W. SCHOTTE
2422
Prediction of Heats of Formation
by W. Schotte Engineering Technology Laboratory, E. I . du Pont de Nemours and Co., Inc., Wilmington, Delaware 19898 (Received Noaember $7, 1967)
Semiempirical quantum-mechanicalequations have been developed to calculate energies of different types of bonds. These equations have been used in Mulliken’s method for the prediction of heats of formation and the energies required to dissociate molecules into gaseous atoms. The scheme has been applied to 24 binary compounds of C, H, C1, and N. Heats of formation of some of these were used to establish parameters appearing in the theory. Good agreement has been obtained between predicted values and the known heats of formation of the other compounds.
I. Introduction Yij = 0.25AIijSij’ (3) Heats of formation of many compounds, particularly where A is an empirical constant, I i j is the average unstable molecules and free radicals, are not known. ionization potential, and Sij is the overlap integral. Group-contribution methods are frequently helpful to To simplify calculations, Pedley used an empirical predict heats of formation of many types of molecules, equation to estimate repulsions between atoms. For but they are usually not applicable for calculations H atoms involving free radicals. Mullikenl proposed a simplified quantum-mechanical procedure to predict the (4) energy of atomization, ie., the energy required to dissociate the molecule into gaseous atoms. Mulliken’s where B is an empirical constant and rHH is the distance method has many good features and explains qualitabetween H atoms. Equation 1 gives good resuhs for tively the various bond energies and repulsions which alkanes and their radicals, such as CH4, CHa, CH2, CH, contribute to the energy of atomization. P e d l e ~ ~ . ~ C2He, CzH6,C3Hs, C3H7, etc. However, one runs into modified Mulliken’s method somewhat to simplify the difficulties when the method is applied to alkenes, calculations and to obtain satisfactory agreement alkynes, aromatics, and nonhydrocarbons, which may between predicted and known energies of atomization. also have double bonds, triple bonds, three-electron Pedley used the following equation for simple hydrobonds, coordinate bonds, and resonance. carbons 11. Bond Energies for Different Types of Bonds 1. Covalent, Two-Electron Bond. This is the usual CCiHz - C C i C , - CY - V (1) type of bond in simple organic molecules. The equation developed by Mulliken’ and also used by Pedley2va X H and X c are, respectively, the energies of carbonis quite satisfactory hydrogen and carbon-carbon bonds; CHlHl is the sum of repulsions between hydrogen atoms on the same carAIizSiz XI2 = bon atom; CHlH2 is the sum of repulsions between (5) 1 s12 hydrogen atoms on adjacent carbon atoms; CC1HZis 8. One-Electron Bond. Mulliken’ proposed that a the sum of repulsions between carbon atoms and nonone-electron bond has one-half of the bond energy of a bonded hydrogen atoms; Cc1c3is the sum of repulsions two-electron bond between nonbonded carbon atoms; and Y is the repulsion between a free electron of an atom and an electron of an adjacent atom. Finally, the total energy, AH,, is corrected by the sum of the valence-stage energies of the atoms, V . This has also given satisfactory results. Pedley adopted Mulliken’s equations for the bond 3. Coordinate Bond. A coordinate bond is formed energy when both bonding electrons come from one atom. For example, there may be a coordinate bond between AIijSij x i j -~ a lone pair of electrons in an orbital of chlorine and a - 1 + sij
+
and made a small change in the empirical constant in Mulliken’s equation for the repulsion between two electrons in orbitals i and j The Journal of Physical Chemistry
(1) R. S . Mulliken, J . Phya. Chem., 56, 295 (1962). (2) J. B. Pedley, Trans. Faraday SOC., 57, 1492 (1961). (3) J. B.Pedley, ibid., 58, 23 (1962).
PREDICTION OF HEATSOF FORMATION
2423
vacant orbital of carbon. A simplified derivation of the bond-energy equation can be made by considering a C atom, a C12+ ion, and two electrons. The wave function for an electron will be approximated as a linear combination of Slater atomic orbitals
+ cz*z
\k = c191
Ei
Hi1
+
d7
$*1~291
'/2(1
+
+
- &)Pi1 '/z(1
+
&)Pi2
(13)
where El = $\klhl*l d r is the kinetic plus the potential energy of electron 1 with respect to the C atom. Similarly H22
=
E2
The electron energy can then be obtained by solving the secular determinant
+
+
dr
$\kz~i*2
'/2(1
- &)Pi2+ ' / 2 ( 1
+
Q)Pz2 (14)
Here, u1 is the potential of electron 2 in the field of a neutral C atom. This is nearly zero: u1 = 0. One can write
It is helpful to define the following symbols =
'/2(H11
+
H22)
(8)
-6 =
'/2(H22
- Hi])
(9)
E
fl = Hi2 - S12E
(10)
$Q?~UZ!P~d r =
$\k1~2\k1
dr
e2 - $\kl\kz-\kl\EZ
dr
r12
where 212 is the potential energy of electron 2 with respect to a C1+ ion. As a rough approximation
which can be used to modify the determinant to
E-E--6 0 + &,(E - E) E - E
where r is the distance between C and C12+. I n that case
+ -6
This can be solved for the electron energy
E = E +
* fld1+ (1 -
-P&2
1-
&22)62/P2
(11) Equations 13 and 14 become
s122
The Hamiltonian for electron 1 can be written as
H
=
hi
+ u2 + uiz
HI1
where hl is the Hamiltonian with respect to the C atom, u2 is the potential energy with respect to C12+, and u12is the potential energy with respect to electron 2. Then
Hi1 = $@ihi\ki dr
+ .f\kiuzql dr + .f\kiu12\ki dr
(12)
=
e2 r
E1 - - - Pl2 +
- &>Pi1+ '/2(1
'/2(1 H22
= Ez
- &)Pi2+ '/z(1
+ '/2(1
Hi1 =
E1
H2z = E2
**
= '/2(1
-
&>*I%
+
'/2(1
+
Q)*z*2
where Q = C22 - C12,which is an indication of the polarity of the bond. Defining also the symbols
&)Pi2
&)I322
The coordinate bond is frequently weak. Both electrons will be largely a part of the C1 atom and Q w 1. Then
where
i\ilulliken4 has approximated \k\k by the expression
+
+
- e2 r
+
P2z
where Ez is the kinetic plus the potential energy of electron 2 with respect to C12+. P 2 2 is the potential energy of electron 2 with respect to electron 1. The sum of these is just the opposite of the ionization potential of C1
E2
+
Pz2
= -IC1
El is the kinetic plus the potential energy of electron 1 with respect to C. This is the electron affinity of C
and e2
Pi2
= J\kl\kz--\ki\k2
E1 = A , dr
Then
r12
one obtains .fq1~12*1
d r = '/z(1
- &>Pi1+ '/2(1
+
which can be substituted into eq 12 to give
Hi1 = A, &)Pi2
- e2 r
Hz2 = -Ici
(4) R.S. Mulliken, J. Chim. Phya., 46, 497 (1949). Volume 76,Number 7 July 1968
2424
W. SCHOTTE
and
Hi1 26 =
- A , + re2 -
-IC1
2E = -IC1
Hzz =
e2
+ A , - r-
+ V(Cl+) - V(C1)
= IC10
+ J*iuz*i
dr
+
2J*iu12*1
dr
Ez
+ J*zui*z
dr
+
2J@zu12*2
dr
Similarly
The ionization potential is the ground-state ionization potential, Icl0, corrected for the difference in valencestate energies between C1+ and C1
IC1
E1
As an example, the system consisting of C+, C12+, and three electrons will be considered. The potential energy of electron 1 in the field of a neutral C1 atom is VZ
+ V(C-)
This is small and will be neglected: vz = 0. Then
- V(C)
The potential energy of electron 2 in the field of a neutral C atom is
+ re2- -
V(Cl+)
+ V(C1) - V(C-) + V(C)
+
- e2 - -
V(C1+)
+ V(C1) + V(C-)
2E = -1~1" A,"
(15)
or J\kzui\Ez d7 = -Pi2
r
- V(C)
As before (16)
p
-l/zAIS (17) The same relationship will be assumed for the coordinate bond. I n that case, eq 11 for the lowerenergy state becomes =
+
f*zuiz~~~ dr
Xz = El0
+ Ez" - 2E
where El0 (=Ezo)is the energy of electron when it is not bonded, i.e., the energy of the electron as part of
c1+
El0 = Ezo = -1~1"- V(Cl+)
+ V(C1)
The equation for the bond energy becomes finally
xz = 26 +
=
-
'/z(1
H = hi
+ uz +
and The Journal of Physical ChemGtry
2 ~ 1 2
&)Pi2
- 2P12 + (1 - &)Pi1+ (1 + &)Pi2 = Ez - Plz + (1 - &)Pi2+ (1 + Q)Pzz
=
H 1 1
Hzz
E1
This gives
- E1 + (1 - 2Q)Piz+ (1
Let 17 = '/Z(P11
+QPzz
- (1 - &)Pi1
+ Pzz) - PlZ
Then
+3 / ~ P ~ ~
26 = Ez - E1 - (1 - 2Q)17 - '/zP11
Usually 17 will be small. Q = 0 for a covalent bond involving one electron from each atom, while Q = 1 when there is no bond. I n the present case, Q will probably be close to 0.5. Therefore, (1 - 2Q)r will be close to zero and 26 = Ez - E1
where 26 can be calculated from eq 15. 4 . Three-Electron Bond. Three-electron bonds are usually weak. They may involve two electrons from a lone-pair electron orbital of one atom and a single electron from an orbital of a second atom. The Hamiltonian for electron 1 is
&)Pi2
Then
AIc1S1z2 - VA21C12S122 (1 - S122)(26)2 26 = Ez 2 0 - s129
The bond energy is
+ '/z(l + + '/z(1 + Q)Pzz
d r = '/z(1 - &)Pi1
J*iuiz*i
Mulliken' found for the covalent bond that p could be approximated by
E = E +
d r = -2P12
J*luz\kl
The final expressions for 6 and E are then 26 = -IC* - A,"
dr
r1z
Similarly
A , = A,O
+ 2J@z-*ze2
= UZ
- '/zPii
+ '/zPzz
The various terms on the right-hand side of the equation can be written in terms of ionization potentials and electron affinities: Ez = -Icl+, the second ionization potential of C1; Ez Pz2 = -1~1; El = -Ic; and E1 P11 = A,. Using ground-state values
+
+
+
26 = '/z[Icl+"- 31~1"
IC'
- A,O + V(C12+) - 4V(C1+) + 3V(C1)
+ V(C+) - V(C-)]
(19)
PREDICTION OF HEATSOF FORMATION
2425
Equation 11 for the electron energy still holds. This gives for the total energy of the three electrons
Et
=
3E
+
-3Ps1z
+ P d 1 + (1 1-
(20)
and the other two give after substitution of the expression for P
E = E +
+
*
3A112812~ d3A21122&22 (1 - 38122)(26)2 (23) 2(1 - 3&2')
SI22
where p = -1/2A112812. In the derivation of the equation for the coordinate bond, E was determined. However, a simplification will be used with eq 20. There is no bond when 812 = 0
There are seven electrons. Six of these can go in the three lower energy orbitals and one is left for the highest energy orbital. The total energy is, therefore
The bond energy is
Xs = El0
+ Ezo + E3'
- Et
Theref ore x3=
-6+
EO
=
7E
+ 36
The bond energy is then The absolute value of 6, calculated from eq 19, should be used. 5. Resonance. Resonance occurs when there are different molecular structures which have nearly the same energy. An example is cc13 where the threeelectron bond between C and C1 may be considered to resonate between the three C1 atoms. Only the three-electron bonds will be considered here. Each C1 atom contributes two electrons and the C atom contributes one electron. There are seven electrons distributed over four orbitals. Each orbital can be represented by an approximate wave function \k =
+ czqz +
ClPl
C3*'
+
c 4 * 4
where \kl, P 2 , \k3, and P4 are wave functions for the atomic orbitals. The secular determinant becomes, after appropriate simplifications
E-6-E
P
P
+ Xiz(E - E )
P
E
+ &z(E - E )
0 0 E+6-E 0 E 0 + 6 - E which can be worked out to give
- E)I
XR2
=
26
+
-2AIc1S122 I = O
+ 6 - E)'[@+ Siz(E - E ) ] ' = 0
This equation has four solutions corresponding to the energies of the four orbitals. Two of these are the same
E=E+6
-6
Resonance can also occur with coordinate bonds, such as in the CClz singlet. In that case
(E - 6 - E ) ( E + 6 - E)' 3(E
+ -6A112S122 + d2A21122S122+ (1 - 2S122)(26)2 (25)
XR3 =
2(1 - 2S122)
+6 -E
+ Xiz(E - E ) 0 + - E) 0 P + Siz(E - E ) P +
where 6 is the absolute value of the result from eq 19. Comparison of eq 21 and 24 shows that the three-electron bond energy with resonance, XR3, is similar to the three-electron bond energy without resonance, X 3 , except that SlZ2 is replaced by 3 S 2 . Another case of interest is that where resonance occurs between two structures, such as in the CClz triplet. In a similar fashion, as shown above, one can derive
(22)
+ d2A21C12X122 + (1 - 2&22)(26)2 1
- 28122
(26)
where 6 is given by eq 15. 6. Double and Triple Bonds. The presence of a r bond in addition to a u bond, such as in a double bond, or two T bonds in addition to a u bond, such as in a triple bond, leads to a complication. Mullikenl calculated separate bond energies for the u bond and the T bond from eq 5 and added these to obtain the total bond energy. However, he found it necessary to use an empirical constant, A,, for the T bond different from Volume 78,Number 7 July 1968
W. SCHOTTE
2426
A , for the u bond to get meaningful results. This is not satisfactory. The double bond between atom A, having orbitals 1 and 2, and atom B, having orbitals 3 and 4, will be considered. There is a u bond, owing to overlap of orbitals 1 and 3, and a n bond, owing to the overlap of orbitals 2 and 4. An approximate wave function for the system of four orbitals and four electrons will be 9
=
CI9I
+
c292
+ C393 +
c494
The total energy of the u and n bonds can be derived in the usual manner, giving
xu,= 1 +
AI13813
~
+ 1+
AI24824
~
813
s24
This would make it appear that it is possible to calculate separate u- and n-bond energies from Mulliken’s equation (eq 5 ) and to add these numerically. It should be noted, however, that the bond energies are considered relative to the energies of the atoms in their valence states. Therefore, the valence-state energies of the atoms are substracted in eq 1. The valencestate energy includes interactions between the electrons which belong to a particular atom. However, the interaction, H,, between the electrons in orbitals 1 and 2 of atom A has been included in the derivation of the equation for the bond energy. The same holds for the interaction, Has, of the electrons in orbitals 3 and 4 of atom B. These interactions should not be included again in the valence-state energies of atoms A and B. However, it is more convenient to use conventional valence-state energies and to apply the correction to the bond-energy equation. The interactions are exchange integrals, l/zKij. Applying the exchange-integral corrections
Xu,= Xu
+ X , - ‘/zKiz - ‘/2K34
(27)
where
discussed earlier. Although the following equation is written for a compound of the type CZCl,, the same type of expression can be used for other compounds as well
+
=
+
X c = X, E X , - ‘ / z C K (30) The coordinate bond is represented by Xz,which, if present, can be calculated from eq 18. However, X R ~ , as calculated from eq 26, should be used if resonance can occur. X S represents possible three-electron bonding. This is calculated from eq 21. Resonance can be included by using X R from ~ eq 24 or 25 in place of X 3 . CllCZl represents a repulsion between C1 atoms on the same C atom. CllC12is a repulsion between C1 atoms on adjacent C atoms, and CIClzis a repulsion between a C atom and a C1 atom on the adjacent C atom. These repulsions can be calculated from expressions of the type of eq 4. Y can be calculated with eq 3. Strictly speaking, CY should include all electron interactions which have not been accounted for in the bonds or in the valence-state energy. However, it is simpler to include only the electron repulsions which cannot be conveniently accounted for otherwise. For example, chlorine-chlorine repulsions have already been accounted for in the CllCll and CllC12terms. Some of the carbon-chlorine interactions can be included in the empirical bond-energy constant, A. However, Y should include repulsions between free C electrons and electrons of the C1 atoms which are bonded to the C atom. Again, C-C1 interactions for nonbonded atoms can be accounted for in the ClC12 terms. Finally, Y should also include electron repulsions of adjacent C atoms. The valence-state energy, V , for C2C1, is = 2V(C) nV(C1) (31) where the valence-state energies of the atoms, V(C) and V(Cl), can be calculated by the method of Van Vleck.6 Many of the atomic constants, needed for the calculation of V , have been listed by Pilcher and Skinner6 or Hinze and Jaffe.’ The constants used in this study were calculated directly from atomic energy levels tabulated by Moore8 using the procedure given by Slater.9 Exchange integrals, K , are used both in the
v
and
For a triple bond, a similar equation is obtained Xu,
=
X,
+ 2 X , - ‘/z(Klz + Kl3
+ + + K46 + K66) K23
K 4 6
(28)
where atom A has bonding electrons in orbitals 1-3 and atom B has bonding electrons in orbitals 4-6. 111. Calculation of Energies of Atomization The general expression for the energy of atomization of binary compounds can be written by expanding eq 1 to include the different types of bonds which have been The Journal of Physical Chemistry
+
EXCl+ x c Ex2 Ex3 - CCllCll CCliClz - CCiCl2 - CY - V (29) where X C Iis a covalent carbon-chlorine bond which can be calculated from eq 5. The carbon-carbon bond, X C , may include both u and n bonds AH,
+
(6) J. H. Van Vleck, J . Chem. Phys., 2, 20 (1934). (6) G. Pilcher and H. A. Skinner, J . Inorg. Nucl. Chem., 24, 937 (1962). (7) J. Hinze and H. H. Jaffe, J . Chem. Phys., 38, 1834 (1963). (8) C. E. Moore, “Atomic Energy Levels,” National Bureau of Standards Circular 467, U. 9. Government, Printing Office, Washington, D. C., 1949. (9) J. C. Slater, “Quantum Theory of Atomic Structure,” Vol. 1, McGraw-Hill Book Co., Inc., New York, N. Y., 1960.
PREDICTION OF HEATSOF FORMATION
2427
valence-state energy calculations and in some of carboncarbon bond calculations (see eq 30). For electrons in orbitals i and j Kij
=
+
c ~ i ~ a j ~ ( FFop’ o ~ ~2FO” 4F2” - 4Gi8’)
+ + aj2)(G1ap- 3Fz’’) + 3FzPp
IV. Results
+
(ai2
(32)
where FOB’, FtP, FOPP, FzPp,and GIBP are the coulomb and exchange integrals for electrons in pure s and p orbitals. The degree of hybridization, ai2, is that from the wave function for the hybridized orbital i \ki
2:
+ d ( 1 - aiZ)*(p)
ai*(s)
(33)
Another convenient method of expressing eq 33 is *i
=
*(SI
d
+ Xi*(p)
(34)
m
Therefore
The detailed calculations of the bond energies, X , and repulsion energies, Y , require a knowledge of overlap integrals, 812, and ionization potentials, 1l2. A general expression for the overlap integral is SI2
= (S(2s, 3s)
+ X1S(2pa, 3s) cos y1 + XZS(2S, 3pa) cos yz
+
X1XzlS(2pa, 3pa) cos y1 cos yz S(2pn, 3pn) sin
~1 sin
1/(1
+
y21j/
+ X12)(1 + X2)
(35)
where y is the angle which a p orbital makes with respect to the bond axis. Equation 35 is expressed in terms of overlap integrals between pure s and pure p orbitals, such as S(2s, 3s). Approximate Slater orbitals have been used in this study. MullikenlO has tabulated equations and numerical data for the appropriate integrals which are needed. As proposed by Mulliken’ and Pedley,2 the ionization potential is the mean value for the two electrons in each of their atomic orbitals 112
=
l/z(ll
+ 12)
are usually fairly well known or can easily be estimated. A small error can be made without significant effect on the prediction of the energy of atomization.
(36)
The determination of the bond-energy constants, A , and the repulsion constants, B remains. They should be calculated from the known energies of atomization (or heats of formation) of some of the compounds. For compounds of the type CZCI,, a bond-energy constant Ac for C-C bonds, Acl for C-CI bonds, a repulsion constant BC for C-CI repulsions, and Bcl for C1-C1 repulsions need to be estimated. This will be explained in more detail below. An approximate estimate of the bond lengths is necessary to calculate the overlap integrals and the repulsions between atoms. Bond lengths
1. System: CH4-CHa-CH2-CH. Calculations for methane and its radicals were made by Pedley3 but were repeated in this study to obtain a consistent set of constants, A H and B H ,for further calculations. Heats of formation listed in the JANAF Tables” were used to obtain energies of atomization and bond distances for CH4d AH,(29S°K) = 17.223 eV/g-mol and TC-H = 1.094 A, 2nd for CH3, AH, = 12.803 eV and TC-H = 1.079 A. These energies of atomization could be used to calculate A H = 1.131 and BH = 2.833. Calculations were then made to predict energies of atomization as a function of the hybridization coefficient, A, for the CHz singlet, the CHz triplet, and CH. Pedley’s finding that the CH2 triplet is the stable state was confirmed and AH, was calculated to have a maximum of 7.862 eV for X = 1 (bond angle of 180O) at a given bond distance of 1.03 A.lz Calculations for CH gave AH, = 3.510 eV for X = 1.84 at a bond distance of 1.113 2. System: CZH~-CZHK-CZH~-CZH~-CZHZ-CZH-C~, This is a more complex system which gave Mullikenl some difficulties and which was not completely covered by Pedley2J because of the n-bond energy. There is a second carbon atom and four empirical constants are needed. The value of AH = 1.131 can be used from the previous calculations. It is not wise to use the previously determined value of B H , since the H-H repulsions will be affected by the electrons of the added C atom. For example, the C electrons may cause some shielding which would reduce B H . A new value of BH should be determined. I n addition, a bond-energy constant, Ac, for the C-C bond and a repulsion constant, Bc, for the repulsions between nonbonded C and H atoms are needed. Values of Ac, B H ,and Bc can be calculated from the known energies of atomization of CZHO,C2H4, and C2H2. The following bond distances were used in the calculations: c256: C-C, 1.54 8; C-H, l.lOO8;l4 C2HS: C-C, 1.54 A; Cl-HI, 1.100 8; C2-H2, 1.095 A (estimated); C2H4: C-C, 1.336 8; C-H, 1.085 A;14 CzH3: C-C, 1.322 A; Cl-Hl, 1.085 8; CZ-Hz, 1.08 8 (estimated); CZHZ:C-C, 1.205 8; C-H, 1.059 A;l4 C2H: C-C, 1.220 A; C-H, 1.062 A;15 Cz: C-C, 1.242 8 . 1 6 (10) R. S. Mulliken, et al., J . Chem. Phys., 17, 1248 (1949). (11) “JANAF Thermochemical Tables,” The Thermal Research Laboratory, Dow Chemical Co., Midland, Mich., 1960-1966. Roy. Soc., A262, 291 (1961). (12) G. Herzberg, PTOC. (13) G. Heraberg, “Molecular Spectra and Molecular Structure,
I. Spectra of Diatomic Molecules,” 2nd ed, D. Van Nostrand Co., Inc., New York, N. Y . , 1950. (14) L. E. Button, “Tables of Interatomic Distances and Configuration in Molecules and Ions,” The Chemical Society, London, 1958. Also, Supplement 1956-1959, publ. 1965. Volume 78, Number 7 July 1068
W. SCHOTTE
2428 Table I: Terms for the Energy of Atomization (298'K) of Hydrocarbons
CzHa CaH6 CzHa CzH8 C2Ha CzH c 2
A1
hl
XHi
XES
Xu
1.94 1.94 1.62 1.62 1.07 1.23 1.27
1.94 1.57 1.62 1.06 1.07 1.06 1.27
6.228 6.228 6.582 6.582 7.376 7.382
6.228 6.610 6.582 7.328 7.376
9.838 10.070 11.416 11.823 12.625, 12.492 12.191
...
...
...
BXa
'/lZR
ZHH
ZCH
ZY
v
A Ha
... ...
...
...
3,584 3.666 8.497 6.334 4.156
1.310 1.378 3.462 2.336 1.199
1.779 1.107 0.573 0.256 0.018
1.938 1.503 1.491 0,999 0.594 0.288
0.505 0.460 0.552 0.475 0.115 0.527 0.985
13.738 13.894 14.083 14.388 14.695 11.395 7.808
29,249 25.011 23.320 18.486 16.989 11.661 6.354
The heat of formation of CZH6 was obtained from API Research Project 4417and was converted to AH,(298"K) = 29.252 eV. Heats of formation of CzH4and CzHz were obtained from the JANAF Tables" and were converted to values of H, = 23.315 eV and AH, = 16.988 eV, respectively. Trial values of Ac, Bc, and BH were tried to maximize AH, as a function of the hybridization coefficients of the C orbitals until calculated and actual AH, values finally matched. This gave AC = 1.535, BH = 2.222, and BC = 7.809. Predicted energies of atomization were then obtained for CZH6, C2H3, CZH, and Cz. The energy terms which appear in eq 29 for AHa, using H in place of C1, are shown in Table I for the optimum values of the hybridization coefficients, A1 and Xz, for the two C atoms. 3. System: CC1~-CCl3-CCl2-CC1. This is a more complex system which has not been covered in the literature, since the C1 atom has 17 electrons, while H has only 1. However, only the 7 electrons in the 3s and 3p orbitals of the valence shell are of interest here. Some hybridization of the bonding p electron and the lone-pair s electrons should be considered, since this may lead to stronger bonding. Again, the bond-energy constant, ACI, and the repulsion constant, BCI,have to be determined from the known energies of atomization of CC14and CCl,. Values of AH, = 13.57 eV and TC-CI = 1.760 A were obtained for CC14 from the JANAF Tables." Results of Farmer,18 Reed,l9 GoldfingerjZ0 and Bensonz2were averaged to obtain the heat of formation of 18 kcal/g-mol (AH, = 10.41 eV) for CC13. A bond distance of 1.74 A was estimated. Values of Acl = 1.237 and Bcl = 34.43 were calculated from the energy equations for CC14 and CC13. Predictions of AH, were then made for the CClz triplet, the CClz singlet, and CC1. Estimated bond distances of 1.71 8 for the CC12triplet and 1.67 A for the CClz singlet were used. Verma and MullikenZ3 have determined a bond distance of 1.645 8 for CC1. Values of the energy terms which make up AHa are shown in Table I1 for the optimum values of the hybridization coefficient, A, of the C atom and the degree of hybridization, 2,of the C1 atoms. For weak hybridization of C1, it is more convenient to use a2 because X goes to infinity when there is no hybridization of the 3p orbital of C1. The Journal of Physical Chemistry
... ...
...
It is of interest to note in Table I1 that the threeelectron bond in CC13 gives negative energy which indicates that there is a slight repulsion. This is not surprising, since three-electron bonds form only when the energy difference, 6, between the electrons is small. The three-electron bond is either weak or gives a repulsion when the energy difference is appreciable. Table I1 shows also that the CClz singlet is more stable than the CCl2 triplet and is, therefore, the ground state, This differs from CHZ where the triplet is the ground state. The difference is largely due to the fact that that a coordinate bond can be formed in CClz but not in CHz.
4.
System:
~
~
~
~
6
~
~
2
~
~
The calculations for this system include again a second carbon atom and require four empirical constants, The bond-energy constant for the C-C1 bonds, Acl = 1.237, and for the C-C bond, AC = 1.535, have already been determined in the calculations for the CC14system and the CZH6 system. The repulsion constants for C1-C1 repulsions, Bcl, and for C-C1 repulsions, Bc, remain to be estimated by substituting known energies of atomization into the equations for C2C16and CzC15. The following bond distances have been used in the calculations: C2C16: C-C, 1.56 A; C-Cl, 1.77 CzCls: C-C, 1.56 A; Ci-ClI, 1.77 A; Cz-ClZ, 1.75 A (estimated); C2C14: C-C, 1.33 A; C-C1, 1.72 A;14 CzCla: c-c, 1.33 A; cl-C11, 1.71 A; Cz-clz, 1.69 A (estimated); C2C12: C-C, 1.195 A; C-Cl, 1.64A;25 CzC1: C-C, 1.22 8; C-C1, 1.64 A (estimated). Puyo28
A;24
(15) M. N. Plooster and T. B. Reed, J . Chem. Phys., 31, 66 (1959). (16) E. A. Ballik and D. A. Ramsay, Astrophys. J., 137, 84 (1963). (17) "Selected Values of Properties of Hydrocarbons and Related Compounds," American Petroleum Institute Research Project 44, Thermodynamics Research Center, Texas A & M University, College Station, Texas, 1962. (18) J. B. Farmer, et al., J. Chem. Phys., 24, 348 (1956). (19) R. I. Reed and W. Snedden, Trans. Faraday Soc., 54, 301 (1958). (20) P. Goldfinger and G. Martens, ibid., 57, 2220 (1961). (21) R. E. Fox and R. K. Curran, J . Chem. Phys., 34, 1595 (1961). (22) 9. W. Benson, ibid., 43, 2044 (1966). (23) R. D. Verma and R. S. Mulliken, J. Mol. Spectrosc., 6 , 419 (1961). (24) A. Almenningen, et al., Acta Chem. Scand., 18, 603 (1964). (25) 0.Hassel and H. Viervoll, ibid., 1, 149 (1947). (26) J. Puyo, et al., Compt. Rend., 256, 3471 (1963).
~
~
~
2429
PREDICTION OF HEATSOF FORMATION
5
"*
.. .. . .
,
. .
,
:?? 1 : : : :
. . :
.. .. .. .. . Z % $ $ - , . . . . : 0? 0? 0? 0? 0? :
and GoldfingerZ0 have given heats of formation of C2Cls for which AH, = 23.849 eV. The heat of formation of CzC16 was obtained from Goldfinger and MartendZ0data and was converted to AH, = 20.726 eV. Using these values, trial-and-error calculations were made to maximize AH, with respect to hybridization of C and C1 orbitals for assumed values of Bcl and Bc until agreement was reached. The results were BCI = 12.295 and Bo = 39.742. Energies of atomization were then predicted for CzC14, CzCla, C2C12, and C2C1. The energy terms of eq 29 are listed for optimum values of X(C) and ap(Cl)in Table 11. 6. System: NH3-NHz-NH. The nitrogen atom has five electrons in the valence shell. Hybridization can take place between the two electrons of the 2s orbital and the three electrons of the 2p orbitals. NHa will have some hybridization of all three 2p orbitals with the 2s orbital, However, NH2 will show hybridization of only two of the 2p orbitals with the 2s orbital, since this requires less valence-&ate energy than for NH3. For the same reason, N H will have hybridization between one 2p orbital and the 2s orbital. The JANAF Tables'l were used to obtain AHa = 12.002 eV for NH3 a t 0°K and a bond distance of 1.012 8. Kerr and coworkers27have determined the heat of formation of NH2 at 0°K for which AH, = 7.653 eV. The bond distance of 1.025 A was taken from the JANAF Tables. Using this information, the following bond-energy and repulsion constants were determined : A = 1.335 and B = 6.576. The energy of atomization of N H was then predicted for a bond distance of 1.038 8 listed in the JANAF Tables. The energy terms are tabulated in Table I11 for the optimum values of X. Table 111: Terms for the Energy of Atomization ( O O K ) of NHs, "2, and N H
NHa NHZ
NH
a
XN
ZXX
ZY
V
AHa
1.546 1.604 1.883
8.630 8.610 8.176
2.336 0.773
0.245 0.721 0.918
11.307 8.073 3.664
12.002 7.653 3.593
...
V. Discussion The utility of these simplified quantum-mechanical calculations can best be determined by comparing the predicted results with measured values or with other predictions reported in the literature. It is easier to compare heats of formation, since these are usually reported. Table IV shows the predicted heats of formation, corresponding to the energies of atomization, and the actual values. The heat of formation of CH2 seems uncertain. Some reviewers, such as Kerr,28 give a value of about 90 (27) J. A. Kerr, et al.,J . Chem. Soc., 3584 (1964).
Volume 78, Number 7 July 1068
2430
W. SCHOTTE
Table IV : Heats of Formation (kcal/g-mol) Temp, OK
298 298 298 298 298 298 298 298 298 298 298 298 298 298 298 298 298 298 298 298 298
0 0 0
Calod
Actual
... .,.
-17.9 31.9 95 142
93.8 142.0
...
... 43.8 97.4
... 25.5 71.8
... 124.9 195.2
-2.1 53.3 36.7 117.0
...
...
81.3
- 26 18 43 96 -20.2 25.7 12.5 72 54.2 123 197 -34.7 8.4 -3.5
...
-9.4 39.3 80
kcal/g-mol. However, a detailed review has been reported in the JANAF Tables” where it was concluded that it should be 95 kcal/g-mol. This value seems better substantiated. The heat of formation of CH is quite well established6qz8a t about 142 kcal/g-mol. Blanchard and Le Goff29 determined bond energies from which a heat of formation of 46 kcal/g-mol can be calculated for CC&. Reed and Snedden’s made similar measurements, which result in a heat of formation of 42 kcal/g-mol. Shilov and Sabirovaaomeasured an activation energy of 47 kcal for the dissociation of CHCI3 to CClz and HCI. This indicates that the heat of reaction will be about 44 kcal which gives AHf(CC12) = 42 kcal/g-mol. An average of the three above values has been selected: AHf(CClZ) = 43 kcal/g-mol. Kuzyakov and TetevskiF have determined a dissociation energy of 104 kcal for CCI, which gives a heat of formation of 96 kcal/g-mol. Ovcharenkoa2found from the CC1 spectrum a dissociation energy of either 76 or 101 kcal. The value of 101 kcal appears to be the better one. This gives AHf = 99 kcal/g-mol. Reed and Snedden’s bond dissociation energy measurement^^^ can be used to calculate a value of 93 kcal/g-mol. An average of the three results gives AHf(CC1) = 96 kcal/ g-mol. Kerr’s28 review shows that the heat of formation of C2H5 is about 25.7 kcal/g-mol. He selected a heat of formation of 64 kcal/g-mol for C2H3, but his review is incomplete. Studies of mercury-photoThe Journal of Physical Chemistry
sensitized reactions of ethylene, carried out by LeRoy and Steacieaaand by Kebarlea4indicate that the C2Ha-H bond-dissociation energy should be about 112 kcal, giving AHf(C2H3) = 72 kcal/g-mol. Steaciea6 has reviewed the reaction of H with C2H4to give C2Haand H. He has concluded that there is an activation energy of 10 kcal. This indicates that the heat of reaction will be about 8 kcal, giving AHf(CzH3) = 73 kcal/g-mol. Lampe and Fieldaa~37 have made a number of electron-impact measurements which have led to heats of formation of 70, 82, and 86 kcal/g-mol. Harrison and Lossing’s3*electron-impact results give AHf = 65 kcal/g-mol and Stevenson’sa9electron-impact measurements give 57 kcal/g-mol. The average heat of formation from the electron-impact studies is 72 kcal/g-mol. This compares well with the values of 72 and 73 kcal/gmol estimated from the kinetic studies and appears to be a better value than that listed by Kerr. A similar uncertainty exists for CzH. Kerr does not list a value, but Knox and Palmer4” have made a review. They concluded that AHf = 116 kcal/g-mol, but this was based on an unpublished prediction and not on experimental facts. Dibeler41 has made electron-impact measurements which give AHr = 122. Coats and Anderson42made electron-impact measurements on a variety of compounds which give heats of formation of 152, 153, 137, and 110 kcal/g-mol and an average of 138 kcal/g-mol. C h e r t ~ n concluded ~~ from photochemical studies that the bond-dissociation energy of C2H-H is 121 kcal, which gives AHf = 123 lrcal/g-mol. The above results suggest that the heat of formation of CzH is probably closer to 123 kcal/g-mol than to Knox and Palmer’s value of 116 kcal/g-mol. Fortunately, the heat of formation of Cz is better known. Brewer44 has calculated a value of 195 kcal/g-mol at 0°K using
(28) J. A. Kerr, Chem. Rev.,66, 465 (1966). (29) L. P. Blanchard and P. Le Goff, Can. J . Chem., 35, 89 (1957). (30) A. E.Shilov and R. D. Sabirova, Russ. J . Phys. Chem., 34, 408 (1960). (31) Y. Y. Kuzyakov and V. M. Tetevskii, I z v . Vyssh. Ucheb. Zaved., Khim. Khim. Tekhnol., 3, 293 (1960). (32) I. E.Ovcharenko, et al., Opt. Spektrosk., 19, 294 (1965). (33) D.J. LeRoy and E. W. R. Steacie, J . Chem. Phys., 10, 676 (1942). (34) P. Kebarle, J . Phys. Chem., 67, 716 (1963). (35) E.W.R. Steacie, “Atomic and Free Radical Reactions,” Vol. 1, Reinhold Publishing Corp., New York, N. Y.,1954,pp 439,440. (36) F. W.Lampe and F. H. Field, J . Amer. Chem. Soe., 81, 3238 (1959). (37) F.H.Field, J . Chem. Phys., 21, 1506 (1953). (38) A. G. Harrison and F. P. Lossing, J . Amer. Chem. SOC.,82, 519 (1960). (39) D.P. Stevenson, ibid., 65, 209 (1943). (40) B. E.Knox and H. B. Palmer, Chem. Rev., 61, 247 (1961). (41) V. H.Dibeler, et al., J . Amer. Chem. SOC.,83, 1813 (1961). (42) F. H.Coats and R. C. Anderson, ibid., 79, 1340 (1957). (43) R. Cherton, Bull. SOC.Chim. Belges, 52, 26 (1943). (44) L. Brewer, et al., J . Chem. Phys., 36, 182 (1962).
KINETICISOTOPE EFFECTS IN
THE
2431
HYDROGEN ELECTRODE REACTION
the most recent spectral data. This gives a heat of formation of 197 kcal/g-mol at 298°K. Puyo26 has determined a value of -3.5 kcal/g-mol for C2C14. Unfortunately, there are no known values of the heats of formation of CZCl3,CzCl2,and CzCl. The heat of formation of n” is somewhat uncertain. The JANAF Tables” list a predicted value of 79.2 kcal/g-mol. Franklin45 found AHr = 81 kcal/g-mol from appearance-potential measurements. Seal and Gaydon4‘jdetermined a value of 90.1 kcal/g-mol from spectroscopic measurements of NH in a shock tube. However, their results conflict with those of a similar study carried out by Harringt~n.~’Calculations based on Harrington’s data show rather good agreement with the thermodynamic properties listed in the
JANAF Tables. The best estimate a t present is a value of about 80 kcal/g-mol. There is good agreement between the predicted and actual heats of formation listed in Table IV. Actually, the differences a,ppear to be smaller than the uncertainties in the measured data ranging from 2 to 5 kcal/g-mol. I t is believed that the present method can give accurate results for a wide variety of molecules and free radicals. I n some cases, however, it may be necessary to derive additional bond-energy equations, since not all possible types have been developed here. (46) J. L. Franklin, et al., J . Amer. Chem. SOC.,80, 298 (1968). (46) K.E.Seal and A. G . Gaydon, Proc. Phys. SOC.,89, 469 (1966). (47) J. A. Harrington, et al., J . Quant. Spectrosc. Radiat. Transfer., 6,799 (1966).
Kinetic Isotope Effects in the Hydrogen Electrode Reaction by J. D. E. McIntyre Bell Telephone Laboratories, Inc., Murray Hill, New Jersey
and M. Salomon School of Chemistry, Rutgers University, New Brunswkk, New Jersey
(Received Decmber 86, 1967)
Partition function ratios for hydrogen and deuterium gases and the lyonium species L~0(1),LsO+, and OL(where L = H or D) have been reevaluated from recent spectroscopic and thermodynamic measurements. These values may be employed in predicting deuterium isotope effects in electrochemical reactions, in estimating the energies of transfer of anions between protium and deuterium oxide solutions, and in establishing an electrode-potential scale for heavy-water electrolytes. The results of measurements of the rate of hydrogen evolution on mercury electrodes in electrolytes of pure and mixed isotopic (H-D) composition are compared and are shown to be in accord with a mechanism involving a rate-determining proton-discharge step.
The effects of H-D isotopic substitution on the kinetics of the electrochemical evolution of hydrogen from aqueous electrolytes have recently been reexamIt was shown ined from a theoretical that by properly accounting for zero-point energies in the activated complex, the low value of the separation factor, SD, observed experimentally with mercury electrodes’o was consistent with that predicted theoretically for a rate-determining proton-discharge step and not with the slow discharge of the molecule ion, Hz+, as previously p r ~ p o s e d . ~ The ~ - ~long-standing ~ disagreement between the results of electrochemical kinetic studies and kinetic isotope effect investigations was thereby resolved. The isotopic exchange reaction Hz(g)
+ 2D30’ + 2HzO(1) DZ(g)
+ 2H30+ + 2D20(1)
(Ia)
may be represented as a hypot,hetical electrochemical cell without transference
(RQHzl H30 +(HzO) I lDaO +(DeO>IDdM)
(Ib)
(1) B. E.Conway, Proc. Roy. SOC.,A247,400 (1969). ( 2 ) B. E. Conway, “Transactions of the Symposium on Electrode Processes,” John Wiley and Sons, New York, N. Y.,1961,Chapter 15. (3) J. 0.M. Bockris and S. Srinivasan, J . Electrochen. Sac., 111, 844 (1964). (4) J. 0.M.Bockrisand S, Srinivasan, ibid., 111, 853 (1964). (6) J. 0.M. Bockris and 8. Srinivasan, ibid., 111, 858 (1964). (6) B. E. Conway and M. Salomon, Ber. Bunsenges. Phve. Chen., 68, 331 (1964). (7) M. Salomon and B. E. Conway, ibid., 69, 669 (1965). ( 8 ) B. E.Conway and M. Salomon, J. Chen. Phya., 41,3169 (1964). (9) M. Salomon and B. E. Conway, DiECUS8wn8 Faraday Soc., 39, 223 (1966). (10) M. Rome and C. F. Hiskey, J . Amer. Chen. SOC.,76, 6207 (1964). Volume Y2, Number 7 July 1068
