where h is enthalpy, p is pressure, and v is volwne. At zero absolute temperature, the enthalpy is zero.
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:
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.
is directly applicable to the "rare gas" elements.
where so is the nearest neighbor distance N is Avogadro's number, and G is a geometric factor. For face-centered-cubic crystals,
For body-centered-cubic crystals,
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:
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
In kN/m² the value for po becomes
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:
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
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.
*Estimated values based on trend line analysis or assumed specific rotational values.