E. Cohen de Lara and J. Vincent-Geisse
1922
Potential Energy af a Molecule Adsorbed in Synthetic Zeolites. Application to the Analysis of Infrared Spectra. 2. Potential Energy of the N 2 0 Molecule Adsorbed in the Cavities of the Zeolite NaA. Interpretation of the Adsorbed N20 Infrared Spectra E. Cohen de Lara* and J. Vincent-Geisse Equipe de Spectroscopie Moleculaire en Milieu Condense, Laboratoire de Recherches Physiques, associe au C.N.R.S., Universite Pierre et Marie Curie, 4, place Jussieu, escalier 22, 75230 Paris Cedex 05, France (Received May 12, 1975)
The potential energy of the N20 molecule in the zeolite NaA has been calculated from the evaluation of its electrostatic, dispersion, and repulsion components for different positions and orientations of the molecule. We thus obtain the positions and depths of the potential wells. The potential barriers to translation and rotation are very high and the molecule performs librational and translational oscillations inside the well, the frequencies of which have been calculated. We also evaluate the lifetime of the adsorption and the time of residence of the molecule in the well. The observed far-infrared spectra may account for the calculated frequencies. As regards the vibrational spectrum, the splitting of v1 is interpreted by the two possible orientations of the molecule N2O: the profile of the v3 band and its variation with temperature can be explained by the frequency distribution due to translation.
I. Introduction In the previous article1 we calculated the electrostatic field in the zeolite NaA. This calculation was an essential stage in the evaluation of the potential energy of adsorbed molecules, which will be dealt with now. The general method used and the approximations made will be described first, after which the application of this method to the case of the N2O molecule will be examined in detail. Finally an attempt will be made to interpret the corresponding infrared absorption spectra in the light of the results obtained by the calculations.
QZZ= -2Q,, = -2Q = 0 we obtain
yy.
Accounting for the relationship div E
(4)
This formula is absolutely general but its application is not very easy when the field possesses no symmetry element and if the molecule is given all possible orientations. On the other hand it can be written more explicitly when the molecule is centered on a symmetry axis of the field, in which case the latter is reduced to its radial component E , and we have
11. Potential Energy Calculation Method This energy may be expressed as a sum of several terms233 @ = @E
+ @D + @R
(1)
where @E, @D, and @R represent the electrostatic, dispersion, and repulsion energies, respectively. (1)Electrostatic Energy @E. This includes a term @pdue to the permanent moments and another 91due to the induced dipole moment. @E = @ p
+ @I
(2)
In the case of a molecule surrounded by a field obeying the Laplace law div E = 0, Buckingham4 gives the following expression for @p: @P =
-
1
- 3 Q,pE’,,
2
with the same summation rules as before. Applying this general relationship to the particular case of a linear molecule of axis Oz the following expression is obtained:
(3)
stopping a t the quadrupolar term. The indices a and p must be replaced successively by x,y , or z , axes of a Cartesian referential linked to the molecule. The terms pa and Qap refer to the components of the dipole moment vector and the quadrupole moment second-order tensor, respectively, the latter being defined by taking the center of mass as origin. E , and E’,@ are the components of the field and its gradient. In each case a summation is made on the indices repeated twice. If the molecule and the field present symmetry elements this expression becomes simpler. In the case of a linear molecule of axis 0 2 , p = pLzand Q = The Journal of Physical Chemistry, Vol. 80, No. 17, 1976
with IC/ the angle of the vectors 8, and ,G. r refers to the abscissa, always positive, of the mass center of the molecule on the cavity radius where it is located. In the case of the induction energy, only the first-order term due to the induced dipole will be retained in its expression. If a0p represents the polarizability tensor of the molecule 1 @I = - a,,E,Ep (6)
@I
1 1 = --ai/ E z 2- - a l ( E Z 2 + E,’) 2 2
(7)
Finally, when the linear molecule is centered on a symmetry axis of the field
- -1 E 2 -1( all + 2 ~ 1 ) 1 (all - a 1 ) ( 3 COS’ $ - 1) (8) 2 ‘ 3 ( 2 )Dispersion Energy @D. Only the first-order terms corresponding to the induced dipole-induced dipole interaction will be considered in this energy contribution. Several formulae have been proposed to express @D and we shall adopt that of Kirkwood and M ~ l l e r , which ~ - ~ involves known or calculable parameters and which has already been used in @I=
+5
N$3 Adsorbed in the Cavities of Zeolite NaA
1923
calculating the potential energy of molecules adsorbed in alkaline halide layers8,g and in faujasite type zeolites."J In the case of atom, ion, or molecule couple this formula is as follows:
where m is the mass of the electron, c the speed of light in vacuo, d the distance between the two particles, a1 and a2 their polarizabilities, and x1 and x 2 their diamagnetic susceptibilities. In the case of an adsorbed molecule the total interaction energy may be considered as the sum of binary interactions between the adsorbate and each of the adsorbant surface ions. For the zeolite NaA we have two kinds of ion in the presence of the molecule: 0°.25ions and Na+ cations. Let M, +, and - be the indices referring to the molecule, the cation Na+, and the ion 0°.25-, respe~tively,~ di the distance from the center of the molecule to the center of the cation i , and dj the distance between the molecule and the oxygen ion j . The dispersion energy of the molecule in the NaA cavity may then be written in the form @D =
A+
d,-6 1
+A-
dj-6
(10)
I
with
A+ = 6mc2
aMa+ aMIXM f a+/X+
(11)
and
A- = 6mc2
aM/XM
+ a-/X-
(12)
The constants A are negative since the diamagnetic susceptibilities are negative. The summations in d-6 can easily be computed since the coordinates of the adsorbant surface ions are known, having been established for the purpose of electrostatic field determination. It should be pointed out that eq 10 does not depend on the orientation of the molecule with respect to the line joining its center to that of each ion considered. This amounts to comparing the N20 molecule to a sphere in the a~ calculation, taking for a the mean polarizability. (3) Repulsion Energy @R. Whereas several formulae are available to account for dispersion forces it is still difficult to express repulsive forces in simple terms and their contribution is often determined empirically. In general the energy of repulsion between two particles is expressed either in the form of an exponential, Be-Cd,8s,gor in the Lennard-Jones form Bd-12. lo We shall adopt the latter which leads to simpler calculations. Applying the additivity principle as in the case of dispersion we obtain an expression of the form (13) but this time the constants B cannot be calculated a priori. They are determined by considering the potential energy of the molecule facing an isolated ion of charge q , placed in its stable equilibrium with respect to this latter. Let CTMbe the collision diameter of the molecule and a1 the van der Waals diameter of the ion; assuming equilibrium between attractive and repulsive forces at the distance ue = ( U M + u1)/2, the energy passes through a minimum and its derivative with respect to d cancels out for d = ue.loJ3 The numerical value of the constant B relative to the couple is then obtained. For repulsion forces also the molecules are likened to spheres. Our
calculation of @D and @R does not account for the orientation of the molecule in the zeolite cavities and hence the variation of these energies with the above factor is neglected. This approximation seems reasonable in the case of the N 2 0 molecule since the electrostatic potential varies considerably with the orientation and even changes sign during a rotation of the molecule (eq 5 ) .
111. Potential Energy Calculation for the N2Q Molecule in the Zeolite NaA (1)General Remarks. The formulae given above involve a number of quantities relative to the adsorbed molecule and the zeolite ions. The numerical values adopted are listed in Tables I and 11. The figures quoted for the 0°.25ion (Table 11)are those of Broier and co-workers.14The dispersion constants given by formulae 11and 12 and the repulsion constants calculated as shown above are presented in Table 111. The potential energy is calculated quite differently according to whether we are dealing with @E or @D and a . ~The . last two are independent of the orientation of the molecule and their expressions are the same a t all points in the cavity, whereas it is different for @E. If the Oz axis of the molecule is directed along any axis it is necessary, to calculate @E from the general expressions 4 and 7 , to know the components of the field and its gradient along Oz i.e., the electrostatic potential value V a t points very close to the center of the molecule and in all directions. Considering that such an estimation would have required a very lengthy calculation it was necessary to make certain choices. A simple examination of the physical problem gives an idea a priori of the equilibrium orientation of the molecule since, @D and @R being independent of this orientation, the energy is a t a minimum when the axis of the molecule follows the direction of the field; along the symmetry axes of the cavity the field is entirely radial. For the other radii considered in NaA the radial component E , is the largest of the field components1 for distances below 4 A from the center. To begin with, 9 will therefore be calculated by displacing the molecule longitudinally along the symmetry axes and certain other radii. The variable is then the distance r from the center of the molecule to the center of the cage. The potential wells having been found, as expected, to be located on the symmetry axes, the molecule will be placed in the deepest well and its orientation, i.e., the angle #, will be varied. ( 2 ) Potential Energy of the N2O Molecule Placed Longitudinally Along a Radius of the Cavity. The molecule is placed along a radius with its more negative pole (oxygen) directed toward the surface ions. Under these conditions # = 0 and eq 5 and 8 give, for the electrostatic energy, the expression
This formula is exact for symmetry axes and approximate for other radii, a term equal to - s a L ( E x E y 2 )being neglected in the latter case because ai is much smaller than all while E x and E y are very probably small with respect to E , since they both cancel out on the symmetry axes 0 2 , ONaI, and ONaIII (Figure 2 of ref 1).Table IV gives, by way of example, the results obtained for each of the @E terms for different r values along the Oz axis. The quadrupolar energy is seen to be always the highest of the three and the dipolar energy the lowest. C€?Dand @R are given by eq 10 and 13 with the constants A and B of Table 111. As an example Table V shows the results obtained in these calculations for the energies @E, @D, and @R and the total energy a, for different r values, in the case of the
+
The Journal of Physical Chernisfry, Vol. 80, No. 17, 1976
1924
E. Cohen de Lara and J. Vincent-Geisse
TABLE I: Physical Constants of NzO Molecule (esu) 0.17 X
-3.0 X 48.6 x 10-25 20.7 x 10-25 30.0 x 10-25 4.4 x 10-8 31.4 x 10-30
a b b b b C
Reference 11. Reference 2. Reference 15.
(1
TABLE XI: Physical Constants of Zeolite Ions (esu) a
U
1.90 x 1.96 x
Na+ Ca*+
2.8 x
00.25-
10-8
10-8 10-8
X
-6.9 x -22.1 x
1.9 x 10-25 4.7 x 1 0 - 2 5 n 14.7 x 10-25 b Q
-17.7
x
10-30 a,c 10-30~~c 10-30 b
Reference 15. Reference 14. e Reference 12. TABLE 111: Dispersion and Repulsion Constants of Different Molecule-Ion Couoles (esu) Couple
A X 1060
B X 10102
NZO-Na+ N20-Ca2+ N20-00.25-
-21.9 -59.4 -121.6
24.2 80.2 15.3
TABLE IV: Electrostatic Energy of NzO in NaA Zeolite Along the OZ Axis of the Cavity (unit ergs)
1
0.01
1.07
1.5
0.12 0.36 0.86 1.82 3.58 6.67
2.95 6.33 12.44 23.41 42.20 71.52
2
2.5 3
3.5 4
0 0.01 0.11
0.62 2.77 10.77
37.36
-1.08
-3.08 -6.80 -13.93 -28.00 -56.55 -115.54
TABLE V: Potential Energy of N2O in NaA Zeolite Along the OZ Axis of the Cavity (unit ergs) r,A 0 0.5 1
1.5 2
@E
+0.28 -0.06 -1.08 -3.08 -6.80
2.5
-13.93
3 3.5 4
-28.01 -56.54 -115.5
@D
-5.67 -5.87 -6.40 -7.29 -8.63 -10.56 -13.36 -17.72 -25.50
@R
0.22 0.27 0.41 0.73 1.44 3.21 9.39 42.51 297.1
@
-5.16 -5.66 -7.01 -9.64 -13.99 -21.28 -31.98 -31.55 +156.1
Oz axis. Figure 1 shows the variation of the potential @ and r along the symmetry axes and certain radii. On Oz a very deep potential minimum is observed, much less pronounced on ONaI and practically nonexistent on ONaIII. It remains to be found out which of these minima correspond to true potential wells (@ minimum for the three variables r , 0, and d,) and this was achieved by following the @ variation in the three planes 4 = O", d, = 22" 5', and d, = 45" varying 0 from 0 to 45" in the The Journal of Physical Chemistry, Val. 80, No. 17, 1976
Figure 1. Potentialenergy @(r)of N20in NaA along the symmetry axis and radii Nall and O02.
plane d, = 0 and from 0 to 90" for the other two planes. The conclusion is that only the minima following Oz and ONaI correspond to potential wells. Figure 2 shows the variation of @ with 0 in the plane d, = 45". This latter is especially interesting because on varying 0 from 0 to 90°, the Oz axis (0 = 0), an ONaII radius (0 IO"),an 002 radius (0 26O), an ONaI axis (0 = 54"),and an ONaIII axis (0 = 90") are encountered in turn. Table VI gives the coordinates of the different potential wells: re refers to the equilibrium position of the molecule along the radius and @, the @ minimum; @O being the energy at the cavity center, $0 - 9,represents the potential barrier to translation along the radius. The deepest well is found on the highest-order symmetry axis. Six of this kind exist, two per OX, OY, OZ axis, and eight wells opposite the NaI cations. (3) Potential Energy Variation of the Ne0 Molecule in a Rotation Around Its Stable Equilibrium Position. The deepest wells have been determined in translation of the molecule along the radii. The mass center of the molecule is now placed at the points found, the nuclear axis making an angle $ with Oz. is then the only variable since the radius considered is an order 4 symmetry axis. Formulae 5 and 8 are applied. For $ = 0 (r;in the direction of the field) and = BO0, two relatively close energy minima are obtained; the difference A@ = 0.56 kcal/mol, represents the variation in dipolar energy which changes sign between the two orientations. For $ = 90°, on the other hand, the quadrupolar energy, very high, becomes positive and the total energy passes through a maximum. Figure 3 shows the @ variation when the molecule turns around an axis perpendicular to 0 2 . The potential barrier to rotation is very high, 5.9 kcal/mol for the equilibrium orientation $ = 0. In other words the molecule has a very small probability of rotation but two possible equilibrium orientations with respect to the field, presenting one or other of its ends opposite the cation. The probability ratio of the two
-
-
+
+
N20Adsorbed in the Cavities of Zeolite NaA
1925
TABLE VI: Coordinates (r,+e) and Depth of Potential Wells of N20 in NaA Zeolite Axis ret
8,
ae,kcal/mol - ae,kcal/mol
2o
40
OZ
ONaI
3.05 -5.0 3.7
3.25 -2.7 1.4
t
I
Figure 3. Variation @(4)of potential energy orientation.
molecule is trapped in a deep enough potential well, it performs translational motion in the case of a potential varying with the position of the mass center of the molecule, or librational motion if an angular potential is involved. The frequencies UT and UL corresponding to these motions can therefore now be calculated from the known potential functions @ ( r )and (a($) (Figures 1 and 3).
20-
Figure 2. Variation of @(r)with 0 in the plane 4 = 45'
orientations is equal to exp(- A + / k T ) , i.e., 0.4 a t room temperature.
IV. Analysis of the Energy Calculation Results The above calculations were aimed at an attempt to interpret the infrared absorption spectra of the NzO molecule adsorbed in the zeolite NaA.16J7 For this it is first necessary to analyze the calculation results and extract certain information for direct comparison with experiment. (1)Calculation of Translation and Libration Frequencies. Energy Levels Corresponding t o These Motions. When a
with m and I, respectively, the mass and moment of inertia of the molecule, and v the wavenumber in cm-l. Table VI1 gives the UTand VLvalues calculated from the curves of Figures 1 (Oz axis) and 3; both are about 60 cm-l. These frequencies are active in the infrared since they correspond to a variation of the dipole moment (permanent induced moments), either in modulus or in direction, with intensities proportional to oi2(dE/dr)2for translation and ( ~ p ~ 1for) libration. ~ To these translation and libration frequencies correspond energy levels. Accounting for the statistical distribution of the molecules on each level we have
+ +
hu 1 +-hv exp(hv/kT) - 1 2 where E represents the mean energy and u the frequency considered (ETand UT for translation, E L and UL for libration). The corresponding levels are placed on the curves @ ( r )(Figure 4) and a($) (Figure 5 ) at two temperatures. The mean translation and libration amplitudes can be read on the figures; the respective values are 0.28 A a t 313 K and 0.20 A a t 173 K for translation, 20" at 313 K and 15" a t 173 K for libration. The E values are given on Table VI1 in kilocalories per mole. (2) Lifetimes Relative to Adsorbed Molecules. Knowing the potential curves it is possible to determine a certain number of characteristic times concerning adsorbed molecules. First of all the adsorption lifetime T A is linked to the energy of the molecule in the well, i.e., @E E . Furthermore the residence
E=
+
The Journal of Physical Chemistry, Vol. 80, No. 17, 1976
E. Cohen de Lara and J. Vincent-Geisse
1926
TABLE VII: Translational and Librational Motion of NzO in NaA Zeolite Translation Frequency, u, cm-I Mean energy, E , 313 K kcal/mol 173 K
0
2 3 . .
66 0.63 0.35
. . .v.. ,
Figure 4. Mean energy levels
58 0.62
0.35
I
.
r
3s
Figure 5. Mean energy levels ELon the curve a($),
time 7~ in a well is connected to the potential barrier to translation, i.e., the energy @E + E - @o. We have18J9
TR
exp(-@E - ET)/kT
7~
=
TT
=
TT
exp(-@E - E T
+ @o)/kT
(18)
(19)
where TT represents the oscillation period of the molecule with respect to the surface, Le., 1/cuT, TT = 0.5 X s. The The Journal of Physical Chemistry, Vol. 80, No. 17, 1976
7A
313 173
5.7 x 10-10 3.7 x 10-7
7R
7FL
7 x 10-11 8.6 X
2.5 x 10-l2
3.4 x 10-12
-
i
/
I
I
T ,K
quantities needed for the calculations appear on Tables VI and VI1 and the results are given in Table VIII. Before the figures obtained are discussed another characteristic time will be calculated, the time of flight TFL between two neighboring sites. We have TFL = l/u with u = (kT/m)1’2, 1 being the distance between the two sites. There 1 6 A. The figures of Table VI11 show that the ratio TA/TFL varies from 30 at room temperature to 2500 a t low temperature. At 173 K the time spent.outside the wells is negligible compared with the residence time inside; at 313 K it is no longer negligible but still short.
on the curve @(r).
I
\
Libration
TABLE VIII: Lifetimes Relative to the Adsorbed NzO Molecule
V. Comparison with Experimental Results In the light of the potential calculation data an interpretation of spectra obtained in the far-infrared17 and in the medium infraredl6 will be attempted. (1)Fur-Infrared Spectrum. The experimental study was carried out17 between 30 and 200 cm-l; great difficulties are caused by the low absorption intensity and a very broad band is observed between 60 and 140 cm-1, hence a t a frequency slightly higher than the calculated value. I t should be noted that these spectra were obtained a t low temperature, the cavities being saturated with NzO which does not correspond to the calculation hypotheses, and furthermore the translation along directions perpendicular to the radius were not taken into account. Finally the calculated frequencies depend on the form of the potential curve near the minimum and in view of the approximations made, the UT value only represents an order of magnitude. The frequencies calculated can therefore account for the absorption observed. (2) Vibrational Infrared Spectrum. The spectra of the parallel bands u1 and us, studied experimentally at two different temperatures,16J7are shown in Figures 6 and 7. Their general aspect pointed to an absence of rotation in the cavities, a conclusion supported by the considerable height of the potential barrier to rotation. A complete interpretation of the spectra would involve the calculation of the frequency shifts and a band profile determination. Where the frequency shifts are concerned the interaction potential would have to be developed as a function of the normal coordinates, but in all the previous calculations the molecule was assumed rigid and hence this point of view will not be pursued. The band VI is a doublet, the two components lying on either side of the gas frequency, while ug is single and its high intensity allowed a precise study of the profile, the width and their variation with temperature. These two phenomena will be examined in turn. (a) Splitting of u1. After discussion of the various likely interpretations it was suggested that this splitting could be explained by the two possible orientations of the NzO molecule.16 The potential energy calculation for these orientations (Figure 3) shows that the intensity ratio of the two components should be 2.5 at 313 K and 5.1 at 173 K. The variation observed is different from that expected. In fact the same would apply if the splitting were interpreted by the existence of two potential wells separated by a barrier, a hypothesis rejected because of the fact that the two components lie on either side of the gas frequency. However, the
N20 Adsorbed in the Cavities of Zeolite NaA
1927
T %
1
I320
1300
I
1270
1250
9 an-'
Figure 6. u1 band of N20 adsorbed on NaA zeolite at two temperatures: (a) 313 K, (b) 173 K.
Log
I, I
(b) Profile of the v 3 Band. This band is very wide, especially at room temperature. A possible reason which may be eliminated straight away is the finite vibration lifetime which correspondsto the residence time TR of the molecule in its well. This lifetime corresponds to a width of l/cl[IrR, Le., 1.5 cm-1 at 313 K and 100 times less at 173 K. Lower values still are obtained by considering the potential barrier to rotation. On the other band, our interpretation of the profile by a vibration frequency distribution is confirmed qualitatively by the potential curves. Referring to Figure 4 we observe that a t room temperature the molecule oscillates between points A and B, and the time it takes to go from one to the other is q / 2 , Le., 25 X s. Since the vibration period u3 is 1.5 X s, the molecule performs 17 vibrations between points A and B. We thus have a slow modulation of the vibration by translation and a frequency distribution is observed, the highest frequency corresponding to point B and the lowest to point A. At 173 K the translation amplitude is smaller and the resulting frequency distribution narrower. However, an examination of the spectrum shows that the limiting frequency on the high frequency side is the same at high and low temperature, which suggests that the rise in potential on the B'B side is really larger than shown on the figure and that an exponential repulsion might be more appropriate. The anharmonic nature of the potential curve moreover explains the mean frequency shifts toward high frequencies when the temperature decreases. On the whole the theoretical results confirm some hypotheses put forward during analysis of the spectral6 and afford a better understanding of the phenomena. Research in progress on other molecules in other synthetic zeolites will extend the possibilities of comparison between theory and experiment, and show more clearly whether the potential calculation should be improved and in what direction. References and Notes
Figure 7. u3 band of N20adsorbed on NaA zeolite at two temperatures: (a) 313 K, (b) 173 K.
importance of this intensity factor should not be over-estimated. Owing to the well-known existence of a resonance between two electronic structures of the NzO molecule,2* one or other of these structures is privileged when the molecule turns one end or the other toward the cation. The intensity depends largely on the variation of the induced moment during vibration and this is not necessarily the same for both structures. In our opinion therefore the interpretation proposed (two orientations) is not contradicted by the potential calculations, but the question can only be settled by a frequency shift calculation.
(1) E. Cohen de Lara and Nguyen Tan, J. fhys. Chem., preceeding article in this issue. (2) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids", Wiley, New York, N.Y., 1954. (3) A. D. Buckingham, "Intermolecular Forces", Interscience, New York. N.Y ., 1967. (4) A. D. Buckingham, "Molecular Properties", Physical Chemistry, Vol. IV, Academic Press, New York, N.Y., 1970. (5) J. G. Kirkwood, fhys. Z.,33, 57 (1932). (6) A. Muller, froc. R. SOC.London, Ser. A, 154, 624 (1936). (7) L. Salem, Mol. Phys., 3, 441 (1960). (8) W. J. C. Orr, Trans. faraday Soc., 35, 1247 (1939). (9) R. Gewirzman, Y. Kozirovski, and M. Folman, Trans.faraday Soc., 65,2206 (1969). (10) R. M. Barrer and R. M. Gibbons, Trans. faraday SOC.,59, 2569 (1963). (11) D. E. Stogryn and A. P. Stogryn, Mol. fhys., 11, 371 (1966). (12) J. H. Van Vleck, "The Theory of Electric and Magnetic Susceptibility", Oxford University Press, London, 1965. (13) D. M. Ruthven and R. I. Derrah, J. Chem. SOC.,Faraday Trans. 1, 68,2332 (1972). (14) P. Broier, A. V. Kieselev, E. A. Lesnik, and A. A. Lepatkin. Russ. J. fhys. Chem., 42, 1350 (1968). (15) "Handbook of Chemistry and Physics", Chemical Rubber Publishing Co.. Cleveland, Ohio. (16) E. Cohen de Lara, Mol. fhys., 23, 555 (1972). (17) E. Cohen de Lara, These, 1975, CNRS RO 10926. (18) J. de Boer, "The Dynamical Character of Adsorption", Oxford Clarendon Press, London, 1968. (19) Frolich (20) L. Pauling, "The Nature of the Chemical Bond", Cornell University Press, Ithaca, N.Y., 1960.
The Journal of Physical Chemktry, Vol. 80, No. 17, 1076
