The Lattice Energy of Nitrogen Pentoxide

3 The Lattice Energy of Nitrogen Pentoxide R. M. CURTIS and J. N. WILSON

Downloaded by EMORY UNIV on February 26, 2016 | http://pubs.acs.org Publication Date: January 1, 1966 | doi: 10.1021/ba-1966-0054.ch003

Shell Development Co., Emeryville, Calif.

A revised value for the heat of formation of nitrate ion has been obtained by re-evaluating the lattice energy of cesium nitrate with consideration of charge distribution in the anion. With the revised value for ΔΗ (NΟ ) the Born-Haber cycle yields a lattice energy of —157 kcal./mole for nitrogen pent oxide. The electrostatic (Madelung) energy of N O has been found to depend significantly on the charge distribution within the two ions. Charge distribution estimated by means of a Hückel molecular orbital treatment leads to a Madelung energy very nearly equal to the lattice enthalpy given above. f

-

s

2

5

À s an ionic crystal, N 0 is unusual i n several respects. It possesses a layer structure i n which each ion is surrounded b y only three nearest neighbors of opposite charge rather than the more usual coordination sphere of six or eight neighbors. The heat of formation is quite small, the vapor pressure is high (50 mm. at 0° C ) , and the gaseous moleoule is covalent rather than ionic. These factors plus the possible effects of charge distribution within both cation and anion indicate N 0 to be a particularly interesting example for applying the ionic model of lattice energy. 2

5

2

5

Heat of Formation of ΝΟ ~ $

In applying the Born-Haber cycle to N O to determine the lattice energy it is found that the heat of formation of the nitrate ion is the only major thermodynamic quantity for which an experimental value is not available. This quantity is obtained from calculated lattice energies of the alkali metal nitrates. Values ranging from —78 (19) to —85 (14) have been reported of which the average of —84 kcal./mole from L a d d and Lee (14) is probably the most reliable. In all these evaluations of the nitrate ion heat of formation, a simplified crystal structure is implied i n which the nitrate group is treated as a point charge ion. The location of 2

s

2 3

In Advanced Propellant Chemistry; Holzmann, Richard T.; Advances in Chemistry; American Chemical Society: Washington, DC, 1966.

24

ADVANCED PROPELLANT CHEMISTRY

the entire unit negative charge of this ion at the position of the central nitrogen atom is unrealistic. A more reasonable assumption involves a charge distribution with nonintegral charges on all four atoms such that the algebraic sum of these charges is equal to the ion charge. Some time ago Topping and Chapman (26) calculated the electrostatic energy of sodium nitrate based on the charge assignment N + 0 ~ , but we have felt it desirable to consider a variable charge distribution in at least one case, C s N 0 , where more recent data are available. Since the nitrate ion is not spherically symmetric, the value calculated for the heat of formation of nitrate ion from a lattice energy w i l l depend i n general on the charge distribution assigned within the ion. If the charge distribution on the ion in the crystal differs from that i n the free ion, then the calculated heat of formation is likely also to be different i n some degree from the true heat of formation of the free ion. Nevertheless, if the charge distribution does not change appreciably from one nitrate crystal to an other, a lattice energy calculated for Ν Ο or some other nitrate from the Born-Haber cycle should still be meaningful Cesium nitrate crystallizes at room temperature i n a hexagonal (29) lattice, but the structure has not been determined. Above 160° C. it exists in a cubic modification containing eight molecules per unit cell. The structure of this form is known (30), having been determined at 167° C , and is the basis for calculation, later correcting to 25° C . The electro static energy of C s N O was computed (JO) for several assumed nitrate ion charge distributions and after correcting for the electrostatic selfenergy of the ion the results were fittted with a second degree equation in x, the nitrogen atom charge, giving the Coulomb energy. 5

3

2


3

2

δ

â

E

e

=

-150.40

+

0.534* -

0 . 9 7 9 * kcal./mole 2

(1)

In this method of calculation the energy of electrostatic interaction among all of the charges i n the structure is determined. The calculated energy thus includes a contribution from the interaction within each poly atomic ion among the charges assigned to its constituent atoms. This "self-energy" must be subtracted from the total calculated energy to ob tain the Coulomb contribution to the lattice energy—i.e., the interaction energy of the ions i n the crystal with each other. It is of course an ap proximation to represent the charge distribution i n a polyatomic ion as a collection of discrete charges centered on each atom; this is equivalent to assuming that the electron distribution i n the ion can be represented by a set of overlapping spherically symmetric distributions, each centered on one of the atomic nuclei. Since covalent bonding between the atoms making up the ion is caused by some concentration of electronic charge density in the region between bonded atoms, this representation cannot be precisely valid. I n the absence of detailed information about the actual charge distribution, however, the procedure used here appears to be a


3.

CURTIS A N D WILSON

Nitrogen Pent oxide

25

useful approximation. Computationally, it appears to be simpler and faster than the alternative method of representing the charge distribution by superposing a point charge and a series of multipoles. Equation 1 is relatively flat between the limits χ = + 1 and χ = 0 which correspond, respectively, to a simple resonance hybrid for the n i trate ion and to unit negative charge distributed over the oxygen atoms with the nitrogen neutral. The exact value assumed for the charge distri bution is thus unimportant. W e shall assume χ = 0.17 based on a H u c k e l molecular orbital treatment (20) which leads to E = —150.3 kcal./mole. This is almost exactly the value E = —150.5 kcal./mole obtained by assuming a simple monomolecular unit cell of the C s C l type w i t h inter atomic distance r = 3.89A. The nonelectrostatic terms include van der Waals, polarization, re pulsive, and zero-point energies. T h e van der Waals energy has been calculated i n several ways with results ranging from 5 to 15 kcal./mole. Examples of these results are indicated i n Table I. I n the first two entries polarizabilities were taken from Bottcher ( 3 ) , and the nitrate group was treated as a single entity with an ionization potential of 99 kcal./mole for the London equation (23) and an effective electron number of 24 for the Slater-Kirkwood equation (23). A simple C s C l type of lattice was as sumed. 0


c

Table I. Van der Waals Energy in C$N0 Gs NOi~ +

Cs+N . 0|-.« +

1 7

London equation Slater-Kirkwood London equation Slater-Kirkwood

3

4.5 kcal./mole 15.3 9.0 13.3

F o r the second two entries the nitrate group was treated i n terms of individual atoms with effective charges as indicated above of +.17 on the nitrogen and —0.39 on each oxygen atom. Effective ionization energies of 190 kcal./mole for oxygen and 390 kcal./mole for nitrogen were ob tained from curves connecting known ionization potentials with ionization state. Similarly, using various calculated and experimental polarizabilities ( I , 6,12,15,17, 21, 2 2 ) , effective polarizabilities of 0.8 A . for oxygen and 1.0 A . for nitrogen were obtained. F o r cesium ion the Pauling (22) value was used. The sum of these atom polarizabilities i n the nitrate group is 3.4 A . , rather close to the nitrate polarizabilities reported b y Tessman, Kahn, and Shockley (25) (3.4-4.0 A . ) and Bottcher (3) ( 3 . 7 3.8 A . ) . F o r the Slater-Kirkwood equation, we have used as electron numbers Cs = 8, Ν = 6, Ο = 6. In summing the van der Waals energy over a l l atom pairs, the lattice sum coefficients for the C s C l structure can no longer be used. Instead the sum has been made over a l l atom pairs out to 5 Α., and this figure was increased b y 1 0 % to allow for more distant 8

3

3

8

3


26


neighbors. O f the values i n Table I, we prefer the Slater-Kirkwood i n dividual atom treatment giving 13.3 kcal./mole. O u r extrapolation of Bridgman's (4) data leads to a compressibility β = 5.0 ± 0.2 χ 1 0 sq. cm./dyne. This applies to the low temperature (hexagonal) form at room temperature. T o apply this to the high tem- 1 2

i

da

perature form at 167° C . we have assumed a temperature coefficient β = 6 Χ 10" d e g . similar to that for the alkali halides and a difference i n β at the transition of 1.7 X 10~ sq. cm./dyne from an earlier paper b y Bridgman (5). The result at 167° C . is β = 7.3 ± 0.5 X 10~ sq. c m . / dyne, a compressibility considerably larger than that used by L a d d and Lee (14). Using the convenient expressions given by L a d d and Lee (13) and assuming 1.0 kcal./mole for the zero-point energy and 13.3 kcal./mole for the van der Waals energy, the repulsive energy is 15.7 kcal./mole and the lattice energy —146.9 kcal./mole. It is necessary to correct the lattice energy to 25° C . and introduce the vibrational energy i n order to obtain ΔΗ for the reaction at 298° K . : 4

-1

12


12

Cs+(g) + N O r ( g ) = CsNO (c) (2) F r o m available heat capacity data (18) and a comparison w i t h related compounds ( K B r 0 , C s C 1 0 ) the incremental enthalpy, (H — H °), was estimated at 25° C . (4.4 kcal. ) and 167° C . (9.3 kcal. ). Cesium nitrate does not appear to fit a simple Debye-type expression, but the vibrational energy has been estimated by difference from integrating C -C = α νΤ/β in which the expansion coefficient, a, was taken from density data (8). A t 167° C . we thus estimate ( E / — E °) = 7.4 kcal./mole. A d d i n g the vibrational energy to the lattice energy and subtracting the difference £Γιβ7° — # 2 5 ° as w e l l as 5 R T for the free ions gives, for E q u a tion 2, ΔΗ = —147.4 kcal./mole. F r o m the Born-Haber cycle using as standard heats of formation A f f , ( C s + ) = 110.1 kcal. (24), A i f / ( C s N 0 ) = —121.5 kcal./mole (24), and the above heat of reaction, it follows that Δ / / / ( Ν 0 ~ ) = —84.2 kcal./mole w i t h an estimated uncertainty of db 2 kcal. If the heats of formation of R b N 0 and C s N 0 are revised (16) by —0.5 and —3.4 kcal., respectively, L a d d and Lee's determinations of Δ ί / / ( Ν 0 - ) become —86.5 and —85.4 kcal./mole. s

3

T

0

4

0

p

v

2

0

3

3

3

3

3

The Lattice Energy of

NO 2

s

A p p l y i n g the Born-Haber cycle to N 0 w i t h Δ # / ( Ν 0 + ) = 233.5 kcal./mole (28), AH (N 0 ) = 10.0 kcal./mole (28), and Δ Η / ( Ν 0 - ) = —84.2 kcal./mole ( a l l at 298° K . ) from above gives as the heat of formation from the gas ions at room temperature a value ΔΗ = —159 ± 2 kcal./mole. The corresponding lattice energy, obtained by subtracting an estimated 5 kcal. for vibrational energy and adding 3 kcal. = 5 RT 2

f

2

5

2

5 c


8

3.

Nitrogen Pentoxide

CURTIS A N D WILSON

27

for the translational energy and PàV term associated w i t h the ions, is -—157 kcal./mole. Based on the known structure (7) the Coulomb energy for Ν Ο was calculated, as with C s N 0 , b y assuming specific charge distributions i n the ions N 0 and N 0 ~ , correcting for the self-energy and fitting the results with a quadratic equation i n X , the nitrate Ν atom charge and Y , the nitronium Ν atom charge: Thus: 2

δ

3

2

E


c

=

+

-150.65

3

-

5.20X -

10.63F + 2.278Z

2

+

1.694ΛΤ -

1.520F

2

(3)

In Equation 3 there is a far greater dependence on charge distribu tion than i n Equation 1. Examples of the Coulomb energy for several conceivable charge distributions are given i n Table II. Table II. Coulomb Energy of N 0 2

Ν Atom Charge in NOt ΝΟΓ (Y) (X) 2.37 2.02 -1 1 0 0 1.67 1 0.17 0.58 +

Configuration Minimum value of —E Point charges Neutral nitrogen Resonance bond Molecular orbital (20)

e

5

Coulomb Energy E e

-143.3 -157.0 -150.7 -172.7 -158.0

The minimum value of — E , while having no apparent physical signifi cance, indicates the least energy that can be associated with this particular hexagonal structure. The most reliable result is probably that from quan tum mechanics, the last entry i n Table I. This energy is surprisingly close to that for the point charge configuration. In deriving Equation 3 it was assumed that the self-energy of each ion, and hence the charge distribution and interatomic distance, are the same i n the free state and i n the crystal. It is quite possible that a real difference exists for the ions i n these two states. There is evidence ( 9 ) , for example, of a charge shift i n the nitrate ions of molten alkali metal nitrates depending on the cation polarizability, and an even greater shift is expected i n going to the isolated ion. Unfortunately, no quantitative estimate of this effect is available, but it should be noted that the results leading to Equation 2 indicate that a difference of only .01 unit i n charge between gas and crystal ions can lead to 10 kcal. difference i n E . Errors arising from such effects are probably small for the nitrate ion since its heat of formation was obtained from a lattice energy. The heat of formation of N 0 + , however, is obtained from the ionization potential and heat of formation of N 0 ( g ) ; in this case no error cancellation occurs. The agreement between calculated and observed heats of formation sug gests that the error from this effect is probably small for both ions. The complexity of N 0 and the absence of compressibility or elastic constant data preclude any reliable calculation of the nonelectrostatic c

0

2

2

2

5


28


terms i n the lattice energy. It would be desirable to sum the repulsive energy over near pairs of atoms, but repulsive parameters for Ν and Ο atoms are not w e l l established and doubtless depend on the charge density at each atom. Consequently, only an approximation is possible. I n the simple Born-Mayer expression (27) for lattice energy the repulsive energy is given by pE /R i n which ρ is the exponential repulsive parameter and R is the interionic distance. Typically (11) ρ is taken equal to 0.345 A . W i t h R equal to the shortest nitrogen-nitrogen distance between ions, 3.12 Α., the repulsive energy is then 17 kcal./mole. W h i l e this value for R is close to the sum of the ion radii, 1.3 A . from Grison et al (7) for N 0 + and 1.9 A . from Waddington (27) for N 0 ~ ~ , it probably results i n too large an estimated energy. There are only three ion neighbors i n the crystal at this distance while the next coordination sphere at 4.53 A . has six neighbors. A mean R for these nine neighboring ions given by 9 / R = 3/3.12 + 6/4.53 yields a repulsive energy of 13 kcal./mole. O u r pre ferred estimate for the repulsive energy is the average, 15 db 4 kcal./mole. The estimated uncertainty arises not only from R but from the rather large variations that have been observed (2) i n p. c

2


3

F o r the van der Waals energy i n Ν Ο we have again adopted the Slater-Kirkwood equation to individual atom pairs. The nitrate group was used with the electron numbers and polarizabilities noted previously. F o r N 0 + , estimated polarizabilities of 0.7 A . for Ν and 0.5 A . for Ο were based on the charges (20) Ν = +.58 and Ο = +.21. Since N 0 + con tains a total of only 16 outer electrons we have arbitrarily chosen electron numbers of 6 and 5, respectively, for Ν and O. The resulting van der Waals energy is 13.2 kcal./mole. 2

δ

3

2

8

2

The combination of Coulomb, repulsive, van der Waals, and zeropoint (estimated at 1 kcal.) energies yields a lattice energy of —158 + 15 —12 + 1 = —154 ± 5 kcal./mole. The discrepancy between this and the "experimental" lattice energy above of —157 ± 2 kcal./mole is w e l l within the expected uncertainty. Incidentally, the alkali halides show a gradual compensation of repulsive and van der Waals energies w i t h i n creasing ion size until with C s l these terms nearly cancel. I n the case of C s N 0 this sum is only 2.4 kcal. so that i n N 0 it may well be that the sum is appreciably less than the admittedly rough estimate of 3 kcal. ob tained here. In addition, the theoretical charge distributions used for the Coulomb energy may be i n error. Raising the Ν atom charge i n N 0 + from 0.58 to 1.0 results i n a 5 kcal. change. Together these two factors can account easily for the difference between "experimental" and theo retical lattice energies. In view of the possibly significant internal energy difference associated with free and lattice bound ions it appears that the ionic model fits Ν Ο extremely well. 3

2

5

2

2

δ


3.

CURTIS A N D WILSON

Nitrogen Ventoxide

Acknowledgment Research i n this paper was supported by the Advanced Research Projects Agency under Contract No. D A-31-124-ARO ( D ) -54, monitored by the U.S. A r m y Research Office, Durham, N . C .


Literature Cited (1) Alpher, R. Α., White, D. R., Phys. Fluids 2, 153 (1959). (2) Baughan, E . C., Trans. Faraday Soc. 55, 736 (1959). (3) Böttcher, C. J. F., Rec. Trav. Chim. 65, 19 (1946). (4) Bridgman, P. W., Proc. Am. Acad. Arts Sci. 76, 1,9 (1945). (5) Bridgman, P. W., Proc. Am. Acad. Arts Sci. 51, 579 (1916). (6) Dalgarno, Α., Advan. Phys. 11, 281 (1962). (7) Grison, E . , Eriks, K., de Vries, V . L . , Acta Cryst. 3, 290 (1950). (8) "International Critical Tables," Vol. 3, p. 44, McGraw-Hill, New York, 1928 (9) Janz, G. J., James, D . W., J. Chem. Phys. 35, 739 (1961 ). (10) Johnson, Q., Lawrence Radiation Laboratory, Livermore, Calif. The I B M 7090 Madelung constant program kindly provided by Dr. Johnson was adapted for 7040 operation by J . H . Schachtschneider. (11) Kapustinskii, A . F., Quart. Rev. (London) 10, 283 (1956). (12) Ketelaar, J. Α. Α., "Chemical Constitution," Elsevier, Amsterdam, 1958. (13) Ladd, M. F. C., Lee, W . H . , Trans. Faraday Soc. 54, 34 (1958). (14) Ladd, M. F. C., Lee, W. H., J. Inorg. Nucl. Chem. 13, 218 (1960). (15) Landolt-Börnstein, "Physikalische-Chemische Tabellen," 6th Aufl., 1,3 Teil, Springer-Verlag, Berlin, 1950, 1951. (16) Lewis, G. N . , Randall, M., Pitzer, K. S., Brewer, L., "Thermodynamics," p. 678, McGraw-Hill, New York, 1961. (17) Matossi F., J. Chem. Phys. 19, 1007 (1951). (18) "Mellor's Comprehensive Treatise on Inorganic and Theoretical Chem istry," Vol. II, Suppl. III, Part 2, p. 2407, John Wiley and Sons, New York, 1963. (19) Morris, D . F. C., J. Inorg. Nucl. Chem. 6, 295 (1958). (20) Mortimer, F . S., Shell Development Co., Emeryville, Calif., private com munications. (21) Parkinson, D., Proc. Phys. Soc. (London), 75, 169 (1960). (22) Pauling, L . , Proc. Roy. Soc. A114, 181 (1927). (23) Pitzer, K. S. Advan. Chem. Phys. 2, 59 (1959). (24) Rossini, F. D., Nat. Bur. Std. (U.S.) Circ. 500 (1950). (25) Tessman, J. R., Kahn, A . H . , Shockley, W., Phys. Rev. 92, 890 (1953). (26) Topping, J, Chapman, S., Proc. Roy. Soc. A113, 658 (1927). (27) Waddington, T. C., Advan. Inorg. Chem. Radiochem. 1, 158 (1959). (28) Wagman, D. D., Nat. Bur. Std. (U.S.), Rept. No. 7437 (1962). (29) Waldbauer, L . , McCann, D . C., J. Chem. Phys. 2, 615 (1934). (30) Wyckoff, R. W . G., "Crystal Structures," Vol. 2, John Wiley and Sons, New York, 1964. ;

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April 27, 1965.


The Lattice Energy of Nitrogen Pentoxide

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