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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  

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

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     kJ   kJ
Element        
 
po
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
 
164.5
 
158
Be
hcp
.752
4
4
1
 
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
 
182.1
 
178
Ti
hcp
.731
4
8
1
 
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
 
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
 
607.5
 
603
Nb
bcc
.770
4
8
5
 
716.1
 
730
Mo
bcc
.770
4
8
2
1
 
669.7
 
658
Ru
hcp
.730
4
8
2
 
662.7
 
650
Rh
fcc
.707
4
8
2
0
 
568.8
 
554
Pd
fcc
.707
4
8
-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
 
421.7
 
431
Ce
fcc
.707
4
4
1
 
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
 
324.1
 
317
Tm
hcp
.679
4
4
1
2
 
237.2
 
233
Yb
fcc
.707
4
2
1
 
146.5
 
154
Lu
hcp
.732
4
4
1
 
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
-3
 
275.7
 
284
Tl
hcp
.690
4
4
1
1
 
189.1
 
182
Pb
fcc
.770
4
4
1
 
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
—————————————————————————–———————

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