Fibonacci numbers and the Pascal Triangle

 

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Electron Configuration and the Pascal Triangle

        

        

The common character of theories of atomic constitution has been the endeavour to find configurations and motions of the electrons which would seem to offer an interpretation of the variations of the chemical properties of the elements with the atomic number as they are so clearly exhibited in the well-known periodic law. A consideration of this law leads directly to the view that the electrons atom are arranged in distinctly separate groups, each containing a number of electrons equal to one of the periods in the sequence of the elements, arranged according to increasing atomic number. In the first attempts to obtain a definite picture of the configuration and motion of the electrons in these groups it was assumed that the electrons within each group at any moment were placed at equal angular intervals on a circular orbit with the nucleus at the centre, while in later theories this simple assumption has been replaced by the assumptions that the configurations of electrons with in the various groups do not possess such a simple axial symmetry, but exhibit a higher degree of symmetry in space, it being assumed, for instance, that the configuration of the electrons at any moment during their motions possesses polyhedral symmetry.
The elecrtron can be analyzed in several ways. Presently, the Bohr model of the atom will be used which uses the shell model for evaluating electrons. The shell model has electrons residing in various energy levels about the nucleus. The first shell (K Shell) can have up to 2 electrons. This shell is filled first with electrons, thus any atom with greater than 2 protons or atomic number will have this shell filled. The next shell is the (L Shell) and can have up to 8 electrons in it. So on with the (M Shell). When the number of electrons starts filling the (N Shell) they then start packing in the inner shells. Electron shells have limited capacity for electrons. The farther an electron shell is from nucleus, the larger it is.

An easy way to calculate the total number of electrons that can be held by a given energy level is to use the formula 2*n2 , where n equals the number of the electron shell. For example, for the 1st electron shell n=1 and 2*12 = 2, telling us that the capacity of the 1st shell is 2 electrons as we have already seen. For the 2nd shell ( n=2 ) and 2*22 = 8. For an atom to fill its 2nd electron shell, 10 electrons would be needed : 2 to fill the 1st shell and 8 to fill the 2nd.

     

Principle energy
level ( n )

Maximum number
of electrons 2*n2

1

2

2

8

3

18

4

32

5

50

6

72

7

98

     

The maximum number of electrons is double square number.The square numbers can be found in the next triangle's form of the natural numbers :

     

1
1
1 2 1
4
1 2 3 2 1
9
1 2 3 4 3 2 1
16
1 2 3 4 5 4 3 2 1
25
1 2 3 4 5 6 5 4 3 2 1
36
1 2 3 4 5 6 7 6 5 4 3 2 1
49

     

We may show a connection between square numbers and the tetrahedral numbers :

     

1
.
.
.
.
.
1
4
.
.
.
.
.
4
9
1
.
.
.
.
10
16
4
.
.
.
.
20
25
9
1
.
.
.
35
36
16
4
.
.
.
56
49
25
9
1
.
.
84

     

As you might expect, the 3rd shell has a total capacity of 2*32 = 18 electrons. But things get a bit tricky here. Electron shells actually have sublevels. The first sublevel ( the s sublevel ) holds 2 electrons. The second, p , sublevel holds 6.The third, d , sublevel holds 10. When levels 3s and 3p are filled, electron shell#3 acts as if it has reached capacity with only 8 total electrons. In other words, in an atom with 20 electrons ( which is the element calcium) the first 2 electrons are located in the 1st shell, the next 8 in shell#2, the following 8 in shell#3 and remaining 2 electrons are located in shell#4.

The number of sublevels that an energy level can contain is equal to the principle quantum number of that level. The first sublevel is called s sublevel. s sublevels have one orbital, which can hold up to two electrons. The second sublevel is called a p sublevel. p sublevels have three orbitals, each of which can hold 2 electrons, for a total of 6. The third sublevel is called a d sublevel and d sublevels have 5 orbitals, for a possible total of 10 electrons. The fourth sublevel is called an f sublevel. f sublevels, with 7 orbitals, can hold up to 14 electrons. Although energy levels that are higher than 4 would contain additional sublevels, these sublevels have not been named because no known atom in its ground state would have electrons that occupy them. The information about the sublevels is summarized in the table below :

     

Orbital and Electron Capacity for the Sublevels

Sublevel # of orbitals Maximum number of electrons
s 1 2
p 3 6
d 5 10
f 7

14

... 9 18
... 11 22
... 13 26
... 15 30

     

The maximum numbers of electrons in sublevels may be represented as the following triangle's form of numbers :

     

 
 
 
 
 
2
 
 
 
 
 
 
 
2
 
 
 
 
2
2
2
 
 
 
 
 
 
6
 
 
 
1
3
2
3
1
 
 
 
 
 
10
 
 
1
1
4
2
4
1
1
 
 
 
 
14
 
1
1
1
5
2
5
1
1
1
 
 
 
18
1
1
1
1
6
2
6
1
1
1
1
 
 
22

     

The law of the electron configurations is the law of the geometrical area. There is a connection between odd numbers and the square numbers :

     

1
 
  
 
 
 
 
 
 
1
3
1
  
 
 
 
 
 
 
4
5
3
1
 
 
 
 
 
 
9
7
5
3
1
 
 
 
 
 
16
9
7
5
3
1
 
 
 
 
25
11
9
7
5
3
1
 
 
 
36
13
11
9
7
5
3
1
 
 
49

     

There is also a connection between odd numbers and the triangular numbers :

     

1
 
 
 
 
 
1
3
 
 
 
 
 
3
5
1
 
 
 
 
6
7
3
 
 
 
 
10
9
5
1
 
 
 
15
11
7
3
 
 
 
21
13
9
5
1
 
 
28

     

The odd numbers , square numbers and pyramidal numbers can be found as columns in Pascal triangle of the second kind :

     

1
1 2
1 3 2
1 4 5 2
1 5 9 7 2
1 6 14 16 9 2
1 7 20 30 25 11 2
1 8 27 50 55 36 13 2
1 9 35 77 105 91 49 15 2


An electron configuration is a method of indicating the arrangement of electrons about a nucleus. A typical electron configuration consists of numbers, letters and superscripts with the following format:
1. A number indicates the energy level.( The number is called the principal quantum number.)
2.A letter indicates the type of orbital: s,p,d,f...
3.A superscript indicates the number of electrons in the orbital.
To write an electron configuration:
1.Determine the total number of electrons to be represented.
2.Use the Aufbau process to fill the orbitals with electrons. The Aufbau process requires that electrons fill the lowest energy orbitals first. In another words, atoms are built from the ground upwards.
3.The sum of the superscripts should equal the total number of electrons.

The periodic table lines up elements in order of the # of protons (atomic number), but the key to the columns is the number of valence (outermost) electrons. To simplify things chemists picture electrons filling shells such that once a shell is filled we only become concerned with electrons outside the full shell. Consider the first two elements hydrogen and helium. With two electrons and the inertness of helium, it has a shell that is full and is stable. Hydrogen can have valences of no electrons, one electron, or two electrons. With the case of hydrogen with two electrons it is pictures as having a full shell (called the "NOBLE GAS ELECTRON CONFIGURATION"). After a pair of electrons, the next shell requires eight electrons (called an 'OCTET'). This element is neon, with atomic number 10, with two electrons in the first shell and the next shell of eight being full. With the shells full, this is what allows neon to be unreactive (and known as a "NOBLE GAS"). Thus a great example of valence electrons dictating an elements postion on the periodic table is the fluorine atom above the chlorine atom. Both of these elements have seven valence electrons outside their respective noble gas electron configurations. Fluorine is the most reactive atom needing one more electron to complete the octet outside the inner electron pair. This is where even with the nine protons (providing an electric charge of positive +9), the fluorine atom requires 10 electrons to be stable (so in water the fluorine anion, F- is how you would find it-resembling the noble gas electron configuration of neon)...Chlorine has 17 protons (and positive +17 electric charge) but would be stable with 18 electrons since two would fill the first helium shell, then eight to fill the neon shell, and with only seven for the next noble gas shell (resembling argon), an extra electron makes the chloride anion stable in water, Cl-, 18 electrons (2,8,8=2+8+8=18e-'s)... In the atomic shell model, the shells are filled with electrons in order of increasing energy until they completely fill a closed shell, producing the inert core of a noble gas. These elements have highly stable properties:

#

Case

  Electron Configuration  

Geometrical Number
of electrons

Real Number
of electrons

1

Geometrical

2

2

  

He

2

  

2

2

Geometrical

2,2-6

10

  

Ne

2,2-6

  

10

3

Geometrical

2,2-6,2-6-10

28

  

Ar

2,2-6,2-6-*

  

18

4

Geometrical

2,2-6,2-6-10,2-6-10-14

60

  

Kr

2,2-6,2-6-10,2-6-*-*

  

36

5

Geometrical

2,2-6,2-6-10,2-6-10-14,2-6-10-14-18

110

  

Xe

2,2-6,2-6-10,2-6-10-*,2-6-*-*-*

  

54

6

Geometrical

2,2-6,2-6-10,2-6-10-14,2-6-10-14-18,2-6-10-14-18-22

182

  

Rn

2,2-6,2-6-10,2-6-10-14,2-6-10-*-*,2-6-*-*-*-*

  

86

   

Geometrical Number of electrons is equivalent with the duble pyramidal numbers :

2,10,28,60,110,182,...= 2*(1,5,14,30,55,91,...)

Real Number of electrons is :

2,10,18,36,54,86,...= 2*(1,5,9,18,27,43,...)

We have series : 2,18,54,118,218...

Magic numbers are featured by the formula:


ZE1m=4*(2*m3+3*m2+m-3)/6

m = 1,2,3 4, 5, 6, 7 ...


And the series : 10,36,86,168...

Magic numbers are featured by the formula:


ZE2m=4*(2*m3+6*m2+7*m)/6

m = 1,2,3 4, 5, 6, 7 ...


The connection between magic numbers of noble gases on one side and pyramidal numbers, on the other, is highly amusing :

     

1
 
 
 
 
=
1
 
2
5
+
1
-
1
=
5
 
10
5
+
5
-
1
=
9
 
18
14
+
5
-
1
=
18
 
36
14
+
14
-
1
=
27
 
54
30
+
14
-
1
=
43
 
86
30
+
30
-
1
=
59
 
118

     

There is a connection between magic numbers of noble gases on one side and pyramidal and square numbers, on the other :

     

1
 
 
 
 
=
1
 
2
5
 
 
 
 
=
5
 
10
5
4
 
 
 
=
9
 
18
14
4
 
 
 
=
18
 
36
14
9
4
 
 
=
27
 
54
30
9
4
 
 
=
43
 
86
30
16
9
4
 
=
59
 
118

     

Here is a connection between real number of electrons and square numbers:

     

1
 
 
=
1
5
-
1
=
4
9
-
5
=
4
18
-
9
=
9
27
-
18
=
9
43
-
27
=
16
59
-
43
=
16
84
-
59
=
25
109
-
84
=
25

Or:

     

1
 
 
 
 
 
 
=
1
 
2
4
1
 
 
 
 
 
=
5
 
10
4
4
1
 
 
 
 
=
9
 
18
9
4
4
1
 
 
 
=
18
 
36
9
9
4
4
1
 
 
=
27
 
54
16
9
9
4
4
1
 
=
43
 
86
16
16
9
9
4
4
1
=
59
 
118

     

There is a connection between pyramidal numbers and odd numbers and the magic numbers of noble gases, as shown in the table below :

     

1
 
 
 
 
 
 
 
 
 
 
 
 
=
1
 
2
5
 
 
 
 
 
 
 
 
 
 
 
 
=
5
 
10
14
-
5
 
 
 
 
 
 
 
 
 
 
=
9
 
18
30
-
7
-
5
 
 
 
 
 
 
 
 
=
18
 
36
55
-
9
-
7
-
7
-
5
 
 
 
 
=
27
 
54
91
-
11
-
9
-
9
-
7
-
7
-
5
=
43
 
86

Here is the alternative form of this connection, as shown in the table below:


1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
=
1
 
2
4
1
 
 
 
 
 
 
 
 
 
 
 
 
 
=
5
 
10
10
4
-
3
2
 
 
 
 
 
 
 
 
 
 
=
9
 
18
20
10
-
4
3
3
2
 
 
 
 
 
 
 
 
=
18
 
36
35
20
-
5
4
4
3
4
3
3
2
 
 
 
 
=
27
 
54
56
35
-
6
5
5
4
5
4
4
3
4
3
3
2
=
43
 
86

     

     

        

  2001-2005 Radoslav Jovanovic                 updated:  November 2004.