods for the estimation of glucose. Ferrous Solutions. The titration of chromate and vanadate after persulfate oxidation and reduction of permanganate by oxalate or for vanadium alone by nitrite followed by urea has been reinvestigated by Enghag (56). Kitrous acid reacts with bichromate and oxalate should be used to destroy permanganate if it is present. Iodine and Iodide Methods. The use of the Karl Fischer method for moisture in food products has been studied by Frediani et al. (61). Titration of bisulfites with or without formaldehyde added in buffered media is described by Bertorelle and Giuffre (16). Knowles and Lowden (109) found the amperometric method more satisfactory than the dead-stop or the derivative method for very dilute iodine-thiosulfate titrations. Tanner and Rentschler ( 1 7 7 )found the dead-stop method suitable for estimation of sulfites with iodine in colored fluids. Swinehart ( 1 7 1 ) found the dead-stop method satisfactory for titration of stannous chloride by iodine, Foster and Smith (68) used a differential null procedure for titration of the branched starch fraction with iodine. -4similar half cell with potassium chloride and potassium iodide present was titrated to restore balance after iodine was added to the starch. The process is a type of concentration cell method, The dead-stop procedure was found by Abrahameon and Linschitz (1)to be useful i n t h r determination of organic peroxides. After tellurium(1V) is treated with excess of thiosulfate, the excess thiosulfate is determined with standard iodine according to Johnson and Frederickson (98). Selenium is determined along with tellurium in the process. Nitrous Acid and Nitrite Processes. The reaction of nitrou; acid with iodide has been studied potentiometrically by Britton and Britton (28). The titration of primary amines has been investigated by several workers: Matrka (118) investigated amines and dihydroxyphenols, etc., with platinum-calomel and with bimetallic systems. Ferrero and Brehain (66) as well as Stone and Scholten have applied the dead-stop method to the diazotization reaction (170). Saeki (154) has also studied the potentiometric method in this application. Mercurous Nitrate as Reductant. Belcher and West ( 1 2 ) have investigated extensively the reducing action of mercurous solutions in presence of thiocyanate. Many of the studies are nonpotentiometric. The determination of iron in ferric thiocyanate in presence of nonreducible coloring material is described (18). Hubicki et al. (91) found that nitrate, sulfate, selenate, and chloride did not interfere in the potentiometric titration of selenious acid with mercurous nitrate. Burriel and Lucena Conde (SO) have also reported on the titration of ferric thiocyanate by mercurous nitrate. Chromous Solutions. Rienacker and Jerschkewitz (160) used chromous solution to titrate copper in presence of iron. Wakkad and Riee (191) found that tungsten could be titrated in presence of phosphate with 5 moles of citric acid per mole of tungsten and i n 36% hydrochloric acid. Muraki (131) has reinvestigated the preparation and standardization of chromous salts. Copper was t h e etandard substance. Chlebovsk9 and Brh6Eek (59)determined copper, tin, and antimony with chromous chloride in strongly acidic medium with much magnesium chloride present. Busev (91) restudied the titration of bismuth with chromous solution. Burriel and Suarez-Acosta (SI) studied chromous and stannous titrations of molybdenum in steel?. Manganese, chromium. and vanadium interfere.
ANALYTICAL CHEMISTRY NONAQUEOUS MEDIA
Lithium Aluminum Amide. Higuchi et al. (89) have reported on the titration of very weak bases with this reagent. Glacial Acetic Acid. Tomicek and associates have made extensive studies of redox reactions in this medium. Tomicek and Heyrovsk9 (186) studied the oxidation of arsenic(III), antimony(III), mercury(I), pyrocatechol, resorcinol, hydroquinone, and diphenylamine with bromine. Chromic anhydride was found unsuitable for arsenic but useful for titrating ant,imony, iron, titanium, and the organic reductants. Sodium permanganate was useful for thanium, iron pyrocatechol, or hydroquinone. Reductions of ferric salts or chloranil with titanous chloride v,-ere studied. Bromine was used by Tomicek and Valcha (187) for titration of a number of organic compounds. Lead tetraacetate was also useful for the oxidation of ascorbic acid, hydroquinone, and pyrocatechol. Tomicek and Heyrovskj. (184) used bromine to titrat,e sulfamides. Tomicek et al. (182)st,udied the titration of unsaturated sutvtances with bromine. Luk'yanits and Sekrasov (114) used the platinum-c:il:~n:elelectrodr system in the titration of organic sulfides with potassium iodate in 90yoglacial acetic acicl. Liquid Ammonia. Watt et al. (192) studied the titration of halides of aluminum, gallium, indium, and thallium with potassium in liquid ammonia. Gallium and indium show three electron changes, but thallium (0j reacts with thalliuln chloride to show int,crnirdiate formation of thallium(1). IVatt and Otto (193) used platinum elertrodes, one in a capillary with small portion of the solution surrounding it, for the titration of polysulfides with potasPiuni in liquid ammonia. LITERATURE CITED
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a9 (94) Ishibashi, M.. and Fujiuaga. T., J . Japan. Chem., 4, 554 (1950). (95) Issa, I. M.,and Awad, S., Analyst, 78, 487 (1953). (96) Jatkar, S. K. K., and Mainkar, V. B., J . Indian Chem. Sac., 28, 497 (1951). (97) Ibid., 28, 645 (1951). (98) Johnson, R. A., and Frederickson. D. R., ANAL.CHEM.,24, 866 (1952). (99) Juliard, A., and Cakenberghe. J. van, Chimie & Industrie, 63, 72 (1950). (100) Kambara, T., Tsukamoto, T., and Tachi, I., J . Electrochem. SOC. Japan, 19, 199 (1951). (101) Kamienski, B., Bull. intern. acad. polon. sci., Classe sci. math. et nat., Ser. .4,1949, 149. (102) Ibid., p. 73. (103) Kamienski, B., Polska Akad. Umiejetnosci Rozprawy Wydzialu Matprzyrod, 74A, 47 (1949). (104) Kate, M., and Glenn, R. A , , ANAL.CHEM.,24, 1157 (1952). (105) Keen, R. T., and Fritz, J. S., Ibid., 24, 564 (1952). (106) Kiess, H. L., Anal. Chim. Acta, 6, 190 (1952). (107) Kilpi, S., J. A m , Chem Soc., 74, 5296 (1952). (108) Kirrmann, A., and Daune-Dubois, N., Compt. rend., 236, 1361 (1953). (109) Knowl-s, G.. and Lowden, G. F.. Analyst, 78, 159 (1953). (110) Kolthoff, I. M., and Kurode, P. K., ANAL.CHEM.,23, 1304 (19.51). 1111) Lee, W.A,, and Peaoocke, -4.R.. J . Chem. Soc., 1951, 3361. (112) LQon.C., Compt. rend., 233, 170 (1951). (113) Li, N. C . , and Doody, E., J . Am. Chem. SOC.,74, 4184 (1952). (114) Luk’yanits. V. G., and Nekrasov. PI. S., Doklady Akad. N a u k S.S.S.R., 90, 1043 (1953). (11.5) SlcBride. W.R., Henry, R. A.. and Skolnik, S., ANAL.CHEM., 23, 890 (1951). (1 16) Ihid., 25, 1042 (1953). (117) hlarkunas, P. C., and Riddick, J. -4.,ANAL.CHEM.,24, 312 (1952). (118) llatrka. hf.. Chernie (Prague). 8, 13 (1952). (119) llatsuno. T.. and Sann, A,. J . Electrochem. SOC.J a p n , 18, 180 (1952). (120) Maror. L., and Erdep. L.. Acta Chim. Acad. Sci. Hung., 2, 331 (1952). (121) Meites. L., J . A m . Chem. SOC..75, 2859 (1953). (122) Xlendle, R. L.. and Henderson, S. R.. ANAL. CHEM.,25. 840 (1953). (123) Sliller, C. 0.. and Furman. N. H.. J . A m . Chem. SOC.,59, 161 I 1 9.77) (124) de Miranda H., and Lemmens, .J F., Perfumery Essential Oil Record, 43, 226 (1952). (125) Vitra R . P., and Mathur, H. B., J . Phys. Cheat., 56, 633 ( 1 952).
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chromatsgraphy And Analogous Differential M i g r a t i o n M e t h o d s HAROLD H. STRAIN, T. R. SATO, and JOHN ENGELKE Argonne National Laboratory, Lemont, 111.
T””
A early reviews in this series were devoted solely to the subject of chromatography. Subsequently chromatographic techniques were recognized as prototypes of numerous analytical methods based upon the phenomenon of differential migration (165, 171 ). Consequently, the scope of this revlew has been extended to include a number of these related, widely applicahle, differential migration technique.. I n the 2 years following the last review in this series (166), there has been an enormous acceleration in the development, the improvement, and the application of differential migration methods of analysis. In many productive research laboratories. several of these methods now serve as adaptable tools for exploration, for control, and for corroboration ( 3 4 , 66, 7 3 , 120, 128, 170). This increase in the number and applications of these basic analytical tools and a concomitant multiplication of the workers versed in their use have stimulated progress in all aspects of science concerned with chemical substances and their reactions. The resultant expansion of knowledge has been so rapid, so great, and so diverse that i t cannot be cited here Even the specialized literature pertaining to the development of the tools themselves ran scarcely be summarized in the space allotted t o this revien,. The examination and correlation of current reports have been complicated tremendously by the description of many minor modifications of earlier methods. Moreover, many of these reports do not include references to the earlier investigations. A note on the separation of chloroplast pigments by paper chromatography (11), for example, gives no reference t o the early reports on the formation of chromatogram with these pigments,
observations which formed the foundation of columnar and paper chromatography (165, 164). For economy, the citations in this review have been restricted to books ( 2 , 6, 19, 26, 42, 4 9 . Y5, 88, 130, 141, 156), surveys (10,16, 27, 51, 32, 35, 41, 64, 74,82, ‘10, 91, 102, 105-107, 109, 118, 127, 129, 131, 158, 160, 164, 165, 171, 174, 182, 184), bibliographies (69, 104, 125), and current reports on basic procedures. This selected material should provide a key to the rapidly expanding literature. Many of the modifications and applications of the techniques can be found only by perusal of the literature of specialized fields. DIFFERENTIAL MIGRATION ANALYSIS
Basic Conditions. For the examination and correlation of various separatory techniques, the concepts of differential migration analysis have now been restudied and extended. I n all these methods of analysis, the migration itself is produced by the application of one or more driving forces, and it is usually opposed by resistive forces. For effective separations, either or both the driving force and the resistive forces must act selectively upon the migrating substances. The effects of a nonselective driving force (gravity) acting upon two different kinds of particles with and without a selective resistive force (viscosity) are illustrated by Figure 1. I n differential migration analysis, the migration conditions must be selected so that the components of each mixture migrate a t different rates. The migration rates are a property of the migration system. They depend upon the properties of the
