THE COHESIVE ENERGY OF THE ELEMENTS AT ZERO TEMPERATURE AND ZERO EXTERNAL PRESSURE

The equation for the internal energy of a substance is

 u = h - pv (1)

where h is enthalpy, p is pressure, and v is volwne. At zero absolute temperature, the enthalpy is zero.

 uo= -pv (2)

For a gas at zero temperature governed by the ideal gas law, the internal energy must also be zero. This is not so with a solid. Larson has shown that the equivalent of an external pressure exists which provides the cohesion of the solid state. This pressure arises from the force of the space-time progression, which is inward directed within the time region. With zero external pressure and zero temperature, the internal energy must equal the cohesive energy. Letting * be the internal pressure in kN/m² and vo be the volume in m³ /mole, and dropping the sign convention, we obtain the cohesive energy in kJ/mole:

 uo = po vo (3)

However, as shown in reference one, motion in the time region (whether inward or outward) is effective only hatf the time. This reduces the cohesive energy given by equation (3) by a factor of two.

 uo = ½po vo (4)

This equation is directly applicable to the "rare gas" elements.
The equation for molar volume is

 vo = GNso ³ (5)

where so is the nearest neighbor distance N is Avogadro's number, and G is a geometric factor. For face-centered-cubic crystals,

 Gfcc= .707 (6)

For body-centered-cubic crystals,

 Gbcc = .770 (7)

For other crystals,

 (GMW/density) x 10-6 G = ———————— (8) so³N

where GMW is the gram molecular weight, density is in grams per cubic centimeter, and so is in meters (10 Angstroms).

In Chapter 25 of reference one, Larson derives the equation for the internal pressure in natural units:

 aZR Po = ———————— (9) 312.89(so/sut)³

where a is the effective displacement in the active dimension, Z is either the electric displacement or the second magnetic disptacement (depending on the orientation of the atom), and R is the number of rotational units. sut' is the time region natural unit of space, given by

 sut = (1/156.44) su (10)

In kN/m² the value for po becomes

 Po = 4.177 x 10-17 aZr kN —— —— (11) So³ m²

Then,

 uo = 12.57GaZR kJ/mole (12)

The parameters a, Z, and R have been deduced by Larson for most of the elements, but not yet for the rare gas elements. Pending this, the value of the internal pressure can be determined as the reciprocal of twice the initial compressibility (equation 25-14, reference one):

 po = 1/2kT (13)

Table I gives the values for po , vo, and uo for the rare gas elements. Overall the values compare within 8% of the experimental values.

Elements other than the rare gas elements have electric displacement and this must obviously have an effect on cohesive energy. The additional energy is given by this expression:

 ut' = INEu (1/156.44)4 (14)

where I is an integer or half integer value, N is Avogadro's number, and Eu is the natural unit of energy. Alternatively from the cohesive energy standpoint, the effective volume, v, may be altered. The factor is the interregional ratio (applicabte to energy, as well as force). I is one for most of the displacement one elements, one and one-half or two for displacement two elements, three or more for displacement three elements, and from 3½ to 5½ for displacement four etements. I can be zero or negative for the electronegative etements. An exact equation for I cannot as yet be given.

The final reduced equation for cohesive energy is

 kJ uo = 12.57 GaZR + 50.31 —— (15) mole

Table II gives the values of G, a, Z R, I, and uo for most of the remaining elements, together with the experimental values from reference two. Usually agreement is within a few percent.

Present atomic theory has nothing comparable to equation (15). The so-called Lennard-Jones potential commonly used is empirically based and has not been deduced from first principles--and even then it has usually been applied only to the noble elements and a few other elements of 1ow atomic weight. Thus we have here a definite advantage of the Reciprocal System over current theory.

References

1. Dewey B. Larson, Nothing But Motion, Vol. 1 of the revised Structure of the Physical Universe, presently in manuscript form.
2. C. Kittel, Introduction to Solid State Phisics, Fifth Edition (New York: John Wiley & Sons, Inc., 1976 , p. 74.

Table 1

 —————————————————————————–——————— kN m³ kJ kJ Element — — — — po m² vo mole uo mole uexp mole —————————————————————————–——————— Helium 8.56x104 1.950x10-5 .835 -- Neon 5.00x105 1.395x10-5 3.488 1.92 Argon 5.33x105 2.227x10-5 5.935 7.74 Krypton 8.93x105 2.806x10-5 12.53 11.2 Xenon 9.52x105* 3.528x10-5 16.79 15.9 Radon 12.30x105* 3.584x10-5* 22.04 19.5

*Estimated values based on trend line analysis or assumed specific rotational values.

Table II

 —————————————————————————–——————— kJ kJ Element Form G a Z R I —— —— uo mole uexp mole —————————————————————————–——————— Li bcc .770 4 1 1 2½ 164.5 158 Be hcp .752 4 4 1 3½ 327.3 320 C dia 1.554 4 6 1 5 720.3 711 Na bcc .770 4 1 1 1 89.0 107 Mg hcp .780 4 3 1 1 157.1 145 Al fcc .707 4 5 1 3 328.6 327 Si dia 1.543 4 5 2 -6½ 448.9 446 K bcc .770 4 1 1 1 89.0 90.1 Ca fcc .707 4 3 1 1½ 182.1 178 Ti hcp .731 4 8 1 3½ 470.1 468 V bcc .770 4 8 1 4 510.9 512 Cr bcc .770 4 8 1 2 410.3 395 Mn cu.com. 1.087 4 8 1 -3 286.3 282 Fe bcc .770 4 8 1 2 410.3 413 Co hcp .696 4 8 1 3 430.9 424 Ni fcc .707 4 8 1 3 435.3 428 Cu fcc .707 4 8 1 1½ 359.9 336 Zn hcp .809 4 4 1 -1 112.4 130 Ge dia 1.541 4 4 1 1 360.2 372 Rb bcc .770 4 1 1 1 89.0 82.2 Sr fcc .707 4 3 1 1 156.9 166 Zr hcp .731 4 6 1½ 5½ 607.5 603 Nb bcc .770 4 8 1½ 5 716.1 730 Mo bcc .770 4 8 2 1 669.7 658 Ru hcp .730 4 8 2 1½ 662.7 650 Rh fcc .707 4 8 2 0 568.8 554 Pd fcc .707 4 8 1½ -1 376.3 376 Ag fcc .707 4 8 1 0 284.4 284 Cd hcp .816 4 4 1 -1 113.8 112 In tet .762 4 4 1 2 253.9 243 Sn dia 1.543 4 4 1 0 310.3 303 Sb rho 1.227 4 4 1 0 246.8 265 Cs bcc .770 4 1 1 1 89.0 77.6 Ba bcc .770 4 2 1 2 178.0 183 La hex .721 4 4 1 5½ 421.7 431 Ce fcc .707 4 4 1 5½ 418.9 417 Pr hex .722 4 4 1 4 346.4 357 Nd hex .698 4 4 1 4 341.6 328 Sm com .716 4 4 1 1 194.3 206 Gd hcp .722 4 4 1 5 396.7 400 Dy hcp .732 4 4 1 3 298.1 294 Ho hcp .732 4 4 1 3 298.1 302 Er hcp .736 4 4 1 3½ 324.1 317 Tm hcp .679 4 4 1 2 237.2 233 Yb fcc .707 4 2 1 1½ 146.5 154 Lu hcp .732 4 4 1 5½ 423.9 428 Ta bcc .770 4 8 2 3 770.3 782 W bcc .770 4 8 3 -1½ 853.8 859 Ir fcc .770 4 8 3 -3½ 677.2 670 Pt fcc .770 4 8 2 0 568.8 564 Au fcc .770 4 8 1½ -3 275.7 284 Tl hcp .690 4 4 1 1 189.1 182 Pb fcc .770 4 4 1 2½ 268.0 265 Bi rho 1.224 4 3 1 1 234.9 210 Th fcc .770 4 8 1 6 586.2 598 U com .998 4 8 1 3 552.3 536 —————————————————————————–———————