proportional to l/[H+], [H202], and [Cu*+-Bipyj, but is probably independent of the concentration of N H 2 0 H (which is, however, consumed during the reaction). The value of YO decreases with increasing [Bipy]. The following expression - analogous to (11) and (15) - was found for the initial rate YO of disappearance of H202 in the same reaction between H202 andNH2OH using free C P i o n as catalyst 190bl: A reaction course as shown in equations (12) and (16) and an active complex as shown in equations (13) and (17) can again be formulated for this system [90a,bl. If the metal ion undergoes a valency change during the catalysis in these reactions, it must be assumed that it occurs between CuI and Cu", and not Cu1II, since o n the one hand Cull is reduced to yellow copper(]) oxide by both NH2NH2 [951 and NH2OH (95.961 in alkaline media, and on the other Cu1 can be reoxidized t o Cu" by H202 [971 (cf. also [ I l l ) . 6. Conclusion The systems discussed here give an idea of the variety of reactions that can occur on decomposition of H202, and the examples in Section 5.3 and 5.4 also clearly [95] 7'.R. Bhat, D . Radhamma, and J . Shanknr, J. inorg. nuclear Chem. 27, 2641 (1965). [96] J. H . Anderson, Analyst 91, 532 (1966); 89, 357 (1964). 1971 A . Zuberbuhler, Helv. chim. Acta 50, 466 (1967). show the relation between catalase and peroxidase activities. Similar active complexes, with their secondary reactions [see equations (13) and (17)], can be formulated for the systems mentioned; one of the H202 molecules can be replaced by an H donor such as HzN-NHz or NHzOH (cf. 190bI). The Cu2+-2,2'-bipyridyl ( 7 : l ) complex can thus catalyze both the catalase (1) and the peroxidase (2) reactions, depending on the reaction conditions. The behavior of this catalyst is thus similar to that found with enzymes (cf. r40,421). A similar relation to that demonstrated in this survey between the structures of Cu2+ complexes and their catalytic activity is also to be expected for complexes of other metal ions. It may be assumed that the conditions that must be satisfied by a Cu2f complex if it is to be capable of catalyzing the catalase or peroxidase decomposition of H202 are also applicable to enzymes containing copper. I am grateful to Pro$ Dr. H . Erlenmeyer for his numerous suggestions. I thank Prof. Dr. S. Fallab, Dr. B. Prijs, and Dr. R. Griesser for their readiness to discuss the problems that arose. Thanks are due to the Schweizerischer Nationalfonds zur Forderung der wissenschaftlichen Forschung for support of our work. Received: April 1, 1968 [A 683 IEl German version: Angew. Chem. 81. 161 (1969) Translated by Express Translation Service, London. Electrochromism and Solvatochromism By W. Liptayf*l The position and the intensity of electronic bands are influenced by an electric fiefa'. Pronounced changes in the position of absorption bands are mainly due to the dipole moment of the molecule in the ground state and the change in the dipole moment during the excitation process, and pronounced changes in intensity are due to the fieId dependence of the transition moment, which can be described by the transition polarizability. The efect of an external electric field on the optical absorption (electrochromism) of suitable molecules can be used to determine the dipole moment in the ground state, the change in dipole moment during the excitation process, the direction of the transition moment of the electronic band, and certain components of the transition polarizability tensor. These data largely determine the strong solvatochromism (solvent-dependence of the position and intensity of electronic bands), which is observed in particular with molecules having large dipole moments. Smaller contributions to solvatochromism result from dispersion interactions, which predominate in the case of nonpolar molecules. The models developed have been experimentally checked and verified by a combination of electro-optical absorption measurements (influence of' an external electric field on absorption) and investigation of the solvent-dependence of the electronic bands. 1. Introduction The absorption or emission spectrum of a substance in the vapor state can exhibit characteristic differences in the positions and intensities of the bands from the spectrum of the same substance in solution; these Angew. Chem. internat. Edit. 1 Vol. 8 (1969) / No. 3 differences depend on the solvent 11-51. This solventdependence of electronic bands is known as solvatochromism. . ['I Prof. Dr. W. Liptay Institut f i r Physikalische Chemie der Universitst 65 Mainz, Postfach 3980 (Germany) [ I ] A . Kundt, Ann. physic. Chem. 4, 34 (1878). 177 The optical properties of molecules can also be influenced by an external electric field; this phenomenon shall be called electrochromism. The fundamental cause of strong solvatochromism, such as is observed in polar dye molecules, is the same as that of electrochromism, i.e. the changes produced in the electronic bands by an electric field. 2. Electrochromism Changes in optical absorption due to the action of an external electric field are basically attributable to three effects, which will be described in the following paragraphs. 2.1. Orientation Effect As our example of a solute molecule, let us consider p-nitroaniline, which exhibits an intense long-wave absorption band (maximum at 28 900 cm-1, extinction coefficient = 13000 lmole-1 cm-1, width at half height = 6000 cm-1) in benzene. The absorption curve can be recorded with natural, unpolarized light or with linearly polarized light. In an ordinary solution the molecules are distributed isotropically, and both methods consequently give the same absorption curve, which can be represented as a plot of the extinction coefficient E ~ O ~ ( G )against the wave number ?. The extinction coefficient is a measure of the intensity of the optical absorption at a given wavelength. The measure of the intensity of an absorption band is the transition moment p","a'of the dissolved molecule, which is related to the molar decadic extinction coefficient by the integral absorption: Band @-+a) The integral should be extended over the whole of the absorption band corresponding to the electronic excitation from the ground state g to a definite excited state a. NA is Avogadro's number, h is Planck's constant, and e is the velocity of light. The transition moment p g l is a vector, which can also be represented by its absolute value p",",' [ and a unit vector rn in the direction of the transition moment. I For a single molecule or for a n assembly of molecules having a definite orientation, the extinction coefficient [2] S . E. Sheppard, Rev. mod. Physics 14, 303 (1942). [3] K . Dimroth, S.-B. Ges. Beford. ges. Naturwiss. Marburg 76, No. 3, 3 (1953). [4] C . Reichardt, Angew. Chem. 77,. 30 (1965); Angew. Chem. internat. Edit. 4, 29 (1965). [S] N . Q. Chako, J. chem. Physics 2, 644 (1934). 178 &orient with light polarized in the direction of the unit vector e is: me is the scalar product of the unit vectors in and e, and is therefore equal to the cosine of the angle y between them. For an assembly of molecules having a definite orientation, the extinction coefficient depends on y as indicated in eq. (3), i.e. on the polarization direction of the incident light wave. The direction rn of the transition moment of molecules having a known orientation can therefore be determined from the dependence of the extinction coefficient on the polarization direction of the light. Partial orientation of molecules in solution can be achieved by means of an external electric field Fa. The direction of the transition moment can be found from the change in optical absorption due to this field. The total electric dipole moment p: of a molecule in the electronic ground state g in the electric field is F P, = P, + ag F, (4) p g is the permanent dipole moment, ug is the polarizability tensor, and Fe is the effective field strength at the position of the molecule. According to eq. (4), the dipole moment of a molecule is influenced by an electric field; the quantitative measure of the dependence is the polarizability ag-For molecules having a sufficiently large permanent dipole moment, p g [ agFe and the polarizability term in eq. (4) may be disregarded. In this approximation the energy EE of a dissolved molecule in the electronic ground state in the electric field is I >I 1, EP' is the energy of the dissolved molecule in the ground state with no external field, and 9 is the angle between the vectors p g and l i e . For molecules having a small permanent dipole moment, on the other hand, the term in eq. (4) that contains the polarizability ug is no longer negligible. Eq. ( 5 ) then also contains an additional term that depends on ug. For the sake of simplicity, however, this will not be taken into account here or in the following equations. According to eq. ( 5 ) the energy E[ depends on the direction of the dipole moment p g in relation to the field Fe. The orientation distribution of the molecules in a homogeneous field, e.g. between parallel electrodes, is therefore no longer isotropic. There will be more molecules with their dipole moments parallel to the field direction than with their dipole moments opposing the field direction. The distribution of the molecules over all possible orientations in the equilibrium state can be determined by means of MaxwellBoltzmann statistics. The extinction coefficient EF of the dissolved molecules in the electric field is found from eq. (3) by multiplication by the Boltzmann factor C exp( -EE/kT) and integration over all orientations. Angew. Chem. infernaf.Edit. / Vol. 8 (1969) J No. 3 The relative change in the extinction coefficient as a result of the orientation effect in an electric field is found to be effect. This hinders the simple determination of the directions of transition moments, but at the same time it allows the determination of additional characteristic molecular data. 2.2. Band Shift Effects is the angle between the direction of the external field Fa and the polarization direction e of the incident light, k is the Boltzmann constant, and T is the absolute temperature. The relative change in the extinction coefficient depends on the square of the field strength, the angle and the angle y between pgaand pg,since h p g = pg cos y. For the case in which the transition moment pgaof a band is parallel to the dipole moment pg of the molecule, and the polarization direction e of the incident light is chosen parallel to the external field Fa, it follows from eq. ( 6 ) that x, and for the case in which pgais perpendicular to pg and e is again chosen parallel to Fa,we obtain The extinction coefficient of an absorption band whose transition moment is parallel to the dipole moment increases in an electric field, while that of a band whose transition moment is perpendicular to the dipole moment decreases. The numerical values given relate to p-nitroaniline (pg = 6.3 D); the field strength used was Fa = 10s V/ cm, i.e. approximately the maximum external field strength that can be achieved in solution, and the temperature was T == 293 OK. Extinction changes of the order of 10-4 due to the orientation effect are thus to be expected in a strong external electric field. To be able to measure the extinction changes with an error of = 1 %, an accuracy of about 10-6 is necessary; this can be achieved with a measuring set-up developed by Labhart L6.71. Measurements on the first intense band of p-nitroaniline showed an increase in the extinction coefficient in an electric field, i.e. the transition moment is parallel to the dipole moment. The orientation effect, which leads to dichroism of the absorbing solutions in a n electric field, was described by Werner Kuhri and his colleagues[sl, and was to be used to determine the directions of the transition moments of electronic bands. However, at least two other effects are superimposed on the orientation [6] H . Labhart, Chimia I S , 20 (1961). [7] W . Liptay, W . Eberlein, H . Weidenberg, and 0 . Elf'lein, Ber. Bunsenges. physik. Chem. 71, 548 (1967). [8] W. Kuhn, H . Duhrkop, and H . Martin, 2. physik. Chem., Abt. B 45, 121 (1940). Angew. Chem. internat. Edit. 1 Vol. 8 (1969) 1 No. 3 The electronic excitation of a molecule may be associated with a change in the dipole moment p a - p g (pa =: dipole moment in the excited electronic state in question). The energy of a molecule in the ground state in an electric field is given by eq. ( 5 ) . In the excited state, EF' is the energy of the dissolved molecule in the excited state with no external field. In the second part of eq. (7) and in the following equations, it is assumed for simplicity that the dipole moment pa in the excited state is parallel to the dipole moment pg in the ground state; this is in fact so in p-nitroaniline and, in general, in molecules having symmetry Czv. For a given transition having the absorption wave number Jp' in g L solution with no external field (he:?' = Eio' - Esol the absorption wave number G: in an electric field is given by The change in the dipole moment (pa - pg) thus leads to a field-dependent shift of the band, which also depends on the angle 9. between the dipole moment and the field direction, as illustrated in the term diagrams of Figure l . a) Fig. I . b) C) d) Dependence of the absorption wave number on the orientation of the dipole moment with respect to the field direction (va ,!vg). Figure l a shows the term diagram of the dissolved molecule with no external field; the difference between the two levels is proportional to G p I . If the dipole moment of the dissolved molecule is oriented parallel to the field (Fig. lb), cos W = 1, so that the energy in the ground state is reduced by pgFe and that in the excited state by vaFe. In the example considered here pa > vg; the distance between the levels EF and Eg" thus decreases, and the absorption wave number G f t (parallel orientation) < G p l . If the dipole moment is oriented perpendicular to the field (Fig. lc), cos W = 0 and the energy levels will be the same as in the absence of an external field; thus G1 (perpendicular orientation) = "0'. Finally, if the dipole moment is oriented antiparallel to the field (Fig. Id), i.e. cos 0 = -1, the energy in the ground state will be increased by ! L ~ Fand , that in the excited state by paFe; in this particular case, therefore G t 1 (antiparallel orientation) > GZOl. 179 cm-1. The width of the band at half its height, on the other hand, is 6000 cm-1. The shift is therefore greatly exaggerated in Figure 2, since for a 1 cm shift of the band, the width at half height should be 6 m. I I I I I I I Fig. 2. Unsymmetrical broadening of an absorption band as a result of the band shift. Figure 2 shows the absorption spectrum ~sol(S)for the band in the absence of an external field, with a maximum at S;ol. For the molecules with p g parallel to Fe,Stt < S p l , i.e. the f at lower wavenumbers. For moleabsorption band ~ f lies cules with pg perpendicular to Fe. the absorption band E~ appears at the same wavenumber as in the absence of an external field. Finally, for molecules with p g antiparallel to Fe, the absorption band E T is~ situated at higher wavenumbers than with no external field. The distribution of the molecules over the various orientations is determined by Boltzmann statistics. There are thus always more molecules with p g parallel to Fe than with pg perpendicular to Fe, and more with p g perpendicular to Fe than with p g antiparallel to Fe. The absorption component E T is ~ therefore greater than the , this in turn is greater than E T J . component E ~ and In this discussion we have picked out three possible orientations. However, the molecules are distributed monotonically over all orientations. The actual course of the absorption curve EF@) in an external electric field is approximately as shown by the broken curve. The band displacement effect leads to broadening of the absorption band in an external electric field. Since this broadening is unsymmetrical, the maximum of the band is also shifted. According t o eq. (8) the maximum shift is Sy' - S t t = ( p a - pg)Fe/hc. In the example of p-nitroaniline pa - pgw 8 D, and SO Sy' - S t t m 10 It can be seen that the shift of the maxima of the broad absorption bands of molecules due to an external electric field, unlike the normal Stark effect in sharp spectral lines, cannot be observed directly. In the case of a constant, very small shift, the field-dependent extinction at a given wave number depends an the slope of the band, and increases with increasing steepness of the band (Fig. 3). The change in extinction due to the band shift can therefore be satisfactorily measured on steep sides of the absorption band. To calculate the relative change in the extinction coefficient in the external electric field as a result of the band shift, it is necessary to start with equations (3) and (8), and again to average over all orientations. This gives The slope of the band (d ln(ESOl/S)/d S ) must be taken at the absorption wave number Ja in question; since the slope is a function of F, the relative extinction change also depends on the wave number. The relative extinction change also varies with the square of the field strength, the angle 1,the dipole moment in the ground state, the change in the dipole moment, and the direction o f the dipole moment with respect to the transition moment. In favorable cases therefore the change in the dipole moment pa - pg and hence the dipole moment pa of the molecule in the excited state can be determined from the experimental quantity corresponding to the band shift. In the example of p-nitroaniline For a steep flank of an absorption band, the derivative of ln(Esol/S) with respect to S may be about 10-3, so that [(EF-ESo')/ES0I]BS may be about 5 x 10-3. In the case of steep bands, the relative change in the extinction coefficient caused by the band shift may be more than ten times as great as that due to the orientation effect. 2.3. Direct Field-Dependence of the Transition Moment y"Fig. 3. Change in extinction in an electric field due to the band shift. absorption curve E S O I ( ~ ) of the dissolved molecule with no external field, - - -: EF(?) in a field. (EF - ESO~)G represents a large change in extinction in the region of a steep rise, and (EF - E S O ~ ) K is a small change in the region of a gradual rise. -: 180 If the wave function of' a molecule is known, the dipole moment can be calculated by an integration process; for example the dipole moment p z of the molecule in the ground state g in an electric field is Angew. Chem. internat. Edit. J Vol. 8 (1969)/ No. 3 ri are the coordinates of the positions of the nuclei and electrons of the molecule, their charges being ei, and $: is the wave function in the electric field. The integration must be carried out over the coordinates of all the particles over the entire space. Similarly, the electric dipole transition moment pza for a transition between the ground state g and an excited state a in an electric field is Thus F P, F = Pgs To find p:, it is formally only necessary to replace the index a in eq. (11) by g, i.e. to use the wave function @ of the ground state instead of the wave function $: of the excited electronic state. The dipole moment p: in an electric field can also be expressed with the aid of the polarizability tensor ag by eq. (4).A completely analogous expression can also be given for the transition moment pza in an electric field: is the transition moment of a molecule in the absence of an external field; it corresponds to the permanent dipole moment, and will therefore be referred to as the permanent transition moment. The tensor aga is a quantitative measure of the effect of an electric field on the transition moment. Owing to the analogy with polarizability, aga will be referred to as the rransition polarizabiliry tensor. According to a quantum-mechanical perturbation calculation, its components (aga)ijare given by the following equation ‘91: transition moment can be influenced by an electric field [eq. (13)], with the result that the intensity of the band also depends on the field. For an electronic excitation with a sufficiently large permanent transition moment pga, pga agaFe and the transition polarizability tensor in eq. (13) may be neglected. In excitations of this nature, the intensity of the absorption band is practically unaffected by an electric field, as is frequently found e.g. in the case of p-nitroaniline. However, there are also electronic excitations for which the permanent transition moment is at least approximately zero, though the excitation is not forbidden by symmetry, r.g. in fluorenone [*01. In cases of this type the effect of an electric field on the absorption is easy to determine, even if the tensor components of agaare very small. Finally, some dyes exhibit electronic excitations for which the tensor components of a g a are so large that the intensity is strongly dependent on the field, despite the high value of the permanent transition moment pga19,111. I I>I 1, To calculate the relative change in the extinction coefficient due to the direct field-dependence of the transition moment, it is necessary to start with equations (3) and (13) and average over all orientations; this gives (neglecting any change in dipole moment on excitation): pga ( p r g ) i and (p& are the i-th components of the permanent transition moments between the states r and g and between r and a; E,, Ea, and Er are the energies of the states. The components of the polarizability tensor ag of a molecule can be expressed by an equation similar to eq. (14), since ag = agg (15 ) i.e. to determine the components (ag)ij of the polarizability, it is formally necessary only to replace the indices a by g. According to equations (14) and (15) the polarizability tensor is symmetrical: (a& = (a&. In general, for the transition polarizability ten- sor, (aga)ij * (aga)ji, i.e. the tensor is not symmetrical [91. According to eq. (l), the intensity of an absorption band is determined by the transition moment. The [9] W. Liplay, B. Dumbacher, and H . Weisenberger, 2. Naturforsch. 23a, 1601 (1968). Angew. Chem. internat. Edit. / Val. 8 (1969) No. 3 The relative change in the extinction thus also depends on the square of the field strength, the angle the dipole moment pg, and the transition moment pE’ of the dissolved molecule, as well as on the tensor agaof the transition polarizability. In all the molecules studied so far, the terms of the type (mga)ij(aag)kl are negligible in relation to the other terms. x, The relative change in the extinction coefficient due to the field-dependence o f the transition moment may be of the same order of magnitude as the changes due to the orientation effect and the band shift. 2.4. Electro-Optical Absorption Measurements The measurable extinction change in the absorption spectrum of a molecule in an external electric field results from the superposition of three effects, i.e. the orientation effect, the band shift, and the direct effect of an electric field on the transition moment. A complete theoretical treatment leads to relations that [lo] W . Liptay, H . Weisenberger, F. Tiemann, W . Eberlein, and G. Kunopka, 2. Naturforsch. Z3a, 3 1 1 (1968). 1111 W . Liptay, H.-J. Schlusser, B. Dumbacher, and S . Hiinig, Z. Naturforsch. 23a, 1613 (1968). 181 permit the evaluation of experimental data ‘12-15~. To obtain relations of this type it is necessary to calculate the effective electric field F, at the position of the dissolved molecule with the aid of the external electric field Fa,which can be determined in the case of parallel electrodes from the applied voltage U and the distance d between the electrodes by means of the expression Fa = U/d. The effective field F, in equations ( 5 ) and (6) can be expressed as the sum of a cavity field Fh and a reaction field F R 116-171: ~ Fh and F R g can be represented as functions of Fa by means of approximations. In the approximations used, the solvent is regarded as a homogeneous dielectric continuum with the dissolved molecule situated in a cavity. In the simplest approximation the cavity is regarded as a sphere ‘14,151; however, comparison of the electro-optical absorption measurements with dielectric measurements showed that this approximation is unsatisfactory in the case of molecules whose shapes are very different from spheres, and that it is necessary to use ellipsoid-shaped cavities [9,1OJ. In the calculation of the effective field Fe in eq. (7)and in the following equations, it must be borne in mind that a change in the dipole moment on excitation is associated with a change in the reaction field, i.e. that the reaction field of a molecule in the electronically excited state may be different from that in the ground state. According to the Franck-Condon principle, electronic excitation leads to a change only in the electronic wave function, whereas the positions of the nuclei in the FranckCondon excited state are, to a good approximation, the same as in the ground state. To calculate the reaction field, therefore, the polarization of the solvent must be separated into electron displacement polarization and atomic and orientation polarization, only the first of these being different in the Franck-Condon excited state from that in the ground state. Finally, the anisotropic polarizability of the dissolved molecules and its change on excitation can also be taken into account. Detailed calculation shows that the change in the relative extinction coefficient in an external field for an isolated absorption band which is not superimposed by another electronic band can be represented in the following form [ l o , 14,151. The quantity L;, which was given explicitly in earlier publications [lo, 14,151, depends on the absorption wavenumber Ga, the angle x between the polarization 1121 W. Liptay and J . Czekalla, Z . Naturforsch. 15a, 1072 (1960). [13] W. Liptay and J . Czekalla, Z . Elektrochem., Ber. Bunsenges. physik. Chem. 65, 721 (1961). [14] W. Liptay, Z. Naturforsch. ZOa, 272 (1965). [IS] W. Liptay: Modern Quantum Chemistry. Academic Press, New York 1965, Vol. 3, p. 45. [16] L. Onsager, J. Amer. chem. SOC. 58, 1486 (1936). [17] C. J. F. Bottchrr: Theory of Electric Polarisation. Elsevier, Amsterdam 1952. 182 direction of the light and the external field direction, the dipole moment p g and change in dipole moment pa-pg, the transition moment pga,and the transition polarizability tensor aga.The polarizability tensor of the dissolved molecule and its change on excitation often play a minor role. For suitable molecules, particularly molecules having (approximately) symmetry CzV,the dependence of L‘X(Ga) on the wave number and on the angle x allows the evaluation of the results of electro-optical absorption measurements and the determination of the direction of the transition moment, the dipole moments p p and pa. and certain components of the tensor Uga. The dipole moment in the ground state can naturally also be obtained from dielectric measurements; comparison of the results of the two methods provides a check in particular on the electro-optical absorption measurements and the models on which they are based (cf. Table 1). So far we have been discussing the effect of an external electric field on the optical absorption. The effect of an external electric field on emission can be treated in a similar manner [27J, except that the final lifetime of a molecule in the excited state has also to be taken into account in this case, so that the excited state need not be identical with the equilibrium state. 0 Table 1 gives the dipole moments and the directions of the transition moments that have been found so far from electro-optical absorption and emission measurements. The dipole moments pg, the changes in dipole moment pa-pg on excitation, and the components (aga)== or (aga)yzand (aga)zy of the transition polarizability are responsible for the strong solvatochromism of polar molecules (see Section 3). These quantities allow a quantitative treatment of the problem. Angew. Chem. internat. Edit. J Vol. 8 (1969) J No. 3 Table I . Dipole moments and transition moment directions from electro-optical absorption and fluorescence measurements. ~ Solvent Compound [a1 i a .10-3 Y (cm-1) ( ”) Ibl [cl Ref. [dl rfl le I - __ __. C H H H H D B B D DME B H B DME B H H D D D H B B B B B B 4-Dimethylamino-4’-nitrobiphenyl B B 4-Dimethylamino-4‘-cyanostilbene D D 4-Amino-4’-nitrostilbene B B 4-Dimethylamino-4’-nitrost~ibene B B 4-Dimeth~lamino-4’-nitroazobenzene B N-(4-Dimethylaminobenz~Iidene)-4-nitroaniline B N-(4-Nitrobenz~lidene)-4-dimethylaminoaniline B 2-Amino-7-nitrofl uorene B B 9H-Carbazole D D 3,6-Dinitro3H-carbazole D 9-Fluorenone D D HP 1-Indanone Xanthone (1) I71 C C D D D D D D D D D D D D 26 36 31 35 42 23 29 29 28 27 26 44 26 26 25 36 36 25 30 25 28 24 19 31 25 25 18 24 19 25 21 24 18 23 18 28 24 22 25 20 29 34 28 23 31 31 31 39 34 29 27 28 31 19 19 20 19 17 18 12 0 90 ~60-90 F90 F40 F 0 ;r0 0 0 0 3.6 4.2 3.1 2.8 2.8 6.6 6.3 5.8 6.2 F O 6.9 6.9 F 90 0 0 F F O 90 F F O 00 0 90 F F F F F F F O O O O O O O F0 F0 F F F F F F F F F O O O O O O O O O small small 0 90 90 90 90 90 90 90 F 10 0 90 90 0 0 0 0 F20 0 0 F30 3.5 %O 3.5 9 12.2 7.1 6.3 13 5.0 6.1 5.7 14 14 15 12 ;rO ;rO ;rO 15 12 5.1 5.0 6.9 6.5 5.6 5.8 5.8 6.1 6.9 7.7 7.7 6.0 6.0 6.0 6.0 6.6 6.6 7.1 7.1 6.5 6.5 7.1 7.1 8.0 8.2 6.6 kI - 5.0 5.2 5.8 ;r0 13 12 13 13 13 12 16 12 14 18 %O %0 -0 ;rO 18 16 14 23 22 24 22 20 21 22 23 26 25 25 23 23 23 5.8 19 1.7 1.7 3.3 3.3 3.1 3.1 3.1 3.6 3.8 7.3 7.3 7.3 12.1 11.5 11.2 8.9 13.0 13.0 14.8 1.9 1.6 6.6 3.0 11.5 11.0 10.7 12.7 12.7 3.1 3. I 14 5.5 4.5 4.4 4.5 3.7 4.4 5 F O F O -0 F O 8 6 -0 F O 9 20 20 21 16 16 17 6 %O 3.0; 6.6 3.7; 6.2 3.5; 7.2 F O F O F O 41 37 38 28 120 %O -25 [a1 H = hexane, H P = heptane, B = benzene, C = cyclohexane, D = 1,4-dioxane, D M E = dimethoxyethane. [bl Average value of the wavenumber range evaluated. Icl Angle between the dipole moment in the ground state and the transition moment. [d] Dipole moment in the ground state from dielectric measurements. [el Dipole moment in the ground state from electro-optical absorption measurements. [I] Dipole moment in the excited State from electro-optical absorption measurements. [gJ Dipole moment in the excited stale from electro-optical fluorescence measurements. 1181 H . Labhart and G. WagniZre, Helv. chim. Acta 46, 1314 (1963). [I91 W. Liptay and B. Dumbacher, unpublished. 1201 J . Czekalla, W. Liptay, and K . - 0 . Meyer, Z. Elektrochem., Ber. Bunsenges. physik. Chem. 67, 465 (1963). [21] J . Czekalla and G . Wick, Z. Elektrochem., Ber. Bunsenges. physik. Chem. 65, 727 (1961). Angew. Chem. internat. Edit. / Vol. 8 (1969) / No. 3 [221 W. Liptay and H . Weidenberg, unpublished. I231 H . Labhart, Helv. chim. Acta 44, 457 (1961). 1241 J . Czekalla and G. Wick, unpublished. [25] W. Liptay and W. Eberlein, unpublished. [261 W. Liptap and H . Weisenberger, [271 W. Liptay, Z . Naturforsch. 18a, 705 (1963). 183 3. Solvatochromism molecule in the same way as an external electric field, i.e. it is capable of causing a band shift and a change It was shown in Section 2 that the position and intensity of an electronic band can be influenced by an electric field. In any solution, molecules having a permanent dipole moment are located in a n electric field, the reaction field of the dissolved molecule, as illustrated in Figure 4. The reaction field acts on the dissolved b) a) in the transition moment, and hence in the intensity of the band. It can be seen in Figure 4 that the reaction field depends on the nature and arrangement of the surrounding solvent molecules. The accurate determination of the (mean) reaction field, which is possible in principle by averaging methods, presents difficulties that have not yet been overcome. Useful approximations have however been developed. In the simplest approximation the solvent is regarded as a homogeneous isotropic dielectric continuum having a dielectric constant EDC. The dissolved molecules are assumed to be acccmmodated in this continuum in spherical cavities having a radius a, or, in a refined version, in ellipsoidshaped cavities [9J. The dipole moment is represented by a point dipole located in the center of the spherical cavity. In this approximation, the reaction field F R of a dissolved molecule in the electronic ground state 116,171 is given by p; is the total dipole moment (permanent plus induced moment) of the dissolved molecule. C) Fig. 4. Reaction field of a dissolved molecule: (a) A molecule in the gas state with a dipole moment 8 , e.g. p-nitroaniline, causes an electric dipole field in its environment, the field lines of which are shown broken. (b) In solution, this dipole field acts o n the surrounding solvent molecules. If the solvent molecules have a permanent dipole moment, they orient themselves with their dipole moment a s nearly a s possible parallel t o the field lines. The orientation is opposed by thermal motion, so that only some of the solvent molecules are ideally oriented at any time. If the solvent molecules have no permanent dipole moment, a dipole moment is induced in the dipole field of the dissolved molecule. In the case of solvent molecules having a permanent dipole moment, the polarizability effect is superimposed on the orientation effect. (c) If the electronic and nuclear configurations of all the solvent molecules are imagined t o be frozen and the dissolved molecule then removed, a cavity surrounded by solvent molecules remains. Each of these solvent molecules has a dipole moment, which is made up from the permanent and the induced moment. Each of the dipole moments of the solvent molecules produces a dipole field in its environment, the field lines of which are shown broken. In the cavity, at the position of the dissolved molecule. the dipole fields of the solvent molecules superimpose and produce a field having the same direction as the dipole moment of the original dissolved molecule; this field IS known as the reaction field PR.If the dissolved molecule is now imagined to be back in the cavity, it is clearly in an electric field. 184 The changes in the position and intensity of an electronic band depend not only on the reaction field in the ground state [eq. (19)], but also on the reaction field in the excited electronic state. When the FranckCondon principle for optical excitation processes is taken into account, the expressions for the reaction field in the excited state contain not only terms similar to that in eq. (19), but also other similar terms that depend on the refractive index n of the solution. The effect of the field on the position and intensity of electronic bands is therefore determined by the dielectric constant and the refractive index of the solution in this approximation. The effect of the reaction field is the main cause of strong solvatochromism, which is observed in particular with molecules having large dipole moments, and hence strong reaction fields. The effect of the reaction field on the electronic bands is much greater than that of an external field, for two reasons. The first is the much greater strength of the reaction field. A molecule having an interaction radius a = 4 x 10-8 cm and a dipoIe moment of only & = 1 D, in a nonpolar solvent having EDC = 2, has a reaction field F Rw ~ 2 x 1 0 6 V/cm; in a polar solvent with EDC = 30, F R ~w 4.5 x 106 V/cm. The maximum field strength that can be achieved with an external field, on the other hand, is not much greater than 105 V/cm. Secondly, the reaction field (on average) is always parallel to the dipole moment of the dissolved molecule. In an external field on the other hand, the molecules are distributed over all possible orientations with respect to the field; at the maximum possible field strength the isotropic orientation distribution is cnly slightly disturbed, i.e. there are relatively few Angew. Chem. internat. Edit. 1 Vol. 8 (1969) / No. 3 ~ more molecules with their dipole moments parallel to the field than with their dipole moments perpendicular or antiparallel to it. The contributions of the individual molecules to the change in the positions and intensities of the bands due to the external electric field thus largely cancel one another out in the average, so that the observable effects are small. 3.1. Solvent-Dependence of the Position of an Electronic Band The solvent-dependence of the positions of electronic bands was discussed in detail in an earlier publicationr281. For a molecule having a permanent dipole moment p g - pgz in the electronic ground state (in the z direction) and a moment pa = pa. parallel to it in the excited state, the solvent-dependence of the wave number Gy' of the absorption maximum is given to a good approximation by the following equation 1 I 1 I Gg is the wave number of the absorption maximum in the free molecule (gas state). Eq. (20) corresponds to eq. (8), except that it also contains a term that depends on a quantity D. This term describes the solventdependence of the position of the absorption band due to the dispersion interactions between the dissolved molecule and the surrounding solvent molecules. For the low electronic excitations D > 0, so that the dispersion interactions always cause a red shift with increasing refractive index of the solution. N o reliable information is available for higher excitations. The effective electric field FRM in eq. (20) is the mean of the reaction fields of the dissolved molecule in the ground state and in the Franck-Condon excited state 1281, i.e. the primary state after the excitation process. For the particular case in question the z component of FRM is: EDK is the dielectric constant of the solution and n is its refractive index. agz is the z component of the polarizability tensor ag of the molecule in the ground state, transformed to principal axes. Any polarizability change on excitation is disregarded. Relations for cases in which p g is not parallel to p a and in which the polarizability change on excitation is taken into account are given in 1281. The solvent-dependence of the position of an electronic band can, according to eq. (20) with the aid of eq. (21), be represented by a function that depends on the properties of the dissolved molecule (pgz,paz, agz,a, [28] W. Lipray, Z. Naturforsch. 20a, 1441 (1965). Angew. Chem. internat. Edit. 1 Vol. 8 (1969) 1 No. 3 D) and on the dielectric constant and the refractive index of the solvent. Since there is no correlation between the dielectric constant and the refractive index uf polar solvents, at least two parameters that depend on the solvent are necessary for the general description of the solvent - dependence of electronic bands. A general description involving only one solventdependent parameter such as has repeatedly been sought 141, is fundamentally impossible. In the case of a dissolved nonpolar molecule, however, FRMdisappears, leaving only the term containing the refractive index n in eq. (20), so that the solvent-dependence of the wave number can now be described by a single parameter (e.g. by n). For molecules having a large dipole moment in the ground state and a large change in moment, the solvent-dependence due to the term in n is small, and the effect of the solvent on the wave number approximately depends only on EDC. The dipole moments in the ground and excited states can be determined for suitable molecules from electrooptical absorption and dielectric measurements. The polarizability agzmay be determined independently or estimated, and the quantity D may be estimated (281 or determined by matching of the experimental data. Using an estimated value a and the values EDC and n, it is then possible to calculate [ a 3 ( F ~ ~ ) Z ( p a z - p giz) 2 hcD(n2 - 1) / (2 d + l)] via eq. (21) for various solvents. A plot of as a function of this expression should give a straight line, from which G g and a can be obtained. This value a is then used to repeat the evaluation until a used for the calculation of the bracketed expression in eq. (21) agrees, within the limits of accuracy, with the value of a obtained as the result. Gr' Corresponding experimental studies were carried out on trans-4-dimethylamino-4'-ni trostil bene 1281, fluorenone 1101, and some dyes [111; the results satisfied the linear relation with reasonable accuracy, and so confirmed the above models. The solvent-dependence of the wave number of the absorption maxima of electronic bands can be described, both for an increase and for a decrease in the dipole moment on excitation, by eq. (20) with the aid of eq. (21). The solventdependence of the position of an absorption band is determined mainly by the dipole moment of the dissolved molecule in the ground state and the change in the dipole moment on excitation. An increase in the dipole moment on electronic excitation leads to a red shift, and a decrease in the dipole moment to a blue shift of the band with increasing dielectric constant of the solution. This change in position, which depends on the dielectric constant, is superimposed on a shift that depends on the refractive index. The latter shift is due to dispersion interactions and the change in the dipole moment on excitation, and, at least at low electronic excitations, always leads to a red shift with increasing refractive index of the solution. In the case of apolar molecules with pg = pa =: 0 the reaction field FRM=: 0 and the second term in eq. (20) disappears. The solvent-dependence is then mainly due to the dispersion interactions. These interactions can cause a strong red shift on transfer of the molecule from the I85 gas state into solution, but because of the smaller range of variation of the refractive index of the solvent, only a weak red shift is generally observed with increasing refractive index of the solvent, e.g. in the case of aromatic hydrocarbons 1281. For molecules having only a small dipole moment in the ground state and a largechange in dipolemoment onexcitation, thereaction field FRMmay be largely determined by the term in eq. (21) that depends on the refractive index. This will also lead to a strong red shift of the absorption band on transfer of the molecule from the gas state into solution, and a weaker red shift with increasing refractive index of the solvent, as is found e.g. in the case of electron donor-acceptor complexes. 3.2. Solvent-Dependence of the Intensity of Electronic Bands In addition to the solvent-dependence of the extinction coefficient, many molecules give a change in the width of the absorption band, with the result that the total intensity varies only slightly with fhe solvent [29,301. Electro-optical absorption measurements have sometimes [9,101 shown a considerable field-dependence of the transition moment. Owing to the solvent-dependence of the reaction field, the intensities of the absorption bands were also expected to depend on the solvent in these cases, and this was in fact confirmed experimentally f10,111. A detailed treatment of the problem I311 showed that the intensity of an optical absorption can be described with the aid of the integral absorption (see eq. (1)); the solvent dependence of the integral absorption is due to the solvent-dependence of the transition moment: pga and pgl are the transition moments in the free molecule (permanent transition moment) and in the dissolved molecule respectively; agais the transition polarizability tensor, the components of which are given by eq. (14). Eq. (22) corresponds closely to eq. (13), except that it also contains a vector W,,, which may be interpreted as the effect of the dispersion interaction on the transition moment. The effective electric field FRM is again the mean of the reaction fields in the ground state and in the Franck-Condon excited state. For the special case of pg parallel to pa (in the z direction), FRM = (FRM),is represented by eq. (21). The vector W,, should generally be very small, so that for molecules having a large reaction field FRM, i.e. for molecules having a large dipole moment, the last term in eq. (22) may be ignored; for molecules having no dipole moment or having a small dipole moment, p?' and hence also the integral absorption should depend only to a small extent on the sol- 1 [29] R . s. I Mulliken and C. A . Rieke, Rep. Progr. Physics 8, 231 (1941). 1301 L. E. Jacobs and J . R . Platt, J. chem. Physics 16,1137 (1948). [31] W. Lipray, Z . Naturforsch. 21a, 1605 (1966). 186 vent [29,301. For molecules having a dipole moment, the integral absorption depends on the reaction field, which can be calculated for various solvents from the data obtained in electro-optical and dielectric measurements (cf. Section 3.1). The theoretical models have been checked and verified experimentally by means of suitable plots L1O3111. It was found that some molecules exhibited an increase and others a decrease in the integral absorption with increasing field strength, and hence with increasing reaction field 1111. The magnitude of the increase or decrease in the integral absorption is determined according to eq. (22) by the transition polarizability tensor aga. The behavior of a given electronic band cannot generally be predicted from other simple molecular quantities. Electronic excitations associated with a very large increase in the dipole moment may be expected to show an increase in the integral absorption, while excitations associated with a very large decrease in the dipole moment may be expected to show a decrease in the integral absorption with increasing field strength. However, this single rule is not generally valid [111. The possibility of simple observation of the solventdependence of the integral absorption of an electronic band depends not only on the transition polarizability Uga (Or more precisely the product ag&M in eq. (22)) but also on the permanent transition moment pga.If pga is large, the solvent-dependence of the integral absorption can be observed only if the components of the tensor a,, are sufficientIy Iarge. If on the other hand pgais very small or zero, the solventdependence of the intensity of the absorption band is clear even when the components of agaare small. Particularly interesting relations are found in fluorenone [lo]. The direction of the dipole moment will be referred to as the z axis. Solutions of this molecule give an absorption band at about 31 000 cm-1 with a transition moment direction in the molecular plane parallel to the y axis. The maximum extinction coefficient cmaxand the integral absorption of this band are strongly solvent-dependent [dimethylformamide (EDC = 36.7): emax = 1320; ether (ZDC = 4.4):cmax = 685; heptane (EDC = 1.97): cmax = 4101. The absorption band can no longer be observed in the gas state, and cmax is certainly less than 50. Fluorenone thus has an absorption band whose intensity is almost zero in the free molecule. The intensity recorded in solution is due to the perturbation of the molecule by its reaction field. The permanent transition moment, and hence the integral absorption of the band in the free molecule, is not zero on symmetry grounds, but disappears purely by chance. The appearance of a field-induced absorption band with a transition moment in the y direction as a result of a reaction field in the z direction will be explained in more detail. In the molecular orbital (MO) scheme the absorption band in question can be interpreted, o n the basis of the experimentally observed transition moment directions and SCF-MO approximation calculations, as resulting from the excitation 9 5 + 98 of an electron from the fifth (filled) r-MO 9 5 into the eighth (empty) x-MO 9 s . 95 belongs to the irreducible representation A2 of the point group CzV, and 9s to the representation B1; the atomic orbital coefficients ~5~ and csp of the MOs in the LCAO approximations are illustrated in Figures 5a and 5b. I n the simple MO approximation the transition moment in question p g a = e 1/? J 9 5 9 s r d T. In the LCAO-MO approximation the equation simplifies further, the integral reducing to a sum: pga = e l/2 Angew. Chem. internat. Edit. 14 p= 1 C~,,CS~~,, 1 Vol. 8 (1969) 1 No. 3 where rp are the position vectors of the atoms p. T o obtain the transition moment pga,it is necessary to form the products cjpcgprpand carry out the summation over all the atoms (p = I to 14). Similarly the components of the transition moment (pga)x,(pga)y, and (pga)zin the directions of the three coordinates are obtained on replacement of the position vectors rp by their components xp. yp, and zp. 2 2 parallel to the electric field, the atomic orbital coefficients of atoms 1, 2, 11. 12, 13, and 14 will increase, while those of atoms 4, 5 , 6 , 7, 8, and 9 will decrease. The contributions to the integral (pga),,. i.e. the areas for the corresponding atoms (Fig. 5d). will thus change. the sum of the positive areas increasing and the sum of the negative areas decreasing, so that a finite transition moment in the y direction arises in an electric field in the z direction. Similar variations of the transition moments and hence of the integral absorptions of electronic bands have also been found for some dye moleculesff11. 3.3. A Relation between the Solvent-Dependence of the Position and the Solvent-Dependence of the Integral Absorption of Electronic Bands 1168751 C) d) Fig. 5. Representation of the molecular orbitals ‘ p ~and ‘pa and of the components (vga)z and ( w ~of~the ) transition ~ moment of fluorenone (for practical reasons, the numbering of the skeletal atoms is different from that normally used): (a) Atomic orbital coefficients csp of the fifth x - M O ‘ps (representation A2). (b) Atomic orbital coefficients cgp of the eighth rr-MO ‘pa (representation B I ) . ( c ) Products cspc8pzp(representation B2). (d) Products cspcgpyp (representation A,). The areas of the circles are proportional to the magnitude of the orbital CspCSpYp) ~ ~ Z for ~ .the atom coefficient (csp, csp) or of the product ( C ~ ~ C p in question. Continuous circles denote positive signs. and broken circles negative signs. The products c ~ ~ arec shown ~ ~ inz Figure ~ 5c. Since the xz plane of the molecule is a plane of symmetry, the magnitudes of the coefficients qPand csP on equivalent atoms, e.g. o n atoms 1 and 1 2 or 4 and 9, must be the same, and so also must Z ~ signs . of the the magnitudes of the products C ~ ~ C S ~The product cspcgpzp for equivalent atoms are always dissimilar, i.e. c54cS4z4 = -cs9csgzg. The sum, and hence the z component of the transition moment, must therefore be zero. The z component of the transition p5 + ps is forbidden in the electric dipole approximation by the symmetry of the molecule and the irreducible representation of the MOs involved. It can similarly be shown that the x component of the permanent transition moment is also zero. Figure 5d shows the products cspCapYp.The y component of the transition moment is proportional to the sum of the areas, and will generally be finite. The optical electronic excitation 95 + ps is therefore generally allowed with a transition moment direction parallel to the y axis. The y component of the transition moment is zero only when the coefficients qP and csp are just such that the sum of the negative terms (sum of the negative areas in Fig. 5d; atoms 2 to 11) is equal to the sum of the positive terms (sum of the positive areas in Fig. 5d; atoms 1 and 12). The optical electronic excitation q5 + ‘ps is then forbidden, but this is accidental, and is not due to symmetry relations. According to experimenlal studies, this situation occurs, at least to a good approximation, in the case of the fluorenone absorption in question in the free unperturbed molecule. The electrons in the molecule are displaced by an external electric field or by the reaction field of the dissolved molecule. On orientation of the molecule with the dipole moment Angew. Chem. internat. Edit. 1 Vol. 8 (1969) 1 No. 3 A plot of the wave number of the absorption maximum of an electronic band of a dissolved molecule as a function of [a3(F~h&(paz- pgJ + 2 hcD (n2 - 1)/(2 n2 + l)] should, according to eq. (20) and eq. (21), be linear. It is in fact found that the experimental points for the various solvents deviate more or less strongly from linearity, and some solvents, such as dioxane and benzene, are quite irregular. The reason for the pronounced scatter is the fact that the reaction field based on the continuum model is only a rough approximation, which fails in particular in the cases of dioxane and benzene 1101. The same deviations are therefore to be expected, and are in fact found experimentally, in suitable plots of the integral absorption as a function of the field FRMin accordance with equations (l), (21), and (22). The relations (20) with (21) and (1) with (21) and (22) can be checked only with the aid of a large number of solvents, so that individual deviations cancel out in the “mean”. According to eq. (20) and equations (1) and (22), both the position of the absorption band and the integral absorption depend on the average value of the reaction field FRM.The reaction field can be eliminated by combination of equations (l),(22), and (20). against a function of the integral abIn a plot of sorption, which is possible in various cases, the deviations due to the poor approximation of the reaction field should be eliminated, as has been confirmed experimentally 110,111. The theory of the effect of a n external field on electronic bands can be checked and confirmed by comparison of the results of electro-optical absorption measurements, electro-optical fluorescence measurements r16,201, and dielectric measurements 19,101. These methods allow the measurement of the dipole moments of suitable molecules in the ground and excited states and certain components of the transition polarizability tensor. These quantities are responsible for the strong solvatochromism of polar molecules, and allow a quantitative description o f the solvent-dependence of both the positions and the intensities of absorption bands. An independent check on the models was provided by various plots of the absorption wave number against functions of the integral absorption. Another possible cause of a strong dependence of the integral absorption on a n external field and on the 187 solvent would be an equilibrium between isomers having sufficiently different dipole moments, e.g. cis-trans isomers. In such cases the relative quantities of the isomers could be strongly field-dependent, and hence solvent-dependent. Since the absorpt'on bands Of isomers are usually different, the extinction of the solution would then depend on the field and o n the solvent. Investigations on a number of dyes that exhibit an extremely strong solvent-dependence have indicated that this possibility is very unlikely, though it cannot be completely ruled out "11. we are to the Fonds der Chemisehen Industrie, the Deutsche Forschungsgemeinschaft, and the StifLlng Volkswagenwerk for their support of' the experimental and theoretical strtdies. Received: May 27, 1968 [A 687 IE] German version: Angew. Chem. 81, 195 (1969) Translated by Express Translation Service, London Steps in Precipitation Reactions By K. H. Lieser[*l The combined application of various methods of investigation (e.g. nephelometry, conductivity measurements, electron microscopy, isotope exchange, BET surface area determinations, paper chromatography, coprecipitation) lead to a refinrd insight into the course of precipitation reactions. The formation of a new solid phase within a solution can, in the case of ionic crystals, be formally described as proceeding via a number of steps - nucleation, growth, ripening, and recrystallization (aging) - which overlap in time. The precipitation of hydroxides is a more complex process since additional chemical reactions (e.g. hydrolysis, condensation reactions! take place within the newly formed solid phase. 1. Introduction Precipitation is a fundamental chemicai operation for the separation and purification of substances. It is well known that precipitations very often give the desired result only if certain well-established procedures are rigidly adhered to, though in many cases the basis of these procedures, which have been evolved empirically, are still far from clear. The principal process in a precipitation reaction is the formation of a new solid phase from a solution. It is convenient to distinguish the following steps: nucleation, growth, ripening, and recrystallization (aging). However, this well-established distinction is only formal, since the various steps overlap in time. The following discussion deals mainly with the precipitation of salts (particularly sparingly soluble salts). Precipitation of hydroxides is a more complex process, since chemical reactions (e.g. elimination of water, conversion of basic salts into hydroxides) also occur during formation of the solid phase, depending on the pH value and time. Consequently, there are as yet hardly any theoretical schemes for the general treatment of the precipitation of hydroxides; the principal features of this process will be discussed in a separate section. [*I Prof. Dr. K. H. Lieser Lehrstuhl fur Kernchemie der Technischen Hochschule 61 Darmstadt, Hochschulstrasse 4 (Germany) 188 2. Steps in a Precipitation Reaction 2.1. Nucleation The formation of submicroscopic particles (nuclei) of the new phase from a supersaturated solution is particularly interesting, but relatively difficult to observe experimentally. It generally begins only when a certain supersaturation has been reached, and proceeds very rapidly. The questions that arise in connection with the formation of a new phase have already been considered in detail by Tanzmann[", Ostwald[21, and Volmer 131. The kinetics of precipitation reactions have also been described by Nielsen [41. Nucleation can be either homogeneous (spontaneous) or heterogeneous. Homogeneous or spontaneous nucleation occurs without the participation of other substances, by combination of the dissolved ions or molecules to form larger particles. Heterogeneous nucleation begins on small particles of foreign matter (seeds), on which ions or molecules are deposited (e.g. by adsorption) until a nucleus has been formed. [I] G. Tammann: Kristallisieren und Schmelzen. J. A. Barth, Leipzig 1903. 121 W. Osfwald Die wissenschaftlichen Grundlagen der analytischen Chemie. W. Engelmann, Leipzig 1897. [3] M . Volmer: Kinetik der Phasenbildung. Th. Steinkopff, Dresden 1939. [4] A. E. Nielsen: Kinetics o f Precipitation. Pergamon Press, Oxford 1964. Angew. Chem. internat. Edit,1 VoI. 8 (1969) No. 3

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