L. A. D'ORAZIO AND R. H. WOOD
2550
Figure 6 represents a scheme of energy levels involved in nitrone -+ oxazirane and cis-trans isomerization. The potential curve for thermal rotation around the C-N double bond is denoted by the curve 1. Excited singlet nitrone produced by S + S* transition, 3 and 3', isomerizes to singlet oxazirane directly, 4 and 4'. The triplet nitrone formed by transfer of triplet excitation, 5 and 5', is deactivated to the ground state along the curves 2 and 2') twisting about the C-N bond.l9 It is concluded that the nitrone gives one of the most distinct examples in which reactions from singlet excited states are entirely different from triplet states.
Acknowledgment. The authors wish to express their gratitude to Dr. Y. Mori for his valuable discussions and to Mr. C. Shin for preparation of the isomers of the nitrone.
(19) The sensitization using tetracene (triplet energy is known to be 28 kcal.) was also carried out. With tetracene excitation, c i s + tram isomerization was observed, accompanied with some side reaction. tram cis isomerization could not be distinguished from trans isomerization sugthe side reaction. This sensitized cis gests that the energy difference between triplet state and cis ground state (nonvertical) is lower than 28 kcal., and the triplet function might fall below the maximum of the ground singlet function.
-
-
Thermodynamics of the Higher Oxides. I. The Heats of Formation and Lattice Energies of the Superoxides of Potassium, Rubidium, and Cesium'
by L. A. D'Orazio and R. H.Wood Department of Chemistry, University of Delaware, Newark, Delaware
(Received January 1.2, 1966)
The heats of formation of KO2, Rb02, and CsOz were determined at 25.0'. They were -68.0 0.4, -68.0 f 0.6, and -69.2 f 0.5 kcal./mole, respectively. The lattice energies of these compounds and the electron affinity of molecular oxygen ( E A = 14.9 kcal./mole) were calc$ated using various approximations. The accuracy resulting from the use of the approximations is assessed. The heats of formation (AHt' = -62 kcal./ mole) and decomposition of hypothetical lithium superoxide are estimated. The results indicate that this compound should be very unstable.
*
Introduction The alkali metals and, to a lesser extent, the alkaline earth metals have the tendency of forming higher oxides. Several of the alkali metals are capable of existence as the peroxide, superoxide, and ozonide. All of the alkaline earth metals can exist as the peroxide and there is evidence that some of them can exist as the superoxide. The thermodynamic data available for these compounds are rather meager and generally old. The following series of papers reports the results of a systematic investigation of some of the thermodynamic properties of these compounds, particularly the lattice The Journal of Physical Chemistry
energies and the heats of formation of the oxygen anions. The alkali superoxides are examples of compounds containing the simplest possible complex ion: that is, a complex ion with a covalent bond between two identical atoms. This presents the opportunity of investigating the accuracy of various approximations used in calculating the lattice energy of crystals containing complex ions. I n particular, it offered the opportunity of (1) This study was supported by the Air Force Office of Scientific Research Grant No. AF-AFOSR-325-63.
THERMODYNAMICS OF THE HIGHER OXIDES
investigating whether alternate methods of calculating the repulsion energy are equally acceptable. This paper is concerned with the experimental determination of the heats of formation of the superoxides of potassium, rubidium., and cesium together with calculations of the lattice energy using various assumptions.
Experimental Preparation of the Superoxides. The superoxides were prepared by oxidizing the liquid ammonia solutions of the alkali metals with oxygen a t -78”. The ammonia used as the solvent was distilled from a storage cylinder into a glass holding vessel and then slowly redistilled through dry potassium hydroxide pellets into the reaction chamber. Alkali metal (99.9% pure) was added to the liquid ammonia under the protection of an atmosphere of dry argon. Oxygen, which was successively passed through anhydrous sodium carbonate, calcium chloride, and phosphorus pentoxide was rapidly bubbled through the solution. As the oxidation proceeded, the dark blue metal-ammonia solutions gradually turned white owing to the formation of the peroxides. For the potassium and rubidium solutions, the white precipitates rapidly became yellow as the superoxides were formed. In the case of the cesium solution, a brown precipitate (giving the solution the appearance of chocolate milk) followed the white precipitate and persisted for a relatively long time before finally giving way to the yellow precipitate characteristic of the superoxide. The ammonia was evaporated from the reaction flask in a stream of dry oxygen. The superoxide remained behind as the residue. The superoxide was baked a t 100’ under 0.1 mm. pressure for 16 hr. The samples were then divided and loaded into weighed fragile glass ampoules in a drybox and stored in a desiccator until ready for analysis and calorimetry. One preparation of potassium superoxide and two each of rubidium and cesium superoxide were made. Analysis of the Superoxides. Five different analyses were carried out in order to characterize the samples. (1) The total metal was determined gravimetrically by precipitation as the tetraphenylborate salt.2 (2) Combined superoxide and peroxide content was determined by measuring the volume of oxygen evolved in the decomposition of the sample with water. A palladiumon-charcoal catalyst was used to ensure complete decomposition. (3) Superoxide content was determined by the method of Seyb and Kleinberg.3 (4)and (5) The samples were titrated with standard hydrochloric acid solution to the phenolphthalein and methyl orange end points. The results are given in Table I. The data have been adjusted according to the usual pro-
2551
cedure for conditioned measurements4 and the error limits are the standard deviations of the average values. ~~~~~
Table I : Analytical Results
KOz RbOz (1) RbOz (2) csoz ( 1) csoz (2)
Mor, %
MHCOs, %
MzCOs, %
94.7 95.5 95.0 94.6 93.5
2.0 3.8 5.0 2.0
3.3 0.7
...
... 3.4 6.5
Calorimetry. The superoxide samples were decomposed in water in a twin-vessel solution calorimeter. This apparatus has been described e1sewhere.j The calorimeter reaction proceeds quantitatively in the presence of a palladium catalyst to form the products, aqueous alkali hydroxide and oxygen. Fragile glass ampoules containing from 1 to 3 mmoles of the superoxide sample were submerged and broken in 675 cc. of M , carbonate-free sodium hydroxide solutions. I n order for the final state of the evolved oxygen to be accurately known, it was necessary to saturate the solutions with oxygen. A blanket of oxygen was kept over the solutions a t all times. Errors in the sample weight were negligible since 85400-mg. samples were used. The uncorrected calorimetric results for each sample are listed in column 1 of Table I1 as calories per gram of sample. The average values and their standard deviations are also given. The heat of ampoule breaking (0.02 cal.) and the correction to infinite dilution were negligible. The corrections for the heat effects due to the impurities and the vaporization of water by the evolved oxygen were not negligible. The corrected results are given in columns 2 and 3 of Table I1 in kilocalories per mole of superoxide. The error limits are the standard deviations of the average values. The heat of formation of water was taken from Circular 500.6 The heats of formation of potassium hydroxide (AHrO = - 115.32) and cesium hydroxide (AHiO = -117.6) are taken from (2) H. Flaschka and A. J. Barnard, “Advances in Analytical Chemistry and Instrumentation,” Vol. I, Interscience Publishers, Inc., Xew York, N. Y., 1960. (3) E. Seyb and J. Kleinberg, Anal. Chem., 23, 115 (1951). (4) A. G. Worthing and J. Geffner, “The Treatment of Experimental Data,” John Wiley and Sons, Inc., New York, N. Y., 1943. ( 5 ) H. S. Jongenburger, Thesis, University of Delaware, Newark, Del., 1963. (6) F. D. Rossini, D. D. Wagman, W. H. Evans, S. Levine, and I. Jaffe, “Selected Values of Chemical Thermodynamic Properties,” National Bureau of Standards Circular 500, U. S. Government Printing Office, Washington, D. C., 1952.
Volume 69, Number 8 August 1966
L. A. D'ORAZIO AND R. H. WOOD
2552
Table 11: Calorimetric Results a t 25" Uncor. heat of reaction of sample, cal./g.
KOz RbOz (1) RbOz (2)
-170.2 -111.0 -111.2 -80.3 -80.9
csoz (1)
csoz (2)
AH of reaction," kcal./mole
f 5.1 (5 runs) f 0.7 (2 runs) f 1 . 5 (4runs) f 2.2 (3 runs) f 1.0 (3 runs)
-13.0 -14.0 -14.2 -14.1 -14.1
The calorimetric reaction was MOz(s) f l / ~ H ~ O (+ l ) MOH(aq)
Gunn and Green7&and Friedman and Kahlweit,lb respectively. The Circular 500 value for cesium hydroxide is not used because it is based on uncertain cell measurements. The value for rubidium hydroxide (AH,' = -116.3) is based on a comparison of Rengade's measurements of the heats of solution of the alkali metals* with the more recent data for potassium and cesium. All other heat data are taken from Circular 500.6 The standard heats of formation of the superoxides calculated from the above data are tabulated in column 4 of Table 11. The limits of error are the standard deviations.
+ 02-(g) +M02(crystal)
where M + is the alkali metal ion. The reactants are considered t o be noninteracting ions at infinite separation and the product is the stable tetragonal alkali superoxide crystal. The energy of the process is equivalent to the enthalpy by virtue of the absolute zero condition. The calculated energy is inserted in the thermodynamic cycle outlined in Figure 1 and the electron affinity of molecular oxygen is deduced. The lattice energy was calculated by the classical
r,,
+
U
02-(g, 0°K.) +-
M02(crystal, 0°K.)
I
nl(crysta1, 2SS0X.)
+ Oz(g, 29S'X.)
Figure 1. Born-Haber cycle.
The Journal of Physical Chemistry
-68.0 i 0.4
-14.1 i 0.2
-68.0 f 0.6
-14.1 i 0 . 2
-69.2 f 0.5
+ a/rOz(g). Born-Mayer method where the energy per molecule, +(r), is given by the equation
M is the Madelung constant, C the van der Waals constant, $0 the zero point vibrational energy, and B(r) the repulsive potential. In the alkali superoxide system, the repulsive potential can be expressed as B(r) = 4c+A exp(r+
+
2c+-b exp(r+
- a/2)l/p r-msj - c / a ) l / p
rFmin
+
+
+
- f i ~ / 2 ) l / P+
+
+ (4/2)c-b exp(2r-,in - 4 ~ / 2 ) l / p+ (8/2)c-b exp(r-,in + -
(8/2)c++b exp(2r+ - 1/21/a2 c 2 ) l / p
The lattice energy of interest in this work is the energy change at absolute zero for the process
i\l+(g, OOX.)
-13.0 f 0.4
(4/2)c++b exP(2r+
Lattice Energy Calculations
M+(g)
f 0.4 f 0.2 & 0.2 f 0.5 i 0.2
Heat of formation, kcal./mole
Value adopted, kcal./mole
AHr' __f
M02(crystal,298°K.)
rWrnsj
where the first two terms represent the interaction of nearest neighbors and the remaining terms give the interaction of next-nearest neighbors. The cij are the Pauling factor^,^ b is a constant evaluated from the equilibrium condition
Mv= vo
= 0
(where U
=
-N+(r))
(3a)
r+ is the cation radius, and r-min and r-maj are the ionic radii of the ellipsoidal superoxide ion. The parameters c and a are the crystallographic unit cell dimensions equal to 27r and Zr, respectively, where r is the nearest neighbor distance and y = c/a. The repulsion constant, p, is determined from the relation
(7) (a) S. R. Gunn and L. G. Green, J . Am. Chem. SOC.,80, 4782 (1958); (b) H. L. Friedman and M. Kahlweit, ibid., 78, 4243 (1956). (8) E. Rengade, Ann. chim. phys., 14, 540 (1908). (9) L. Pauling, 2. Krist., 67, 377 (1928).
THERMODYNAMICS OF THE HIGHER OXIDES
2553
and also applying eq. 3b (with the assumption of isotropic compressibility), we find
The resulting equation is
Me2 C o 4(r) = -- -- - + 4
+
To6
where Ta
+
where A = 4 c + J exp (r+ r- - a/2)/p, B = next term in B(r) of eq. 2, etc., and 7
==
.-To
-+-
dB(ro) Me2 dro
= -a
bB(a,c) ba
( ~
)j
bB(a,c) Tc
= - C ( b c ) ,
Now, expanding eq. 8 in terms of (y -1) and ignoring terms higher than (y - 1)2,we find that
6C
1 B(r) = - [ r 2
To6
U
+ r a r c ( y- l)']
(9)
+
The zero point energy is considered to be independent of r . This approximation does not appreciably affect the results. The lattice energy was also calculated from a potential function containing a much simpler expression for the repulsion energy. Ladd and Leelo have expressed the repulsion potential as
B(r) = b exp
(
9
--
After applying 'conditions (3a) and (3b), the Ladd and Lee potential fmction reduces to
Since the higher alkali superoxides have tetragonal crystal structures and are therefore anisotropic, we have used an alternate repulsion potential which will allow us to take advantage of anisotropic equilibrium conditions. The repulsion may be expressed as
B(r) = B(a,c) = 2b exp
(--2cp) + 4b'exP(-;)
where the first term corresponds to the repulsive interaction between an ion and its two nearest neighbors along the crystallographic c axis and the second term corresponds to the interaction of the ion with its four nearest neighbors along the a axes. After applying the conditions
(2)= o
where T = T , rC. It is shown, then, that our modification reduces to the Ladd and Lee equation for a cubic crystal and that for a tetragonal system the deviation in the repulsion is only second order in (y - l). It can also be shown that the coefficient of the r 2 / r term in eq. 4 takes on values close to unity for systems with cubic symmetry and small like-ion repulsions. We should, therefore, predict little difference in the various calculations for the lattice energies of the superoxides. The technique reported by Baughanl' for calculating lattice energies was unsuccessfully applied to the superoxide system. Only data for three salts were used for solving his eq. 29 and these showed considerable scatter about the least-squares solution. This fact, coupled with the long extrapolation required to deduce the electron affinity, makes his method unsuited to the present study.
Data Table I11 contains the data used for the calculations along with references to the sources of this data. The Madelung constant, M a , was interpolated from the data reported by Wood.12 The characteristic distance, 6 , is the cube root of the molecular volume. It was evaluated for an 0-0 bond distance of 1.28 8. and a charge distribution of 0.5 electronic unit per oxygen atom. This charge distribution corresponds to a quadrupole moment of -1.97 e.s.u. Errors in the quadrupole moment of 50% result in errors in M 8 of only 1%. This Madelung constant, based on a given charge distribution in the superoxide ion, includes higher pole interactions and eliminates the need for calculating separate summation constants for quadru-
a=ao
):(
(7)
=o a
c = co
(10) M. F. C. Ladd and W. H. Lee, J . Inorg. S u c l . Chem., 11, 264 (1959). (11) E. C. Baughan, T r a n s . Faraday Soc., 5 5 , 736 (1959). (12) R. H. Wood, J . Chem. Phys., 37, 598 (1962).
V o l u m e 69, Number 8
A u g u s t 1966
L. A. D'ORAZIO AND R. H. WOOD
2554
Table 111: Data for the Calculations Ma M6atz = 0 a, 10-8 om., a t 298OK. 10-8 om., a t 298OK. ao, 10-8 om., at 0°K. co, 10-8 om., at OOK. T + , 10-8 cm., at 0°K. r-msj, 10-8 om., at OaK. r-min, 10-8 om., a t OOK. y (Le., c / a ) c,
2
a, 10-5 deg.-1 (expansivity)
8, 10-12 cm.z/dyne c++ (Pauling factor)
(Pauling factor) c+- (Pauling factor) e--, 10-60 erg cm.8 C + + , 10-60 erg om.' e+-,10-60 erg cm.6 S-- (van der Waals sum) S++ (van der Waals sum) S+- (van der Waals sum) AHf", kcal./mole IP, kcal./mole AHaub, koal./mole CpMoz,cd T , kcal./mole e--
$7
KO1
RbOz
CSOZ
2.1888 2.2287 5.704 6,699 5.654 6.641 1.32 2.00 1.51 1.1744 0,0955 10.5 3.86 1.05 1.25 0.85 26.5 24.3 23.3 8.67 3.65 59.5 -67.9 100.0 21.5 3.9
2.1917 2.2280 6.00 7.03 5.95 6.97 1.47 2.00 1.51 1.1717 0.0910 10.1 4.80 1.05 1.25 0.85 26.5 59.4 37.4 8.46 3.64 59 -. 6 8 . 0 96.3 20.5 3.9
2.1929 2.2232 6.28 7.24 6.23 7.18 1.68 2.00 1.51 1.1529 0.0884 9.8 5.70 1.05 1.25 0.85 26.5 152 48.7 8.32 3.76 59 -69.2 89.7 18.8 3.9
Ref.
I
(I
b b C C
d e e
b b, m 2 2
f f 2
2 2 1
2 2 2
Q h
i
$oZa8
Cpar,s dT, l.:cal./mole
1.5
1.5
1.5
j
'"0$
Cpo2,g d T , kcal./mole
2.1
2.1
2.1
k
a See ref. 12. Ma is the Madelung constant defined in terms S. C. Abraof V1l3,the cube root of the molecular volume. hams and J. Kalnajs, Acta Cryst., 8,503 (1955); L. I. Kazarnovskaya, Russ. J . Phys. Chem., 20, 1403 (1946). Calculated from the expansivity. Goldschmidt radii. Estimated from the differences between the interatomic distances and the GoldThe Pauling factors were calculated Schmidt cation radii. from the approximation formula cij = 1 (Zi/Ni) (Zj/Nj), where Z i = va.lence of ith ion and Ni = number of electrons in W. Finkelnburg and W. Humthe outer shell of the ith ion. W. H. Evans, et al., J. Res. bach, Naturwiss., 42, 35 (1955). Nutl. Bur. Std., 55, 83 (1955). Estimated from the data of S. S. Todd, J. Am. Chem. SOC., 75, 1229 (1953). Assumed to be the ideal value, 6 / * R . H. W. Woolley, J. Res. Nail. Bur. Std., 40, 163 (1948). See text. The definition of z is in terms of the 0-0 distance, R ( 0 - 0 ) = 2zc.
'
+
'
+
'
pole-point and quadrupole-quadrupole interactions. The Madelung constant for the superoxide ion considered as a point charge was also taken from the same paper. The difference between these two constants enables the calculation of the higher pole interaction energy. The equation of Slater and Kirkwood13has been used to estimate the van der Waals interaction coefficients
E
3efi 49m1l2R6(ax)'/' NA
= --
+ (T~) -
CYA a B
1/2
=
CAB
- - (10) R6
where N , the effective number of electrons, is used as an empirical constant. The Journal of Physical Chemistry
Huggins and Sakamoto14have used a procedure which is equivalent to assuming that N112is a linear function of the nuclear charge in an isoelectronic series. Morris16assumed that N is the same for chloride and sulfide. For polyatomic systems it has been shown that the assumption of an independent center of attraction at each nucleus gives reasonable results.16 Accordingly, the equation of Slater and Kirkwood has been used to calculate the interaction of the oxygen atoms using a polarizability of each oxygen atom (a = 1.32) taken from Kazarnovskii's value of the polarizability of the superoxide ion ( C Y = 2.64). l7 The number of effective electrons for each O-'/' group (N = 1.93) was calculated for the values for the fluoride ion (N = 1.61) and the sodium ion (N = 3.13) assuming that N'I2is a linear function of charge. The fluoride and sodium ion values were calculated from Mayer's accurate evaluation of the van der Waals energyI8 and Pauling's values of the p~larizabilities.~gThe interaction constants for the positive ions were taken from Mayer while the interactions of the positive and negative ions were calculated from the above values and eq. 10. The final values are given in Table 111. The van der Waals sums were calculated using an IBM 1620'I computer to perform a direct sum.20 .4 sum over shells of unit cells was used following the electrostatic energy method of Wood.21 The program calculates the sum of R-6 for a series of nonequivalent atoms. The van der Waals energy is then
where the sums are over-all types of atoms j and i and over-all radii, Rk, between a central atom j and an atom of type i in the shells being calculated. These sums are grouped according to the coefficients of the van der Waals interaction C++, C+-, and C- t o give (13) J. C. Slater and J. G. Kirkwood, Phys. Rev.,37, 682 (1931). (14) M.L. Huggins and Y. Sakamoto, J . Phys. SOC.Japan, 12, 241 (1957). (15) D.F.C. Morris, Acta Cryst., 11, 163 (1958). (16) L. Pauling and M. Simonetta, J. Chem. Phys., 20, 29 (1952); K. Pitaer and E. Catdano, J . Am. Chem. SOC., 78, 4844 (1956); K. Pitaer in "Advances in Chemical Physics," Vol. 11, I. Prigogine, Ed., Interscience Publishers, Inc., New York, N. Y., 1959, p. 59; S.Kimel, A. Ron, and D. F. Hornig, J . Chem. Phys., 40, 3351 (1964). (17) I. A. Kaaarnovskii and S. I. Raikhshein, Russ. J . Phys. Chem., 21, 245 (1947). (18) J. E.Mayer, J . Chem. Phys., 1, 270 (1933). (19) L. Pauling, Proc. Roy. SOC. (London), A114, 191 (1927). (20) The authors wish to thank the University of Delaware Computing Center for the use of their facilities. A Fortran listing of the program and a set of directions can be obtained by writing t o R. H. Wood. (21) R. H.Wood, J . Chem. Phys., 32, 1690 (1960).
THERMODYNAMICS OF THE HIGHER OXIDES
E
2555
= ~:l,rnCl,m(Sl,m/~‘)
where the sum is taken over all pairs of atoms and SI,, is a reduced sum using the cube root of the molecular volume, 6, as a characteristic distance. The results of the calculations are given in Table 111. The zero point energy of interest in this work does not include the internal vibrational energy of the superoxide ion and is given by the relation22
40= ‘/she
(12)
where 4ois the energy per constituent ion and 0 is the Debye temperature. The compressibility and zero point energy were obtained from the Debye temperature using Blackman’s equation23 and eq. 12, respectively. The compressibility agrees to within 3% with the value calculated from Huggins’ equation.24 The fact that these equations apply to cubic crystals was neglected. The Debye temperature (0 = 279°K.) wm calculated from the specific heat measurements of Todd. 25 The expansivity of potassium superoxide was estimated from a consideration of its size, binding energy, and heat capacity. The estimated value is in agreement with the value calculated from the equation of Joshi and Mitjra.26 The expansivities of rubidium and cesium superoxide were then estimated from the change in expansivities in the K-Rb-Cs series of the isostructural, homologous families in the alkali halides. The hypothetical lattice distances at absolute zero were then calculated, the slight decrease in expansivity with temperature being considered.
Results and Discussion The heat of formation of potassium superoxide has been accurately determined by Kazarnovskayan (A H f o = -67.9 f 0.1 kcal./mole) and by Gilles and Margravez8 (-67.6 f 0.8 kcal./mole). The result of this work (-68.1 f 0.4 kcal./mole) is in excellent agreement with these values. The present results are also very close to the results of de F ~ r c r a n d but , ~ ~this is fortuitous since corrections for changes in supplementary data reveal differences of 5 to 10 k~al./mole.~ Kraus and Petrocelli30have calculated the heats of the reactions
+ Rb202 +Rb2O +
RbO2 +‘/zRbzOz
‘/202
‘/202
from equilibrium pressure measurements at 280 to 360’. Calculation of the standard heat of formation of rubidium superoxide from these data and estimated heat capacities yielded AHfo(RbOz) = 51 kcal./mole, a figure which is irreconcilable with the result of this work. It is interesting to note that the entropy of the
first reaction reported by Kraus and Petrocelli, 2 e.u., is considerably smaller than the value for the same reaction of the sodium analogs, 19 e.u. The individual value for the Coulomb energies, higher-pole interaction energies, London energies, and zero point vibrational energies are given in Table IV. The lattice energies and electron affinity are also given in Table IV. Kazarnovskii3’ has calculated the lattice energies and found values 6 to 9 kcal./mole lower. Kazarnovskii’s values were based on a quadrupole moment for the 02- ion of 3 X e.s.u. cm.2. This corresponds to a +0.76 charge on the oxygens and a -1.52 charge a t the center of the molecule. Recent measurements of the quadrupole moment of the oxygen give a charge of less than 0.1 on the oxygens. This indicates that a - l / Z charge on each oxygen in a much better estimate of the charge distribution on 02-since 02-has one more antibonding electron than 02. Table IV : Results of Calculation
Coulomb energy, kcal./mole Higher pole interaction energy, kcal./mole London dispersion energy, kcal./mole Zero point energy, kcal./mole Repulsion energy, eq. 5 , kcal./mole Repulsion exponent, eq. 5 , A. Lattice energy, eq. 5 , koal./mole Electron affinity of oxygen, kcal./mole
KO1
RbOa
CSOZ
197.0 -3.6 8.7 -1.2 -23.8 0.274 177.1 12.6
187.1 -3.0 10.0 -1.1 -24.3 0.296 168.7 16.4
179.5 -2.5 10.9 -1.0 -24.7 0.317 162.2 15.8
Evans and U r i ’ P value for the lattice energy is based on an incorrect Madelung constant and a spherically symmetric 02-ion. Yat~imirskii’s3~ values for the lattice energies were calculated using Kapustinskii’s equation. Both results differ by. +2 to - 5 (22) R. H. Fowler, “Statistical Mechanics,” 2nd Ed., Cambridge University Press, London, 1936, p. 123. (23) M.Blackman, Proc. Roy. Soc. (London), A M I , 58 (1942). (24) M. L. Huggins, J . Chem. P h y s . , 5, 143 (1937). (25) S. S. Todd, J . Am. Chem. SOC.,75, 1229 (1953). (26) S.K. Joshi and S. S.Mitra, 2.p h y s i k . Chem., 29, 95 (1961). (27) L. I. Kazarnovskaya and I. A. Kazarnovckii, Zh. Fiz. Khim., 25, 293 (1951). (28) P. W. Gilles and J. L. Margrave, J . P h y s . Chem., 60, 1333 (1956). (29) de Forcrand, Compt. rend., 150, 1399 (1910); 158, 991 (1914). (30) D. L. Kraus and A. W. Petrooelli, J . P h y s . Chem., 66, 1225 (1962). (31) I. A. Kazarnovskii, Dokl. A k a d . N a u k SSSR, 59, 67 (1948). (32) W. V. Smith and R. Howard, P h y s . Reu., 79, 132 (1950); R. Anderson, W. V. Smith, and W. Gordy, i b i d . , 8 2 , 264 (1951).
S.
(33) M.G. Evans and N. Uri, T r a n s . Faraday Soc., 45, 224 (1949). (34) K. B. Yatsimirskii, Khim. i Khim. Tekhnol., 4, 480 (1959).
V o l u m e 69, N u m b e r 8 A u g u s t 1966
L. A. D'ORAZIO AND R. H. WOOD
2556
kcal./mole from the present values. This is surprisingly close for approximate equations. It is instructive to examine the dependency of the total lattice energy on its various components. Equation 6 can be rewritten in the form
Analogous expressions derived from eq. 4 and 9 yield results similar to eq. 13. For the alkali superoxides, the parenthetical factor in the repulsion term does not differ by more than 8% for the three calculations. The repulsion term in (13) consists of a sum of terms each of which is one of the attractive energy components Kr-n multiplied by a factor -n/r, Now for these salts p / r takes on values between 1/9 and l/ll. This is in the same range as p / r for the alkali halides, cuprous halides, and silver halides. I n general, most salts that have been reported in the literature have p / r values in the 1/6 to 1/12 range. This means that potentials varying between and r-12 are largely cancelled in the empirical repulsion term. The general principle can be stated: the lattice energy becomes insensitive to a small energy term, E ( r ) ,as
+
E(r) +B (r)
dE(r) dr
d
a
dr
In the usual case of E(r) varying as r-n and B(r) as exp( - r / p ) , the lattice energy becomes insensitive to the minor term E(r) as n + r / p . Since p is somewhat constant, this criterion can be used a priori to determine the effect of a proposed potential term on the resulting lattice energy. For example, in the superoxide and 1/11, we case with p / r having values between may completely neglect the London r-8 and r-l0 terms. This is a fortunate aspect of the lattice energy calculation since those potentials varying with high negative powers of the distance are generally very difficult to determine and contain large uncertainties. This criterion also shows why the lattice energies of the superoxides (and in general any complex system) are considerably more sensitive to the higher pole interaction energy than the London energy. The superoxide ion is a quadrupole and gives rise to a quadrupole-point charge interaction varying as r3 and a quadrupole-quadrupole interaction varying as T - ~ . The repulsion energies and repulsion exponents ( p ) for eq. 5 are given in Table IV. Equations 2 and 8 yielded energies which were within 0.3 kcal./mole of the reported values and repulsion exponents which differed by less than 5%. The negligible differences beThe Journal of Physical Chemistry
tween the Ladd and Lee method (eq. 6) and the BornMayer method (eq. 4) are to be expected for crystals which are not very asymmetric and for which like-ion repulsion is small. The situation concerning the dependency of the lattice energy on the compressibility is not so favorable. The effects of the compressibility can be seen by expanding the Ladd-Lee repulsion term
-
ro
t-tr2 Np
The analogous expressions from eq. 4 and 9 yield results which are within 1% of eq. 14. For the superoxides the term 9V/Np amounts to approximately 75% of the denominator in eq. 14. This means that approximately 75% of the uncertainty in /? will be carried into the repulsion energy. This is surely the major uncertainty in this work and contributes an uncertainty of 2 kcal./ mole to the lattice energy and electron affinity. That this is a general case for all lattice energy calculations can be shown by observing that the ratio, (9V/NB)/u, will be sizeable for most salt systems. It is 0.7-0.8 for the alkali superoxides, 0.6-0.8 for the alkali halides, and 0.3-0.5 for the alkaline earth chalogenides. In fact, if the lattice energy of a system increases, its compressibility decreases. This ensures that the ratio will never become small and that accurate lattice energy calculations by the usual techniques will always require reasonably good compressibility data. In summation then, this investigation indicates several factors that stand out in applying the classical treatment of lattice energies to cornplex systems. First, although the Coulomb interaction presents no problem in that the lattice sums can be quickly and accurately determined on modern electronic computing equipment, the charge distribution of the system can become critical, especially in the cases of the larger complex ions and the asymmetric ions, Here the higher pole interactions become important and can seriously affect the lattice energy. Second, the availability of accurate compressibility data has been shown to be of paramount importance. This holds even for simple inorganic systems. Since there is a paucity of accurate compressibility data, this appears to be the weakest link in the calculation. Third, a more simplified expression of the repulsion energy is apparently suitable for crystals that approach cubic symmetry and do not have large lilieion repulsion. This reduces the labor of calculation. In calculating the electron affinity of oxygen from the superoxides the approximate contributions to the total
2557
THERMODYNAMICS OF THE HIGHER OXIDES
error are 1 koal./mole from the total uncertainties in the Born-Haber cycle, 2 kcal./mole from the uncertainty in the compressibility, and 3 kcal./mole from the uncertainty in the model itself. Accordingly, the electron affinity of molecular oxygen is 15 kcal./mole with an estimated uncertainty of 5 kcal./mole (0.65 f 0.22 e.v.). This value is in reasonably good agreement with those previously calculated by the thermochemical t e ~ h n i q u e . ~They ~ ~ ~ range ~ , ~ ~from 0.65 to 0.96 e.v., the present value, 0.65, being the most carefully determined. Physical techniques for measuring this quantity (e.g., photodetachment experiments) have indiM ~ I l i k e nhas ~ ~ atcated a lower value of 0.15 tempted to resolve this situation by a semitheoretical argument. He contends that the lower value, 0.15 e.v., is slightly more consistent with his estimate of the superoxide bond dissociation energy. This being established, he then adopts the lower value on the grounds that considerable errors are possible in the thermochemical value. The present work results in a lower electron affinity than previous thermochemical values. However, the value 0.15 derived from photodetachment experiments is still well outside the estimated limits of error. The X-ray data for potassium superoxide have been re-examined by Halverson. 37 He interprets the data in terms of a crystal in which the oxygen atoms are revolving around the c axis in a circle of 0.15 to 0.34 A. radius with a resulting coupling of nuclear and electronic motions. Since the higher pole energy term is only of the order of 5 kcal./mole, it is unlikely that the change in electrostatic energy due to the distortion could be large enough to account for the 0.5e.v. discrepancy between the thermochemical and the photochemical values of the electron affinity. It is also
doubtful that coupling between the nuclear and electronic energies would be large enough to account for the difference. In conclusion, this work confirms the inconsistency between the thermochemical and photodetachment values of the electron affinity of molecular oxygen. Finally, the lattice energy of Li02 can be estimated from a plot of lattice energy vs. alkali metal for the superoxide family. This type of plot performed for various alkali metal salts (e.g., halides, cyanides, sulfides, azides, etc.) yields a family of curves of similar slope. The estimate from the extrapolation is U(Li0,) = 210 i 10 kcal./mole. From this value the heat of formation is calculated to be AHfO(Li02) = -62 f 10 kcal./mole. This value indicates that Li02 is unstable with respect to decomposition to Li202and Li20. The following properties can be estimated from available data3* 2Li02 2Li02
-
Liz02
+ +
AH = -28 kcal./mole AF = -28 kcal./mole
0 2
a/2o2
AH = -18 kcal./mole AF = -25 kcal./mole
This explains the great difficulty that has confronted the attempted preparation of Li02.
Acknowledgments. The authors wish to thank Messrs. Ronald W. Smith and Henry L. Anderson for assistance with the calorimetric measurements. (35) D. 5. Buroh, S. J. Smith, and L. M. Bransoomb, Phys. Rev.. 112, 171 (1958). (36) R.S. Mulliken, ibid., 115, 1225 (1959). (37) F. Hdverson, J. Phys. Chem. Solids, 23, 207 (1962). (38) L. Brewer, Chem. Rev., 52, 1 (1953).
Volume 69,Number 8 August 1966
