885
SPECTRAL SHIFTS IN POLYMERIZATION STUDIES OF METALIONS
K+, Br-, and C104- are structure breaking. Comparing the polarizability of eaq- with that of F- would suggest that eaq- is a more powerful “structure maker” than is F-. One would expect very extensive structuring in solutions containing eaq- since F- is known to be a very efficient structure maker. This effect would increase the diffusion coefficient of HzO* since the mobility of the species is dependent on the ease of orientation of the water dipoles. Furthermore, if it is proposed that H20* reacts with earl-, then the structuring effect of eaq- would greatly increase the reaction radius. The increase in both DAB and TAB in eq 1 would mean that a diffusion-limited rate constant for the reaction of eaq- with H20* could exceed 10” M-l sec-’. Consideration of the experimental data indicates that the observed decay of eaq- is adequately explained by previously studied reactions in acidic solution and in
Ba(OH)2 at pH 9.1 when 3 X 11 or more KBr is present. Each of these solutions contains an excess of structure-breaking ions over structure makers. I n the cases where the experimentally observed decay of eaq- does not agree with the calculated values, an excess of structure-making ions is present. It should be emphasized that the work reported discusses only the disappearance of the optical absorption due to eaq-. The reason for the disappearance, whether it be due to chemical reaction or simply the removal of some of the waters of hydration brought about by promotion of the species to an excited state, is not considered. Acknowledgment. The authors gratefully acknowledge the technical assistance of Thomas L. Smith, U. S. Army Xuclear Defense Laboratory, and stimulating discussions with Professor R. H. Wood, University of Delaware.
Implications and Use of Spectral Shifts in Polymerization Studies of Metal Ions by Jack I. Morrow and Joel Levy Department of Chemistry, T h e City College of The City University of N e w York, N e w York, N e w York (Received July 28, 1967)
loodl
For polymers which conform to SillCn’s hypothesis of cores and links, a method was developed to determine both the value of t in the links (OH)lMand the degree of polymerization (average number of monomers present in all species). It is based upon the spectral shift caused by hydrolysis and polymerization of the metal ion, which in this study was Cr(II1). As the a value (moles of added OH- per mole of Cr(II1)) was increased, the wavelength of the first maximum in the visible region shifted from 408 to 422 mp. The degree of polymerization in the wavelength span from 410 to 415 mp changed from 1.20 to 1.62. The value of t in the links vvas shown to be 2.
The ultimate object in most polymerization studies is the identification of species present. Acidity measurements, molecular weight determinations, and equilibrium centrifugation are among the most commonly employed techniques used in achieving this object. Accompanying the polymerization there is usually a red spectral shift when briding occurs through hydroxo or oxo groups, and the peak originates from d-d splitting. Thus, red-purple solutions of monomeric Cr(II1) solutioas turn green when polymerization occurs to an appreciable extent.’ Dimerization of Fe(II1) solutions is accompanied by a color change from yellow to red.2 The wavelength of the principal charge-transfer band is also sensitive to polymerization.3J The direction of the shift, however, would depend upon the direction of electron transfer.
With regard to the treatment developed in this paper, the direction of the shift is not important. Of importance is that the position of maximum absorbance must change monotonically as the degree of polymerization increases. The implications and use of spectral shifts in polymerization studies, unfortunately, have not been developed, and it is to this that we turn our attention. Consider a series of solutions of a given metal ion, AIZ+, at a total metal concentration, T M . To each (1) C. L. Rollinson in “Chemistry of the Coordination Compounds,” J. Bailar, Ed., Reinhold Publishing Corp., New York, N. Y., 1964. (2) H. Schugar, C. Walling, R. B. Jones, and H. B. Gray, J . A m e r . Chem. Soc., 89, 3712 (1967). (3) C. Altman and E. L. King, {bid., 71, 425 (1961). (4) C. Berecki-Biedermann, Proceedings of the 7th International Conference on Coordination Chemistry, Stockholm, 1962, p 161.
Volume 72, Number 9 March 1968
886
JACKI. MORROW AND JOEL LEVY
solution in this series a different amount of hydroxide is added, allowing us to obtain solutions of various a values, where a = T O H / T h l (moles of OH- added per mole AIz+). Spectral analysis would then show that as a increases, the position of maximum absorbance, A,, changes owing to increasing polymerization.6 If this procedure was followed using solutions of different TM values, it would be observed that A,, is a function of both T,\I and a. I n order to obtain solutions having the same, , ,A (isolambda solutions) for different T M values, their a values must of necessity be different. The mathematical argument given below indicates that isolambda solutions should have a similar fractional distribution of species. Let C1, Cz, C3, etc. be the concentrations of the various species in equilibrium for a given T M , and al, a2, a3, etc., their molar absorptivities. The fractional concentration of the ith species is Ct* = C ~ / T M . In applying Beer's law, a l-cm cell length is assumed. The measured absorbance, A , at any wavelength is A = alCl azCz a3C3 . . . or A = Td"1*al T M C ~ * U ~ThrC3*a3 . . . . At, , ,A only
+
+
+ +
+
Kt
(2)
The value of t in reaction 2 is most frequently 2 or 3.
AI''+
+ (OH) lfil+z-tKz
+ n(OH) ,;\I+Z-t
AIz+
M(0H) ,A~1+2z--1
(3)
K(fL+l)
A/l((OH)Jl),L+n(z-c)+z
(4) Reaction 4 is the over-all reaction involving n links, thereby yielding the n 1 polymer. The minimum value of n is 1, corresponding to dimer formation (eq 3). I n subsequent equations, the charges of the species while not indicated are understood. The equilibrium constants for the several reactions are
+
K, =
[MOH][H+]
--__
+
and for the n The last equation may be more conveniently written in the form of a summation,
+ tH+
AI''+ J_ RI(0H)
1
(5)
+ 1 polymeric species Kn+1 =
[M((OH) [AI'][(OH),141"
(7)
The material balance equations are for total metal
[MI+ [AJOH] + [,1I(OH),] + 2[A'I(OH),N] + , . . + (n + l ) [ l l ( ( O H ) , l l ) n ] (8)
Tx The values of batlbA, like at, are independent of Ci*and are fixed for a given wavelength. -4s they are slopes, they may have positive, negative, and zero values. It must be noted that these fractional concentrations of species are related to one another through their equilibrium constants. It will be shown in the Theory that the limiting C,* values for isolambda solutions, as 1/ [H+] approaches zero, yields information concerning both the extent of polymerization and the nature of the polymers formed. The use of Cr(II1) solutions in this study was dictated both by the availability of the spectra of several polymeric species and by the fact that the composition of these polymers6 is in accord with SillBn's hypothesis of cores and l i n l i ~ . ~ 'The * hypothesis states that the predominant species are members of a homologous series (which does not necessarily include all possibilities). The polymers AI [(OH)J1ln are composed of the core, 11, added to the links, (0H)JI. For Cr(C104)3solutions, the value of t is 2.6,9
General Theory The reactions occurring are written to illustrate and conform to SillBn's hypothesis.
W+1 ' lIOH+Z-l + H + KO
The Journal of Physical Chemistru
(1)
=
for total hydrogen ion
+ [H+] - [OH-] = [NOH] + t[RI(OH),3 + t[RI(OH),111 + . . . + n t [ M ( ( O H tl\l)n] ) (9)
TOH
and for total concentration of all species
ZCi
+ [MOH] + [lf(OH)t] + [RI(OH),M] + . . . + [bI((OH),PII),]
= [hi]
(10)
By combining eq 5 , 6, and 7 with each of the material balance equations, the latter equations may be written as
T M = [X]{1
K, Ka ++-,+ [H+l [H+l
(5) An excellent discussion on the relationship between a values and polymerization is given by L. Pokras, J. Chem. E d u c , , 33, 223 (1966). (6) G. Thomson, AEC Accession, No. 35255, Report No. UCKL11410, University of California, Berkeley, Calif., 1964. (7) L. G.Sillh, Acta Chem. Scand., 8, 299, 318 (1954). (8) 6. Hietanen, and L. G. Sillbn, 8, 1607 (1954). (9) R. W. Kolaczkowski and R. A. Plane, Inorg. Chem., 3, 322 (1964).
887
SPECTRAL SHIFTSIN POLYMERIZATION STUDIES OF METALIONS
+ [H+] - [OH-] =
The reciprocal of eq 16 yields
TOH
1
- = [M]*R DP I n applying eq 12 to the Cr(II1) study, [OH-] was neglected, since :measured pH values were in the range 2-4.
When equilibrium is attained, the concentration of each polymeric species (and of course the monomer) relative to that of the monomer is fixed by the ratio [M]/[H+]t. This is not so with the monomeric hydroxides, whose concentrations relative to the monomer are determined only by [H+]. The basic assumption of this method is that isolambda solutions have the same value of [hI]/[H+]' (= [M(OH),]/K,). This means adherence to "core plus links" behavior, and the plausible assumption that A,, must change monotonically with increasing n. For isolambda solutions presumably having a fixed value of [MI/ [H+]l, eq 13 may be written as
The degree of polymerization, DP, may now be defined as D p = - -T- M -
m,
DTl(R
+
SI+ &) TM
K
-
1
+-}K ill* R + { ];:I [H+],
___
If isolambda solutions have similar [MI* values, then a plot of 1/DP vs. K,/[H+] should be linear, with a slope of [MI* and an intercept of [M]*R. I n terms of experimentally measurable quantities, with the exception of [RIOH], the D P may be written as
To allow comparison of terms in eq 18 with those of eq 16, eq 18 may be written as DP
=
1
therefore, R of eq 16 is R =
where
]*K, + [AI[H+l
~ TM TOH- [H+] tTM [MI*
t
- 1 K, t [H+l
Although the value of [MI* is unknown, a decrease in { t T M - TOH - [ H + l j / t T ~with increasing [H+] should be observed to counterbalance the decrease of [(t - l)/t]K,/ [H+],thereby maintaining R constant for isolambda solutions. Careful scrutiny of eq 11 reveals that [MI* for isolambda solutions should also vary slightly with [H+], but becomes constant as K,/[H+] approaches zero. This is seen in eq 20, obtained by rearrangement of eq 11
~
Many metal ions,including Cr (111), have separated hydrolysis constants, thereby allowing to be simplified to
1
As K,/[H+] approaches zero, the limiting value of [PI]* for isolambda solutions is obtained.
1
For Cr(II1) ion, K,
=
1.5 X
and K, (where
2 = 2) = 1.5 X 10-ll.lo Under the experimental
conditions, the measured pH values were less than 4; therefore, K,[H+] >> K,, and the simplification introduces no measurable error. Ions having hydrolysis constants not so well separated might require the use of the more rigorously derived equation.
Equation 20 indicates that a proportionality exists between [MI and T M for isolambda solutions a t high [H+]. We, therefore, may write (10) J. E. Earley and R. D. Cannon in "Transition Metal Chemistry, A Series of Advances," Vol. 1, R. L. Carlin, Ed., Marcel Dekker Inc., New York, N. Y.,1965. (11) C. Postmus and E. L. King, J . Phys. Chem., 59, 1208 (1955).
Volume 76,Number S March 1068
888
JACKI. MORROW AND JOEL LEVY [MI ma[H+1"TM
constant
or -log T M = tpH
+ log (constant)
(21)
A plot of -log TM us. pH should be approximately linear with a constant limiting slope of t at lower pH values. Once the value of t is determined, the degree of polymerization may be calculated using eq 18, but neglecting the [hIOH]/t term. This should not produce any error in the (DP)ovalues obtained by extrapolation of the data to K,/[H+] = 0 (eq 17), since [MOH] becomes effectively zero at high [H+].
rrc
1 0.11
I
I
oa
I
04
4.3
Results and Discussion It was observed that as
a increased for a given T x value, the maximum in the visible spectrum of Cr(II1) The Journal o j Physical Chemistry
I
0,b
l
47
l
0.8
I
l
l
0.9
1.0
1.1
a &Mi)
Experimental Section Sulfate-free Cr(C104)3,>99.95% pure, obtained from K & K Laboratories, Inc., was used. Solutions of Cr(C104)3were analyzed by comparing their absorbancies to those on a Beer's-law curve and were prepared by reducing acidified KzCrz07 solutions with HzOz and driving off the excess H202. A Beckman Rlodel DL spectrophotometer was used for these measurements. A Beer's-law curve was also run on It was observed that acidified C I - ( C ~ O ~ solutions. )~ freshly prepared solutions did not obey Beer's law, having a wide scattering of points. After the solutions stood for 2 weeks, depolymerization of Cr(C10J3 occurred and these same solutions obeyed Beer's law. The a value of each solution was set using a Na2C03-free NaOH solution. Hydrogen ion concentrations were measured using an L &: S pH meter. The spectra of all solutions, with the exception of those used for standardization, were obtained using a Cary 14 recording spectrophotometer. In preparing the chromium solutions of various a values, the reactants were mixed rapidly using a constant method of addition. Homogeneity mas attained within 2-3 sec after addition. Schwartzenbach12 has recently used fast-flow apparatus to obtain complete mixing within several milliseconds, thereby allowing him to study the very rapid formation of monomeric hydroxo complexes uncomplicated by the formation of polynuclear species. We prepared a limited number of solutions using the Aminco-Chance rapid-mixing apparatus (time of mixing = 0.02 sec) and observed no significant difference in results between the rapid mixing and the addition method. All solutions were maintained at a temperature of 25.0' until the pH in each solution was constant. It was assumed that equilibrium had been attained when the pH remained constant (within kO.01 pH unit for 2 weeks). The ionic strength was held constant using NaC104at p = 1.O in all solutions.
I
Figure 1. Variation of the wavelength maximum with increasing a value for several different total metal concentrations.
-====-
a*
t
0.1
I
410
l
4'20
l 430
1
440
1 JJ
x (yu)
1 570
1
580
1
590
1
bQ0
1
610
for both peaks in Figure 2 . Variation of Lax visible region: upper spectrum has a = 1.0; lower spectrum has a = 0.0. The total metal concentrations for both spectra are the same, T M = 2.11 X M.
ion shifted toward the red (Figure 1). It should be noted that chromium has two peaks in the visible region, one in the vicinity of 408 mp and the other around 575 mp. Both peaks showed a red shift, (Figure 2)) with the shift of the former peak being larger than that of the latter peak. The 408-mp peak was also the sharper of the two and so was used in this study. The experimentally measurable quantities, T M ,TOH, and [H+],wereusedin eq 19 and 21 and results compared only for solutions exhibiting the same A, value, since, as was shown earlier, isolambda solutions most likely have a constant value of [M]/[H+]'. I n many instances, the values of TM, T ~ Hand , [H+]used were obtained by interpolation of our data in the following manner. Isolambda lines were drawn through (12) G.Geier, J. Littler, and G. Schwartzenbach, HeZw. Chim. Acta, 45, 260 (1962).
SPECTRAL SHIFTS IN POLYMERIZATION STUDIES OF
METALIONS
889
lambda solutions were calculated using eq 18; their reciprocals were plotted against K,/ [H+] (Figure 4). The plot is linear and positive in slope as required by for K , was used." eq 17. A value of 1.5 X Table I lists the (DP),values obtained by extrapolation to K,/[H+] = 0. The third column of this table contains DP values calculated from the work of Thomson,'j by using our spectra and her molar absorptivities for monomer, dimer, and trimer which she had separated using ion exchange. The two methods yield comparable results. The last two columns contain the [MI* (slopes) and R values for the several isolambda series of solutions. PH Figure 3. Determination of t from the limiting slope at low pH values.
Table I mr
(DP)o
DP (G.T.)
[MI*
R
410 411 412 413 414 415
1.20 1.25 1.35 1.43 1.54 1.62
1.12 1.20 1.25 1.38 1.46 1.55
0.40 0.37 0.34 0.30 0.27 0.25
2.1 2.1 2.2 2.3 2.4 2.5
A,
-- *-
*IO "p
-a-
4LI
D-
411 " p 412 .p
-.-
-)i-
-0-
.f .f
414 *p
4L5
s P
I n partially hydrolyzed solutions [Crz(OH)2+4] >> [Cr(OH)+z];13-1e the equilibrium constant,lO Kzz,for Kza
Figure 4. Variation of 1/DP with K,/[H+]. Extrapolation to &/[H+] = 0 yields l/(DP)O.
this "true" dimerization, ~ C I - ( O H ) e ~ +C I - ~ ( O H ) ~is~ + , Kzz = lo4. This permits the [MI* values in Table I to be tested for correctness by calculating the equilibrium constant, Kd, for the dimerization 2Cr3+
the lines in Figure 1 at wavelengths of 410, 411, 412, 413, 414, and 415 mp. From this, the a values required to give the wavelength maxima above were obtained. Graphs of T M us. a were also prepared, since there was a slight dilution factor which was taken into account even though it was small. A graph of pH vs. a was also prepared so that [H+]could be calculated. Determination of t. Equation 21 predicts an approximately linear relationship between -log 7'D.I and pH with a limiting slope of t at lower pH values. The results for the six isolambda series of solutions are shown in Figure 3 (A-F). Figure 3G serves as a comparison between the 414 mp ( t I I ) isolambda series and the series of solutions ( t I ) to which no hydroxide was added, a = 0. These solutions do not exhibit the same A, as is evident from Figure 1. The values of t for the isolambda solutions are in good agreement with each other and with the reported ~ a l u e . ~The , ~ value of t r in Figure 3G is considerably different from the other six values and is well outside of experimental error. Degree of Polymerization. DP values for iso-
+ 2H20
Kd
+
C ~ Z ( O H ) ~ ~2H+ +
(assuming the only two chromium species present are monomer and dimer) and comparing this value to that obtained from the product of Ka2Kzz(=Kd), where K , = 1.5 X and Kzz = lo4. The following relationships were employed [Cr3+] = [M]*TM (see eq 20) [Crz(OH)z4+] = ~/z(TH - [Cr3+])= ' / z T ~ ( l- [AI]*) and Kd =
[Cr2(OH)z4+][H+]2 - (1 - [MI*) [H+]2 [Cr3+I2 2 [hI]*ZTM
Values of Kd were calculated for each of the 415 mp isolambda solutions using [MI* = 0.25, and for each of the 410-mp isolambda solutions using [At]* = 0.40. The 415-mp series yielded a value of Kd = 3.6 X (13) J. Faucherre and R. Schaal, Compt. Rend., 225, 118 (1947). (14) R . Schaal and J. Faucherre, Bull. SOC. Chin. France, 927 (1947). (15) P. Souchay, ibid., 143 (1948). (16) J. Faucherre, ibid., 253 (1954).
Volume 72, Number 9 March 1968
890
0.3 X the 410 mp series yielded a value of = 3.3 X These are in excelf 0.3 X lent agreement with each other and with the value of Kd = 2.3 X (=Ka2K22). The R values listed in Table I may also be tested for correctness by comparing them with the values obtained using eq 14a i
Kd
Assuming only dimer formation, that is, n = 1, this may be written as
where Kd = K 2 K , . The 415-mp series gave an average value of R = 2.1; the 410-mp series, R = 1.8. The only significance attached to these results is their demonstration of internal consistancy.
Conclusion A method for determining the stoichiometry of the links for polymers conforming to SillBn's hypothesis was devised and tested and found to yield good results for the particular metal ion, Cr(III), studied. Degrees , also of polymerization, defined as DP = T M / Z C ~were calculated and were in agreement with results obtained using a different method. This method should be useful in determining whether a homologous series is formed; for if it is not formed,
The J O U Tof~Physical ChemistTy
JACKI. h,~ORROwAND JOELJJEVY the t values for all isolambda solutions should not be in agreement. For those who might desire to use this method, some words of caution resulting from our observations should be noted. First, as degree of polymerization increases values, the peaks tend to corresponding to higher A, broaden, as might be expected. This broadening makes it increasingly more difficult to obtain accurate For this reason we reported results values of A., for isolambda solutions up to 415 mp, although A, values up to 422 mp were observed. Second, increased accuracy and reliability in determining both the t value and the (DP)ovalues can be obtained by extending the range of T Mvalues covered to include the highest values (absorbance) measurable. This is predictable from the theoretical treatment, since as the TM value increases, for isolambda solutions, the pH decreases (eq 21), and hence the importance of [MOH] in the many equations in which it appears is lessened. The last limitation and precaution imposed upon this method is that no ligands, other than water and its conjugate base, capable of complexing can be present. For this reason, the method is probably limited to studies of metal ion perchlorates and nitrates.
Acknowledgment. The authors gratefully acknowledge the support of this research by the National Science Foundation Undergraduate Research Participation Program and the City College Faculty Research Committee. Discussions with Professors F. Condon and 34. Green are also gratefully acknowledged.
