PAUL A. GIGUEREAND OSIASBAIN
340
a function of the ratio of the vibration frequencies, the relationship being the more direct the smaller the value of m J w . At the two so1,utions corresponding to &,z = 0, a/2, where the directions of the normal modes coincide with the axes of the elliptical contours, the eccentricity is a maximum (for v-) or a minimum (for v f ) (for nomenclature see ref. (3)). The defect of the direction of the normal mode vector from the principal axis is dependent on both vibration frequency and massratio. For HzO the directions nearly coincide for all solutions. The defect increases through SO2 to Cl20, and is always greater for the mode of higher frequency (Le., nearer to the short axis). I n Fig. 6, the variation of axis-length is shown as a continuous plot against d12,and in Fig. 7 the directions of the axes are similarly plotted. The dependence of this plot on the mass-ratio is rather striking; in C120 the type of curve is essentially different. However, this does not affect the continuity of the axis-length plots. V. Conclusion The above treatment of the symmetrical tri-
Vol. 56
atomic system gives a clear picture of the implications of any particular potential function; from the two directions allowed to the normal modes the potential gradient opposing motion along any other direction has been deduced. Such a treatment, or its inverse, would probably be found convenient in quantum mechanical calculations of potential functions, but the results have some intrinsic interest. The fact that all directions other than those corresponding to the normal modes are in general “unobservable” should be remembered; there is some evidenceD that the value of a given force constant may be dependent on the directions of displacement. But this type of variation should be only of the order of an anharmonicity correction. The above work has been carried out as part of the programme of fundamental research undertaken by the British Rayon Research Association. The author is grateful to Dr. Orville Thomas, of the Edward Davies Chemical Laboratories, University of Wales, Aberystwyth, for checking the calculation for Fig. 7 (e). (9) P. Torkington, Proc. Roy. Soc., in the press.
FORCE CONSTANTS I N HYDROGEN AND DEUTERIUM PEROXIDES BY PAULA. GIGUSREAND OSIASBAIN Department of Chemistry, Lava1 University, Quebec, Canada I
Received December $6, 1861
The infrared spectrum of hydrogen peroxide was re-examined and that of deuterium peroxide was measured for the first time in the vapor state. For the isotopic molecule five bands were observed at 480, 947, 1944, 2680 and 5236 Fm.1. The observed frequencies were combined with those .of the Raman spectra to calculate the force constants of the various bonds.
Introduction Hydrogen peroxide has a rather weak, 11011revealing infrared spsctrum.1n2 Only two bands are prominent,, corresponding to the asymmetric OH bending and stretching modes. The 0-0 vibration is hardly noticeable in the vapor and the overtones of O-H bands show multiple overlapping due to doublet splitting so that, even under high resolution, very little of the rotational structure is visible. Finally the torsional oscillation, to be expected from the model of Penney and Sutheris very elusive; a broad absorption band at 16 p in the spectrum of the liquid has been assigned to that mode4 although i t could arise merely from molecular perturbation in condensed phases. Therefore, a study of the isotopic molecule DzO2 was desirable to provide further information on the structure. The Raman5 and infrared4 spectra of this compound in the liquid phase have been investigated but no work has been reported yet on the vapor. It was hoped that such a study would throw further light on the controversial question of the 0-0 bond energy68’; as it turned out, however, (1) P. A. GiguBre, J . Chem. Phys., 18, 88 (1950). (2) L. R. Zurnwalt and P. A. GiguBre, ibid., 9,458 (1941). (3) W.G.Penney and G. B. B. M. Sutherland, Trans. Faraday Soc., 80, 898 (1934); J. Chem. Phrs., 3, 492 (1934). (4) R. C. Taylor, ibid., 18, 898 (1950). ( 5 ) F. FehBr, Ber., ‘la, 1778 (1939). (6) P. A. Giguare, Can. J . Research, B28, 17 (1950).(7) A. D.Walsh, J. Chem. Sac., 831,388 (1948).
’
there still remains some uncertainty on the 0-H bond length in hydrogen peroxide. Discussion of Results The vibrational spectrum of deuterium peroxidc vapor was measured in the range 2-25 p with a prism instrument. Experimental details, as well as related studies on the absorption of alkaline solutions of hydrogen peroxide, will be reported elsewhere. For the present purpose the observed frequencies are listed in Table I together with the Raman shifts5 for both molecules. The coarse rotational structure of the band a t 3.7 p was partly resolved and from the 10 or 12 observed maxima the major rotational constants of D202 were calculated: A’ - B’ = 4.8 cm.-I, A” B” = 5.1 em.-’. Because of the limited resolution these figures are less accurate than those for the two overtone bands of hydrogen peroxide. The assignment v~ v6 (B) now seems the only possible one for the unexplained band a t 2630 cm.-l in H202 and 1944 cm.-l in D202; anharmonicity requirements indicate that the frequency of v2 in the vapor must be slightly higher than had been thought previously. The broad band of medium intensity a t about 660 crn.-l in the spectrum of crystalline H202 (at -70”) is shifted to 480 cm.-l in the isotopic compound. The ratio of these frequencies (1.37) is further evidence that they belong to the torsional oscillat,ion.
-
+
.
March, 1952
FORCE CONSTANTS IN HYDROGEN AND DEUTERIUM PEROXIDE
341
Isotope Effect.-Application of the Teller-Redlich product rule to hydrogen and deuterium peroxides leads to the following equations for the symmetric and the antisymmetric vibrations
now differ by less than 1% and the ratio ( I ~ I ~ ) / ( I is J zfound ) to be 2.113 f 0.02. Moments of Inertia.-From the ratio of the moments of inertia obtained by means of equation 3 an attempt was made to determine the range of possible values for the azimuthal angle 4 between the two H-0-0 planes. These calculations were made using the equations derived by Wilson and Badger for H & P and verified following Hirschfelder's general method1' of evaluating the moments of inertia of molecules for which the orientation of the TABLE I ,three principal axes is not ,obvious. The 0-0 VIBRATIONAL SPECTRUM OF HYDROGEN AND DEUTERIUM distance was taken as 1.49 A. from X-ray12 and PEROXIDES electron diffraction datal3 and from application of -HzO----Infrared Raman Badger's rule to the 877 cm.-l frequency; the 0-H (vap.) (lid distance and the 0-0-H angle were varied from (em.-1) Av (em.-]) 480' (m.) their va.lue in 520, 0.957 A. and 105.5', respec660° (m.) 877 (V.8.) 877 (v.w.) 877 (v.s.) tively, to 1.01 A. and 96", the latter limits being 947 (V.8.) 1266 (v.s.) set by the measured rotational constants of hy1420 (K.) 1009 (w.) drogen peroxide.' The same parameters were 1944 (w.) 2630 (w.) 3414 (rn.) 2510 (rn.) used for deuterium peroxide in accordance with 2680 (s.) 3614 ( 8 . ) what has been found in other isotopic molecules. 7036.6 (v.w.) 5236 (v.w.) The reylts of these numerous calculations are 7041.8 (v.w.) summarized in the curves of Fig. .1, from which it 10283.7 (v.w.) 10291.1 ( V . W . ) appears that the azimuthal angle is comprised 2vi + vh(B) within the range 82 f 20'. Previously it had been 0
In the solid at -70".
where the symbols are those used by Herzberg (reference 8, p. 232). The superscript D refers to the isotopic molecule; I , is the moment of inertia about the Cz symmetry axis and 11,12 are respectively the small and large moments about the other two axes. Since the symmetric vibrations have been observed only in the Raman spectrum their frequencies are not known accurately because of molecular association in condensed phases; consequently equation 1 cannot provide much significant information. The ratio I:/Is = 1.12 obtained from the Raman shifts in the two molecules is that to be expected within the limits of accuracy of experimental data. . If the observed frequencies are used in equar, on 2 the left-hand side will be greater than the righthand side due to the anharmonicities of the 0-H and 0-D stretching vibrations. To correct for the use of first-order, instead of zero-order frequencies the left-hand side was multiplied by the factor
2.22 I 2.20 2.18 2.16 2.14 '
d
2 2-12 2
22.10 Lu 2.08 2.06 2.04 2.02 2.00 30 60 90 120 150 180 Azimuthal angle ("). Fig. 1.-Variation of ratio 1 ~ 1 ~ / 1 1with 1 2 the azimuthal angle for the following values of the parameters: 0
as suggested by Hedberg and Badger.s The force constants IC and IC: were calculated from the antisymmetric vibrations and appropriate structural parameters as described below. The choice of the latter has no great importance in the final result as the correction term amounts to less than 1.7%.of the whole. The two sides of the modified equation
Cg)"' (3)
(8) G.Herrberg. "Infrared and Raman Spectra of Polyatomic Molecules," D. Van Nostrand Co., h e . , New York, N. Y.,1945. (9) K. Hedberg and R. M. Badger, J . Chem. Phys., 19, 508 (1951).
A
0.957
101.2
0
concluded that this angle should not be between and 95" because of irregular intensity dis-
85'
(10) M. K. Wilson and R. M. Badger, ibi3., 17, 1232 (1949). (11) J. 0. HirRchfelder, ibid., 8 , 431 (1940). (12) S. C. Abrahams, R. L. Collin and W. N. Lipscomb, Acta Crust.. 4, 15 (1951). (13) P. A. Giguere and V. Schomaker, J . Am. Chem. SOC.,68,2025 (1943).
PAULA. GIGUI~RE AND OSIASBAIN
342
tribution near the center of the combination band of H202 in the photographic infrared.2 Angles of 94' and 106" have been found in the crystal of hydrogen peroxidela and its addition compound with ureal4 but these variations are easily understood in view of packing constraints. Figure 2 shows the locus of possible values for the three uncertain parameters. c
peroxides the following quantities are obtained readily ~ O - H ~ ' o - H = 7 . 2 9 X 1W dynes/cm.
180.1
+ +
7 . 5 6 X lo6 dynes/cm. 0.848 X 106 dynes/cm. 8oon f 6'00~ = 0.858 X 106 dynes/cm. ko-n
~'o-D =
~ O O H f 6 ' 0 0 ~=
The following data mere used in calculating these force constants: Ro = I .49 A. ro = 0.97 A.
I
I-
Vol. 56
A.
(7r
-
102O = 82"
=
01") c$
+
'
The values of ( ~ o - H k6-1.1) and ( ~ o - Dand k b - ~ ) are insensitive to the assumed azimuthal angle. Thus for 4 = 90' there obtains
+
k ; ) - ~ )= 7 . 2 9 X 1 0 6 dynes/cm. (ko-D f k & ~ ) = 7.57 X lo6 dynes/cm. ( 6 0 0 ~f &OH) 0.841 X 10' dynes/cm. (600~ 4- &OD) = 0.844 X lo6dynes/cm. (~o-H
=.
.e3
e2
I
I
I 95.
I
I
I
I
I IO0
I
'
I
'
" I05
OOH ANGLE ('1. Fig. 2.-Locus of possible 0-H distances and 0-0-H angles for 0 ",90' and 180' azimuthal angles fitting the rotational constant Xolo9.23 cm.-l for HzOz.
On the other hand the solution of the secular equation corresponding to the symmetric vibrations is mtubh less accurat,e because of the uncertainby on the frequency of vl, v2 and va. ( v 3 is certainly very close to 877 cm.-l in the vapor of both HzOz and DzOz). The' fundamental frequencies listed in Table I1 were selected as the most probable for calculating the force constants.
Force Constants.-A quadratio potential funcTABLE I1 tion has been assumed by Morino and Mizushima16 for the S2Clzmolecule of C2 symmetry, which in the FUNDAMENTAL FREQUENCIES AND FORCECONSTANTS OF case of HzOzbecomes HYDROGEN A N D DEUTERIUM PEROXIIIES Force _.__ ~
Y
1
-k 3 6r;[(Aor1)'
+ ( A a # ] - ~ ' $ A W A W 4--yr: (A+)'
(4)
The symbols have the following meaning: AR, Ar,, Ar2 are the changes in the 0-0 and the two 0-H distances, ro being the equilibrium 0-H distance, Aal and A m are the changes in the angle between the 0-H, bonds and the extension of the 0-0 bond, and A+, the change in azimuthal angle. k , 6 and y are the force constants for the squared terms, and k' and 6' are those for the crossed terms. This potential function has proved unsabisfactory in the case of Dz02 yielding imaginary values for ko-o and ( k o - ~ k ' o - ~ ) . It was therefore necessary to add to the potential function an interaction term of the form (k"roARAa) where k" is a nondiagonal element which cannot be eliminated in the d matrix of Torkington.16 The elements of the A matrix of Torkington associated with the appropriate internal coordinates have been given by Decius.17 The secular equation may be factored into two equations, one corresponding to the symmetric, and the other to the antisymmetric vibrations. As the antisymmetric vibrations have been measured directly in the vapor of hydrogen and deuterium
+
(14) C. 8. Lu. E. W. Hughes and P. A. Giguhre, J . Am. Chem. Soc., 63, 1507 (1941). (15) Y . Morino and 9. Miiushims, Sci. Papers I n s t . Phys. Chem. Research, 32, 220 (1937). (16) P. Torkington, J . Chem. Phus., 17, 1026 (1919). (17) J4 C , Degius, i b i d , , 16, 1025 (1948),
(om.?)
HzOp
DzOn
PI
3610 1315 877 660 3614 1266
2670 1000 877 480 2680 947
~1 ut u4
vs U6
constsnts (dynes/om.) X 10-6
ko-H kb-H
LO-o ~OOH
660~ Y
k"
DiOt
Ha02
7.28
