Atom shell

Energy - deriving

Ionization energy of electrons
Given real (meassured) ionization energy of atoms E:
Z r=1
H  1 13.598 r=2
He  2 54.416 24.587 r=3
Li  3 122.451 75.638 5.392 r=4
Be  4 217.713 153.893 18.211 9.322 r=5
Standardized energy - values En = E/E1, where E1 = E(H) ~ 13.6 eV:
Z r=1
H  1 1.000 r=2
He  2 4.001 1.8079 r=3
Li  3 9.004 5.5616 0.3965 r=4
Be  4 16.008 11.3157 1.3390 0.6853 r=5
Standardized charge of nucleus - value  V = √ En
V r=1
H  1 1.0000 r=2
He  2 2.0003 1.3446 r=3
Li  3 3.0007 2.3583 0.6297 r=4
Be  4 4.0010 3.3639 1.1572 0.8278 r=5

Standardized charge of nucleus - value V: V = √En , where En = E/13.6, table V(r,t)
r\t t=1 t=2 t=3 t=4
r=1 1.000 2.007 3.007 4.001
r=2 1.344 2.358 3.364
r=3 0.630 1.157
r=4 0.828
Effective charge of nucleus - value Zef= F(n,V) = n * V = n √En. Let E1 = E(H) = 13.53 and En = E/E1.  E.g. for Li is E(Li) = 5.41, En = E/E1 = 5.41/13.53 = 0.40 V=√En= 0.630, Zef = nV = 2*0.630 = 1.260.
H He LI Be B C N O F Ne Na Mg
n 1 1 2 2 2 2 2 2 2 2 3 3
r 1 2 3 4 5 6 7 8 9 10 11 12
V(r,r) 1.000 1.344 0.630 0.828 0.781 0.910 1.034 1.001 1.132 1.259 0.615 0.750
Zef 1.00 1.37 1.26 1.65 1.56 1.84 2.07 2.00 2.27 2.52 1.85
Placement of electrons in the atom of the same layer with the higher charge nucleus Z.
By changing nucleus is proportionately increased nZ’: F(n,Z’) = F(n,Z’-1) + 1
1s1 2s1 3s1 4s1
n 1 2 3 4 5
Z=1 1.0
Z=2 2.0
Z=3 3.0 1.26
Z=4 4.0 2.31
Z=5 5.0 3.34
Excitation of electrons to higher layers - Effective number of layer nef = 1/Z‘.
Moving electron does not change his shading: nef(Z,n) = nef(Z,n-1) + 1 (pro n > n*)
1s1 2s1 3s1 4s1
n 1 2 3 4 5
Z=1 1.0
Z=2 0.5
Z=3 0.333 1.589 (= 1/0.630) 2.596 3.598
Z=4 0.25 0.864
Z=5 0.20 0,599

Quantization of standardized charges

We will observe the ratios of normalized charges V(t,r).

Values V of the first and second (s) electron: - First electron: V(t, 1) = t = (3t) / 3 - Second electron: V(t,2) = t-2/3 = (3t-2) /3 (přesněji ne 2/3, ale 0.64)

V(r,t)
r\t 1 2 3 4 5 6 7 8 9 10 11 12
1 1.000 2.007 3.007 4.001 5.000 6.002 7.003 8.005 9.061 10.004 11.010 12.013
2 1.344 2.358 3.364 4.367 5.369 6.370 7.373 8.419 9.377 10.379 11.382
Value V of the third and the fourth (s) electron: - Third electron: V(t, 3) = (4t - 7) / 4 - Fourth electron: V(t,4) = (4t - 10) /4 V(r,t)
r\t 1 2 3 4 5 6 7 8 9 10 11 12
3 0.630 1.157 1.670 2.177 2.683 3.187 3.689 4.193 4.696 5.199
4 0.828 1.360 1.876 2.386 2.894 3,399 3.904 4.407 4.911
Value V of the fifth, sixth and seventh (p) electron: - Fifth electron: V(t, 5) = (5t - 18) /5 - Sixth electron: V(t,6) = (5t - 22) /5 - Seventh electron: V(t,7) = (5t - 26) /5 V(r,t)
r\t 1 2 3 4 5 6 7 8 9 10 11 12
3 0.630 1.157 1.670 2.177 2.683 3.187 3.689 4.193 4.696 5.199
4 0.828 1.360 1.876 2.386 2.894 3,399 3.904 4.407 4.911
The ratio of the values V of r-th electron of the elements Z = r a Z = r-1.
r\t V(r,r) V(r,r+1) V(r,r+1)/ V(r,r)=p/(r+2) p – r
-1 3/1 4
0 H 4/2 4
1 H 1.000 He 2.007 6/3 5 1s/1
2 He 1.344 Li 2.358 7/4 5 1s/2
3 Li 0.630 Be 1.157 9/5 6 2s/1
4 Be 0.828 B 1.360 10/6 6 2s/2
5 B 0.781 C 1.339 12/7 7 2p/a1
6 C 0.910 N 1.475 13/8 7 2p/a2
7 N 1.034 O 1.607 14/9 7 2p/a3
8 O 1.001 F 1.604 16/10 8 2p/b1
9 F 1.132 Ne 1.736 17/11 8 2p/b2
10 Ne 1.259 Na 1.865 18/12 8 2p/b3
11 Na Mg 20/13 9 3s/1
12 Mg Al 21/14 9 3s/2
13 Al Si 23/15 10 3p/a1
14 Si P 24/16 10 3p/a2
15 P S 25/17 10? 3p/a3
16 S Cl 27/18 11 3p/b1
17 Cl Ar 28/19 11 3p/b2
18 Ar K 29/20 11? 3p/b3
19 K Cu 31/21 12 4s/1
20 Cu 32/22 12 4s/2

Quantization of effective charge

We are going to observe values of effective charges n.V =n.√ En, see the following tables..
E.g.. For Li it is:
    E(Li) = 5.392
    En = E/E1 = 5.392/13.598 = 0.3965
    √ En =√ 0.3965=0.6297
    n.√ En = 2*0.6297 = 1.2594.

S-orbits
                           s1                           s2
n X Z n.E* r-0.25 r=(n+1)/2 r-0.25 n.E* Z X
1 H 1 1.000 0.750 1.000 1.250 1.344 2 He
2 Li 3 1.259 1.250 1.500 1.750 1.656 4 Be
3 Na 11 1.845 1.750 2.000 2.250 2.250 12 Mg
4 K 19 2.260 2.250 2.500 2.750 2.680 20 Ca
5 Rb 37 2.775 2.750 3.000 3.250 3.235 38 Sr
6 C 55 3.216 3.250 3.500 3.750 3.724 56 Ba
7 Fr 87 3.787 3.750 4.000 4.250 4.361 88 Ra

P-orbits
                           p1                      p2                 p3
n X Z n.E* 2n/3 X Z n.E* (2n+1)/3 X Z n.E* (2n+2)/3
1 - -
2 B 5 1.562 1.333 C 6 1.820 1.667 N 7 2.068 2.000
3 Al 13 1.989 2.000 Si 14 2.322 2.333 P 15 2.649 2.667
4 Ga 31 2.656 2.666 Ge 32 3.044 3.000 As 33 3.396 3.333
5 In 49 3.260 3.333 Sn 50 3.665 3.667 Sb 51 3.975 4.000
6 Tl 81 4.020 4.000 Pb 82 4.333 Bi 83 4.667
7 4.667 5.000 5.333

                           p4                      p5                 p6
n n' = n +1 X Z n.E* 2n'/3 X Z n.E* (2n'+1)/3 X Z n.E* (2n'+2)/3
1 2 - -
2 3 O 8 2.000 2.000 F 9 2.264 2.333 Ne 10 2.516 2.667
3 4 S 16 2.616 2.667 Cl 17 2.934 3.000 Ar 18 3.222 3.333
4 5 Se 34 3.388 3.333 Br 35 3.732 3.667 Kr 36 4.048 4.000
5 6 Te 52 4.050 4.000 I 53 4.375 4.333 Xe 54 4.705 4.667
6 7 Po 84 4.667 At 85 5.000 Rn 86 5.333
7 8 5.333 5.667 6.000

D-orbits
          d1            d2          d3       d4           d5
n (3n-1)/4 X Z n.E* X Z n.E* X Z n.E* X Z
n.E* X Z n.E* 3n/4
3 2.000 Sc 21 2.085 Ti 22 2.127 V 23 2.112 Cr 24 2.115 Mn 25 2.217 2.250
4 2.750 Y 39 2.788 Zr 40 2.860 Nb 41 2.844 Mo 42 2.896 Tc 43 2.916 3.000
5 3.500 Lu 71 3.520 Hf 72 3.180 Ta 73 3.760 W 74 3.830 Re 75 3.805 3.750
6 4.250 4.500
          d6            d7          d8       d9           d10
n 3n/4 X Z n.E* X Z n.E* X Z n.E* X Z
n.E* X Z n.E* (3n+1)/4
3 2.250 Fe 26 2.286 Co 27 2.280 Ni 28 2.247 Cu 29 2.259 Zn 30 2.493 2.500
4 3.000 Ru 44 2.944 Rh 45 2.964 Pd 46 3.132 Ag 47 2.988 Cd 48 3.252 3.250
5 3.750 Os 76 4.000 Ir 77 4.110 Pt 78 4.060 Au 79 4.115 Hg 80 4.380 4.000
6 4.500 4.750

Detailed tables

Meassured energy
Given real (meassured) ionization energy of atoms:
Z r=1
H  1 13.598 r=2
He  2 54.416 24.587 r=3
Li  3 122.451 75.638 5.392 r=4
Be  4 217.713 153.893 18.211 9.322 r=5
B  5 340.217 259.368 37.930 25.154 8.298 r=6
C  6 489.981 392.077 64.492 47.887 24.383 11.260 r=7
N  7 667.029 552.057 97.888 77.472 47.448 29.601 14.534 r=8
O  8 871.387 739.315 138.116 113.896 77.412 54.934 35.116 13.618 r=9
F  9 1103.089 953.886 185.182 157.161 114.240 87.138 62.707 34.970 17.422 r=10
Ne 10 1361.164 1195.797 239.090 207.270 157.930 126.210 97.110 63.450 40.962 21.564 r=11
Na 11 1648.659 1465.091 299.870 264.180 208.470 172.150 138.390 98.910 71.640 47.286 5.139 r=12
Mg 12 1962.613 1761.802 367.530 327.950 265.900 224.940 186.500 141.260 109.240 80.143 15.035 7.646
Standardized energy
Let E1 = E(H) ~ 13.6 eV. Values En = E/E1:
Z r=1
H  1 1.000 r=2
He  2 4.001 1.8079 r=3
Li  3 9.004 5.5616 0.3965 r=4
Be  4 16.008 11.3157 1.3390 0.6853 r=5
B  5 25.016 19.0712 2.7882 1.8493 0.6103 r=6
C  6 36.028 28.8221 4.7412 3.5199 1.7926 0.8279 r=7
N  7 49.049 40.5809 7.1956 5.6949 3.4875 2.1757 1.0691 r=8
O  8 64.0726 54.3614 10.1556 8.373 5.690 4.0393 2.582 1.0013 r=9
F  9 81.1095 70.8740 13.610 11.551 8.398 6.407 4.607 2.5713 1.2810 r=10
Ne 10 100.086 87.9263 17.5801 15.2404 11.6125 9.2801 7.1404 4.6654 3.0119 1.5856 r=11
Na 11 121.225 107.727 22.0493 19.4250 15.3287 12.6581 10.1757 7.2728 5.2676 3.4769 0.3779 r=12
Mg 12 144.310 129.544 27.0243 24.1140 19.5515 16.5397 13.7132 10.3868 8.0324 5.8929 1.1055 0.5621

Standardized charge
Values  V = √ En
Z r=1
H  1 1.0000 r=2
He  2 2.0003 1.3446 r=3
Li  3 3.0007 2.3583 0.6297 r=4
Be  4 4.0010 3.3639 1.1572 0.8278 r=5
B  5 5.0016 4.3671 1.6698 1.3599 0.7812 r=6
C  6 6.0023 5.3686 2.1774 1.8761 1.3389 0.9099 r=7
N  7 7.0030 6.3703 2.6825 2.3864 1.8675 1.4750 1.0340 r=8
O  8 8.0045 7.3730 3.1868 2.8936 2.3854 2.0098 1.6069 1.0007 r=9
F  9 9.0061 8.4187 3.6892 3.3987 2.8979 2.5312 2.1464 1.6035 1.1318 r=10
Ne 10 10.0043 9.3769 4.1929 3.9039 3.4077 3.0463 2.6722 2.1600 1.7355 1.2592 r=11
Na 11 11.0102 10.3792 4.6957 4.4074 3.9152 3.5578 3.1899 2.6968 2.2951 1.8646 0.6147 r=12
Mg 12 12.0129 11.3817 5.1985 4.9106 4.4217 4.0669 3.7031 3.2229 2.8341 2.4275 1.0514 0.7498

Standardized charge - table
Table V(r,t)
r\t 1 2 3 4 5 6 7 8 9 10 11 12
1 1.000 2.007 3.007 4.001 5.000 6.002 7.003 8.005 9.061 10.004 11.010 12.013
2 1.344 2.358 3.364 4.367 5.369 6.370 7.373 8.419 9.377 10.379 11.382
3 0.630 1.157 1.670 2.177 2.683 3.187 3.689 4.193 4.696 5.199
4 0.828 1.360 1.876 2.386 2.894 3,399 3.904 4.407 4.911
5 0.781 1.339 1.868 2.385 2.898 3.408 3.915 4.422
6 0.910 1.475 2.010 2.531 3.046 3.558 4.067
7 1.034 1.607 2.146 2.672 3.190 3.703
8 1.001 1.604 2.160 2.697 3.223
9 1.132 1.736 2.295 2.834
10 1.259 1.865 2.428
11 0.615
12 0.750
Column differences
Differences of charges - column values:
Z r=1
H-#  1-0 1.0000 r=2
He-H  2-1 1.0003 1.3446 r=3
Li-He  3-2 1.0004 1.0137 0.6297 r=4
Be-Li  4-3 1.0003 1.0056 0.5277 0.8278 r=5
B-Be  5-4 1.0006 1.0032 0.5126 0.5321 0.7812 r=6
C-B  6-5 1.0007 1.0015 0.5076 0.5162 0.5577 0.9099 r=7
N-C  7-6 1.0007 1.0017 0.5051 0.5103 0.5286 0.5651 1.0340 r=8
O-N  8-7 1.0015 1.0027 0.5040 0.5072 0.5179 0.5340 0.5729 1.0007 r=9
F-O  9-8 1.0016 1.0457 0.5027 0.5051 0.5125 0.5222 0.5395 0.6028 1.1318 r=10
Ne-F 10-9 0.9982 0.9582 0.5052 0.5052 0.5098 0.5151 0.5258 0.5565 0.6037 1.2592 r=11
Na-Ne 11-10 1.0059 1.0023 0.5035 0.5035 0.5075 0.5115 0.5177 0.5132 0.5596 0.6054 0.6147 r=12
Mg-Na 12-11 1.0027 1.0025 0.5028 0.5032 0.5065 0.5091 0.5132 0.5261 0.5390 0.5629 0.4367 0.7498
These differential values of elements - approach to 1/m (m is integer).