J. Phys. Chem. 1982, 86, 3489-3491
reasonably well. Slight deviations,at high temperatures can be accounted for by the change in heat capacity for the reaction and by deviation from ideal gas behavior.
3489
Acknowledgment. We express our appreciation for financial assistance from the Energy Laboratory a t the University of Houston.
Random Siting of Alumlnum in Faujasite Alan W. Peters W. R. &ace &
Co.,Davison ChemlcalDlvislon, Columbla, Meryland 21044
(Received: January 26, 1982)
High-resolutionsolid-state "Si NMR experiments by several groups have led to the determination of the relative concentrations of five chemically nonequivalenttetrahedral silicon atoms in zeolites. Tetrahedrally coordinated silicon may bond to zero, one, two, three, or four 0-A1 groups in the zeolite framework leading to in each case a different chemical shift. These experiments can be used as evidence for ordering in zeolites provided the results expected of a random distribution are available for comparison. This letter illustrates a procedure for calculatingthe appropriate distribution using standard statistical methods. The application to faujasite described here involves maximizing the distribution probability of the aluminum atoms subject to constraints consistent with Lowenstein's rule. The random distribution of silicon atoms is close to but not identical with published experimental results obtained by Ramdas, Thomas, Klinowski, Fyfe, and Hartman.
Introduction Recent high-resolution solid-state "si NMR experiments have led to the determination of the relative concentrations of five chemically nonequivalent tetrahedral silicon atoms in mlites.l2 Tetrahedrally coordinated silicon may bond to zero, one, two, three, or four 0-Algroups in the zeolite framework leading to in each case a different chemical shift. The results of these experiments have been used to support a particular partially ordered model of faujasite initially proposed by Dempsey3 to explain the apparent breaks in the Si/Al vs. cell size curve.4J However, to establish ordering, it is useful to be able to compare experimental results with results expected of a disordered model. The purpose of this work is to calculate the distribution of the five types of chemically nonequivalent silicon in a faujasite where the A1 atoms are randomly distributed consistent with the constraint of Lowenstein's rules6 Previous work'ss concerned with aluminum siting is not directly applicable to the W i NMR data. The present method obtains the random distribution in an analytical form from a statistical calculation of the expected distribution of silicon and aluminum atoms on localized independent sites subject to appropriate constraints. (1)S. Ramdas, J. M. Thomas, J. Klinowski, C. A. Fyfe, and J. S. Hartman, Nature (London), 292,228 (1981). (2)E. Lippmaa, M. Magi,A. Samoson, G. Engelhardt, and A. R. Grimmer, J. Am. Chem. Soc., 102,4889 (1980). (3) E. Dempsey, J. Catal., 33,497(1974);39,155 (1975). (4)E.Dempsey, G. H. Kuhl, and D. H. Olsen, J. Phys. Chem., 73,387 (1969). (5) D.W.Breck and E. M. Flanigen, "Molecular Sieves",Society of Chemical Industry, London, 1968,p 47. (6)W. Lowenstein, Am. Mineral., 39,92(1954). (7)R. J. Mikovsky, J. F. Marshall, and W. P. Burgess, J. Catal., 68, 489 (1979). (8)R.J. Mikovsky and J. F. Marshall, J. Catal., 44,170(1976);49,120 (1970).
Discussion Faujasite can be considered to be built of linked hexagonal prisms. A prism contains a total of 12 atoms of Si or A1 bonded through oxygen and may contain 0-6 A1 atoms according to Lowenstein's rule. A randomly sited faujasite will contain a distribution of these prisms subject to the conditions no + nl
+ n2 + n3 + n4 + n5 + n6 = N nl + 2nz + 3n3 + 4n4 + 5n6 + 6n6 = 12N/(R + 1)
(1)
(2)
+
where N = total number of prisms and 12N/(R 1) = number of Al atoms. The distribution numbers no, nl, ..., n6 are the total number of prisms containing 0, 1,...,6 Al atoms, and R is the silicon/aluminum atomic ratio. Although the distribution numbers are generally unknown, a most probable distribution can be calculated by using the LaGrange method of undetermined multipliers to maximize the log of the distribution probability subject to constraints 1 and 2. The log of the probability is
where wk is the number of different ways that a prism can have k aluminum atoms. The result is the set of distribution numbers
nk = wkeaek@ k = 0, 1, 2, 3, 4, 5, 6
(3)
where a and @ are multipliers to be determined by conditions 1 and 2. Substitution of nk into eq 1and 2 gives respectively 6
C wkek@= Ne-*
k=O
0022-3654/82/2006-3409~0~.25/0 0 1982 American Chemical Society
(la)
3490
The Journal of Physical Chemistry, Vol. 86, No. 17, 1982
Peters
TABLE I: Number of Ways of Arranging k Aluminum Atoms ( w h )in Faujasite Hexagonal Prisms Where k = 0, 1, 2, 3, 4, 5, and 6 k" wk 0 1 2
3
4
1 12
Aliprism Aliprism Aliprism one in each ring both in one ring: para both in one ring: meta total Aliprism three in one ring two in one ring: para two in one ring: meta total Aliprism three in one ring two in each ring: para, para two in each ring: para, meta two in each ring: meta, meta total Aliprism Aliprism
6
(24
6
1 12
kWkeka k=O
= - 12
(4)
R + l
c k=O 6
Wkekp
Provided we know wk, R is given as a function of /3 since there is only one solution for both R and e@greater than zero. The wk can be obtained by inspection of the seven possible prisms. There is only 1 way that a prism can contain no Al, and 12 ways of containing one Al atom. For two to six A1 atoms, the possibilities are listed in Table I. In each prism, three of the four neighbors of each atom can be identified (Table 11). The fourth neighbor is determined by the way that the prisms are packed together. According to Lowenstein's rule, A1 must attach to Si. Si may attach to either Si or Al. The number of Si atoms available for attachment to Si, NS?, is
48 4 24 48 76 12 6 12 18 48 12 2
12 2
+ 1)
Substitution of eq l a into eq 2a gives
30 6 12
5 6 a k = number of A1 atoms per prism.
kwkeka= 12Ne-*/(R
k=O
Ns? = Nsi - NM
TABLE 11: Number of Silicon Atoms N ( m k j ) with m = 0, 1, 2, and 3 OAl Neighbors for Each Distinct Prism Structure, Si(OAl),(OSi),, Where m + n = 3
Since
NM = N/(R
+ 1)
Nsi = RNM = RN/(R
+ 1)
ha 0 1 2 one in each ring
3
4
two in one ring: meta t w o in one ring: para three in one ring two in one ring: para two in one ring: meta t w o in one ring: meta two in one ring: meta three in one ring two in each ring: meta, meta two in each ring: meta, meta two in each ring: meta, para two in each ring: para, para
5 6
2 1
1 1 2
3 2 2 1 4 4 2
3 6 2 4 6 3 5 2 7 2 1 2
6 1
3 6
12 1 8 1 2 4 1 8 6 1 2 5 1 2 4 6 3 4 2 2 4 4 1 2 1 1 2 3 2 4 2 1 2 2 1 2
1 1 1 2 3 4 1 2 3 4 5 1 2
2
6
3
2
5
12
4
4
4
6
5
1 12 2
1 1
3
All of the A1 atoms are available for bonding to Si NAt = N/(R + 1) The probability that a Si will attach to A1 is
I'AI-s~ = NAt/(NSiA+ NAIA)= 1 / R and that Si will attach to Si is Psi-si = Ns?/(Nsi* + N M ~=) 1 - 1/R We can now calculate the fourth-neighbor population for each of the prisms containing various amounts of Al. For example, an arrangement such as five A1 atoms/prism contains one Si having no A1 neighbors (Table 11). If R = 2, half of these Si atoms will have no A1 neighbors and half will have one Al neighbor. The complete calculation is as follows: (1)Prepare a table of R as a function of /3 from eq 4. (2) Calculate populations nk for an appropriate R (0) from eq 3. (3) Calculate the numbers of Si with zero
k = number of A1 atoms per prism.
TABLE 111: Numbers N o , N , , ..., N , of Prismsa Having 0, 1, ..., 6 A1 Atoms as a Function of Silicon/Aluminum Atom Ratio R Calculated for Random A1 Distribution from Eq 3
R No 10.08 270 7.04 5.11 60 3.85 2.98 5 2.36 1.89 0.1 1.56 1.33 0 1.193 1.112 0 1.065 4.0 1.038 0 Normalized t o 1000 total prisms. 3 -2.0 -1.5 - 1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
N,
N2
438
237
N3 51
265
390
227
60
24 1
5
N,
N5
N6
0.1
0
53
5
0.3
383
24 1
60
10
50
214
368
250
113
0.1
3
40
187
345
425
0
0.1
3
44
219
7 34
0
0
0.2
7
98
895
4
J. Phys. Chem. 1982, 86, 3491-3492
3481
TABLE IV: Comparison of Calculated Distributions in Faujasite of Silicon Atoms Having n = 0 , 1, 2, 3, and 4 Nearest-Neighbor OAl Groups with Experimental Distributions from Ref 1 at Various Si/Al Atom Ratios R(Si/Al) R(Si/Al) = 1.19 Si(OAl), n= n= n= n=
0 1 2 3
n=4
R(Si/Al)= 1.35
R(Si/Al) = 1.67
R(Si/Al) = 2.0
R(Si/Al) = 2.45
ref 1
calcd
ref 1
calcd
ref 1
calcd
ref 1
calcd
ref 1
calcd
0 0 0 7.4 18.6
0.3
1.0
1.7
2.3
1.8 7.0 15.2
6.1 8.8 9.8
0.8 2.9 4.0 9.3 11.0
1.3 4.6 9.9 8.7 5.5
2.0 5.3 7.4 9.6 5.7
3.1 6.2 13.4 7.6 3.1
3.5 8.1 9.9 8.6 3.3
3.1 12.5 13.7 4.7 0
5.4 10.5 10.4 6.2 1.5
Conclusions and Results Partial results of the first two steps are given in Table 111. Table IV lists distributions of chemically inequivalent Si in faujasite calculated by this method and determined experimentally by ref 1. The random distributions are close to but not identical with the experimental ones. In general, the method described here need not be limited to faujasite, and Lowenstein’s rule need not be incorporated in the constraints. All that is required is that one be able to identify some relatively simple structural unit in the zeolite lattice.
COMMENTS Comment on “Photosensitization of Titanium( I V ) Oxide with Tris( 2,2‘-bipyridine)ruthenlum( I I ) Chloride. Surface States of Titanium( I V ) Oxide”
Sir: In a recent investigation1 of the photosensitization of titanium(IV) oxide by tris(2,2’-bipyridine)ruthenium(II) chloride, the fast-rise-time anodic photocurrent has been analyzed on the basis of the following reaction scheme: A
-
diffusion & B (electrode) k4
k
B+ + e-
where A corresponds to R ~ ( b p y ) ~B~to + ,the excited state, B+to R~(bpy),~+, and e- to the electron transferred to the solid and giving rise to an anodic current. The differential equation for the concentration b(x,t) of the excited species B was therefore given by
D(6b/6x),,o = kb(0,t) (5) the authors obtained an expression for b(x,t) and for the anodic photocurrent i, According to their analysis,l for times t such that t 1 the current was shown to decrease as t-ll2. This last result seems surprising in so far as it contradicts the assumption of a constant rate of generation of B by light, represented by the constant last term of eq 2. Indeed under such conditions one would expect that for long times the anodic photocurrent should reach a plateau. We suggest that this difficulty stems from the fact that b(x,t) as obtained in eq 10 of ref 1 satisfies only the first boundary condition (eq 4) and not the second one (eq 5). It may be shown that the following expression, different from the original one, for the concentration of the excited species
6/k2
b(x,t) = bo[l - exp(-kft)] where x is the distance from the electrode and where the last term bokf = $laax (3) represents the rate of generation of B by light of intensity I falling on species A of concentration a (4 is the quantum efficiency for conversion to the lowest triplet excited state and axthe extinction coefficient). Due to the fact that the lifetime of B is relatively short, only light absorbed close to the surface is efficient for the photocurrent. Therefore, the rate of generation of B was assumed to be a constant. From the above differential equation (eq 2) and the boundary conditions (4) b(x,O) = 0
+
satisfies both conditions 4 and 5; this expression also obeys the time evolution equation (eq 2). The anodic photocurrent is then easily deduced and is given by
i, = FAkbo
1”” dy exp[(p - l)y] erfc 0
(8)
where the notation is as in ref 1. Performing the integral over y leads to a more convenient expression 1
P-1
(pz)’/2[exp(p - l)z] - 1 + p1i2erf (z)’/~)(9)
J. B.Goodenough,J . Phys. Chem., 83,3280 (1979).
0022-365418212086-349 1$0 1.2510
(6)
with F(x,t? = exp(-kft?(exp[k(x + kt?/D] erfc [k(t’/D)1/2 ~ / ( 4 D t ? ’ / ~-] erfc [ ~ / ( 4 D t ? l / ~(7) ])
i, = FAkbo-(erfc (1) A. Hamnett, M. P.DareEdwards, R.D.Wright, K.R.Seddon, and
+ bokfLtF(x,t?dt’
0 1982 American Chemical Society
