This paper presents the first rational calculation of the dissociation energy of diatomic molecules. Quantum mechanics does not have such a calculation, even in principle. The importance of this calculation is that it provides additional quantitative verification of the molecular force and energy concepts of the Reciprocal System.
Dissociation energy is the change in energy (usually expressed in kcal per mole) at absolute zero temperature in the ideal gas state for the reaction
| A-B —> A + B |
(1)
|
the products (atoms A and B) being in their ground states and the reactant (molecule A-B) in the zeroth vibrational level. Note that dissociation energy is slightly different from bond energy, which is defined as the standard enthalpy change at 25º C for the ideal gas reaction given above. Calculating dissociation energy rather than bond energy frees us from having to consider molecular thermal energy.
Now let us proceed to the derivation of the expression for bond dissociation energy from the principles of the Reciprocal System. A diatomic molecule, as a unit, exists in the time-space region. However, the two individual atoms of the molecule, relative to one another, exist in the time region because the interatomic distance is less than one space unit; hence, time region expressions apply to the attributes of the bond. To quote Larson,
The motion in time which can take place inside the space unit is equivalent to a motion in space because of the inverse relation between space and time. An increase in the time aspect of a motion in this inside region (the time region, where space remains constant at unity) from 1 to t is equivalent to a decrease in the space aspect from 1 to 1/t. Where the time is t, the speed in this region is equivalent space 1/t divided by time t, or 1/t²[Ref. 1].
Thus,
| v = 1/t² |
(2)
|
In the Reciprocal System, energy is the reciprocal of speed. Hence, in the time region,
| E = t² |
(3)
|
This energy equation gives the proper dimensional form of the expression for dissociation energy. It can be generalized to
| E = ta * tz |
(4)
|
In application to the problem at hand, ta and tz refer to the rotational time displacements of the atoms of the molecule, where ta is the primary magnetic displacement or the secondary magnetic displacement and tz is the second magnetic displacement or the electric displacement. To justify this intepretation, let us recall that the two atoms of the molecule are in translational equilibrium; in the Reciprocal system this means that the scalar translational repulsion effect of the rotational force of the atoms is equal and opposite to the cohesive translational force of the space-time progression; the magnitude of the force is thus equal to the translational equivalent force of the rotational force of the atoms and so the required dissociation energy must equal the rotational energy. Because of the discrete unit postulate, less than this amount of energy would be ineffective.
As it stands, equation (4) expresses the energy in natural units of the time region. We have to convert the equation to an equivalent expression for the time-space region so that we can compare calculated to observed results. First of all we must apply the fourth power of the interregional ratio, 1/156.44, to the equation, just as is done in the atomic force equation.
| E = (1/156.44)4 * ta * tz |
(5)
|
This is the energy in natural units as would be observed in the time-space region. To convert this to conventional units of measurement we multiply by the value of the natural unit of energy expressed in conventional units, Eu.
| E = (1/156.44)4 * ta * tz * Eu |
(6)
|
The experimental values are expressed as kcal/mole so we must multiply the right side of the equation (6) by a conversion factor, k, and by Avogadro’s number, N.
| E = (1/156.44)4 * ta * tz * Eu * k * N |
(7)
|
Next we must append a factor of ½ to the expression to account for the inherent vibrational nature of the time region motions and a factor of 1/3 to the expression to reduce the energy to one dimension. So now we have
| E = (1/156.44)4 * ta * tz * Eu * k * N * (1/6) |
(8)
|
From Ref. 1, Eu is 1.49175 x 10-3 ergs and N is 6.02486 x 1023. k is 2.389 x 10-11 kcal/erg. The final working equation is
| E = 5.9747 * ta * tz kcal/mole |
(9)
|
Displacement ta can range from 1 to 4 and displacement tz can range from 1 to 8. Table I lists the possible values of E for the various combinations of ta and tz.
I have applied equation (9) to 18 diatomic molecules of the elements. The theoretical and experimental results are given in table II. Let t1 symbolize the primary magnetic displacement of an element, t2 the secondary magnetic displacement, and t3 the electric displacement. It is clear from the table that
| ta = t1, or ta = t1 + 1, or ta = t1 - 1, or ta = t2, or ta = t2 + 1, or ta = t2 - 1 |
(10)
|
And
| tz = t2, or tz = t2 + 1, or tz = t2 - 1, or tz = t3 |
(11)
|
For electronegative elements, the 8-t3 rule applies:
| tz = 8 - t3, or tz = 8 - t3 + 1 |
(12)
|
Generally, only one (if any) of the two displacements has to be incremented or decremented by 1 to obtain a good fit with the experimental data; the other displacement equals the rotational displacement (or 8 minus the rotational space displacement) as the theory requires. Elements that require an increment of displacement usually have low atomic number; elements that require a decrement of displacement usually have high atomic number.
The values of ta and tz thus fit the normal variations in the elements that have appeared in other Reciprocal System calculations. This, together with allowance for experimental error, allows us to conclude that we have good agreement between theory and reality.
A future paper will apply equation (9) to diatomic molecules of unlike atoms.
References
- Dewey B. Larson, Nothing But Motion (Portland, Oregon: North Pacific Publishers, 1979), p. 155.
- John A. Dean, ed., Lange’s Handbood of Chemistry, Eleventh Edition (New York: McGraw-Hill Book Company, 1973), pp. 3-123 to 3-127.
|
E kcal/mole
|
ta | tz | ta | tz | ta | tz | |
|
5.9747
|
1 | 1 | |||||
|
11.9494
|
1 | 2 | 2 | 1 | |||
|
17.9241
|
1 | 3 | 3 | 1 | |||
|
23.8988
|
1 | 4 | 2 | 2 | 4 | 1 | |
|
29.8735
|
1 | 5 | |||||
|
35.8482
|
1 | 6 | 2 | 3 | 3 | 2 | |
|
41.8229
|
1 | 7 | |||||
|
47.7976
|
1 | 8 | 2 | 4 | 4 | 2 | |
|
53.7723
|
3 | 3 | |||||
|
59.7470
|
2 | 5 | |||||
|
71.6964
|
2 | 6 | 3 | 4 | 4 | 3 | |
|
83.6458
|
2 | 7 | |||||
|
89.6205
|
3 | 5 | |||||
|
95.5952
|
2 | 8 | 4 | 4 | |||
|
107.5446
|
3 | 6 | |||||
|
119.4940
|
4 | 5 | |||||
|
125.4687
|
3 | 7 | |||||
|
143.3928
|
3 | 8 | 4 | 6 | |||
|
167.2916
|
4 | 7 | |||||
|
191.1904
|
4 | 8 | |||||
Table II: Calculated and Observed Values of Dissociation Energy
|
Molecule
|
Displacement
|
Method
|
ta
|
tz
|
Ecalc.
|
Eobs.
|
||
|
|
||||||||
|
As-As
|
3-3-(3)
|
t2 8-t3
|
3
|
5
|
89.62
|
91
|
||
|
Cs-Cs
|
4-3-1
|
t2-1 t3
|
2
|
1
|
11.95
|
10.4
|
||
|
Cl-Cl
|
3-2-(1)
|
t1 t2+1
|
3
|
3
|
53.77
|
57.07
|
||
|
Cu-Cu
|
3-3-(7)
|
t1 t2
|
3
|
3
|
53.77
|
48
|
||
|
F-F
|
3-2-8
|
t1 t2
|
3
|
2
|
35.85
|
36
|
||
|
Ga-Ga
|
3-3-(5)
|
t1 t2-1
|
3
|
2
|
35.85
|
35
|
||
|
Au-Au
|
4-4-(7)
|
t2 8-t3+1
|
4
|
2
|
47.80
|
52
|
||
|
D-D
|
2-1-(1)
|
t1 8-t3+1
|
2
|
8
|
95.60
|
105
|
||
|
I-I
|
4-3-(1)
|
t1-1 t2-1
|
3
|
2
|
35.85
|
35.55
|
||
|
Li-Li
|
2-1-1
|
t1 t2+1
|
2
|
2
|
23.90
|
25
|
||
|
P-P
|
3-2-(3)
|
t1+1 8-t3
|
4
|
5
|
119.49
|
116.0
|
||
|
K-K
|
3-2-1
|
t2 t3
|
2
|
1
|
11.95
|
11.8
|
||
|
Se-Se
|
3-3-(2)
|
t2-1 8-t3
|
2
|
6
|
71.70
|
65
|
||
|
Ag-Ag
|
4-3-(7)
|
t2 8-t3+1
|
3
|
2
|
35.85
|
39
|
||
|
Na-Na
|
2-2-1
|
t2+1 t3
|
3
|
1
|
17.92
|
17.3
|
||
|
S-S
|
3-2-(2)
|
t2 8-t3+1
|
2
|
7
|
83.65
|
83
|
||
|
Te-Te
|
4-3-(2)
|
t1-1 t2
|
3
|
3
|
53.77
|
53
|
||
|
Sn-Sn
|
4-3-(4)
|
t2-1 8-t3
|
2
|
4
|
47.80
|
46
|
||
Note: the observed values, Eobs., come from Reference 2.