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## The quantum numbers and principles of distributing electrons

**Quantum numbers **

**The mathematical solution of Schrodingers equation introduced four numbers that were called quantum numbers , to determine the energy of an electron in multi-electron atoms , we should know the four quantum numbers which describe it .**

**These four quantum numbers are :**

**The principal quantum number ( n )**

**Bohr had used this number in explaining the spectrum of hydrogen atom , it is given the symbol ( n ) and it is used to define the following :**

**The order of principal energy levels or electron shells , Their number in the heaviest known atom in its ground state is seven .**

**The number of electrons (e) required to fill a given energy level which equals the formula 2n ( two times the square of the shell number ) .**

**The principal quantum number has whose number values 1 , 2 , 3 , 4 , . etc , excluding zero , Each value is expressed by an alphabetical letter that represents a principal energy level , Energy levels increases from K to Q , K < L < M < N < O < P < Q .**

**The rule 2n isnt applied to the energy levels higher than the fourth level , because the atom becomes unstable , if the number of electrons exceeds 32 electrons in any level .**

**The subsidiary quantum number ( l )**

**It is given the symbol ( l ) , It determines the number of sub-levels in each principal energy level , Each principal energy level consists of a number of energy sub-levels equals to its principal quantum number .**

**The energy sub-levels take the symbols and values , Subsidiary quantum number ( l ) values [ 0 : ( n1 ) ] : Symbols of sub-levels , s = 0 , p = 1 , d = 2 , f = 3 )**

**There is a small difference in the energy of the sub-levels , They can be arranged according to increasing their energy , s < p < d < f .**

**Examples : What are the probable ( l ) values when n = 3 ?**

**Each principal energy level consists of a number of sub-levels which equals its numerical value , Number of sub-levels = 3 .**

**The probable ( l ) values range between [ 0 : ( n1 ) ] = [ 0 : ( 31 ) ] = 0 , 1 , 2 **

**The magnetic quantum number ( ml )**

**It determines the number of orbitals within a certain energy sub-level from the relation ( 2l + 1 ) .**

**It determines the spatial orientation ( orientation in space ) of orbitals .**

**It is represented by the whole number values ( odd ) ranging between (l , . , 0 , .. , + l ) .**

**The orbitals of the same sub-levels are equal in energy , but differ in direction and shape in space , Any orbital cant be occupied by more than two electrons .**

**The p-sublevel is completely filled with 6 e ^{} , while the d-sublevel is completely filled with 10 e^{}, Because the p-sublevel contains 3 orbitals , while the d-sublevel contains 5 orbitals and each orbital is filled with 2 e^{} .**

**Example : What are the probable ( m _{l }) values when ( l = 2 ) ? **

**The probable ( m _{l} ) values ranges between l ,, 0 , .., + l **

**The probable( m _{l} ) values :2 ,l , 0 , +1 , +2**

**Example : Which of the following probabilities of the quantum numbers of a certain electron includes a mistake ?
**

**The choice ( c ) , because when ( n = 1 ) the probable l and m _{l} values are ( 0 ) .**

**The spin quantum number ( m**_{s} )

_{s})

**Since any orbital can not be occupied by more than two electrons , each electron spins on its own axis during its orbit around the nucleus as the spinning of the Earth on its own axis during its rotation around the Sun .**

**The spin quantum number determines the type of spin motion of the electron around its axis in the orbital , which is either :**

**The spin motion of the two electrons of the same orbital around their own axis arises a magnetic field in two opposite directions ( spin-paired state ) () .**

**Orbitals have three different possibilities depending on the number of electrons located in them as follows :**

**Although the two electrons of the same orbital carry the same negative charge , they do not repel with each other , because the magnetic field arises from the spinning of one electron is in a direction opposing the direction of the other magnetic field arising from spinning of the other electron and that decreases the repulsive force between the two electrons .**

**Summary of the relationship between the principal energy level , sub-levels and orbitals .**

**Each principal energy level :**

**Each sublevel consists of a number of orbitals which equals ( 2l + 1 ) .**

**Each orbital occupied by 2 electrons : ( s = 2 , p= 6 , d = 10 , f = 14 ) .**

**Principles of distributing electrons **

**Paulis exclusion principle**

**Aufbau ( building-up ) principle **

**Hunds rule**

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