The quantum numbers and principles of distributing electrons
The mathematical solution of Schrodinger’s 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 ) : It describes the distance of the electron from the nucleus .
- The subsidiary quantum number ( l ) : It describes the shapes of electron cloud in the sub-levels .
- The magnetic quantum number ( ml ) : It describes the shape and the number of the orbital in which the electron exists .
- The spin quantum number ( ms ) : It describes the spin of the electron .
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 2n² ( 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 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 : ( n − 1 ) ] : 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 : ( n − 1 ) ] = [ 0 : ( 3 − 1 ) ] = 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 can’t 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 ( ml ) values when ( l = 2 ) ?
The probable ( ml ) values ranges between −l , …… , 0 , …….., + l
The probable ( ml ) values : −2 , −l , 0 , +1 , +2
Example : Which of the following probabilities of the quantum numbers of a certain electron includes a mistake ?
- n = 3 , l = 2 , ml = − l
- n = 4 , l = 3 , ml = − 2
- n = 1 , l = 1 , ml = + l
The choice ( c ) , because when ( n = 1 ) the probable l and ml values are ( 0 ) .
The spin quantum number ( ms )
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 :
- Clock wise ( ↑ ) with ms value equals to ( + ½ ) .
- Anticlockwise (↓) with ms value equals to ( − ½ ) .
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 :
- ( ) Empty orbital .
- ( ↑ ) Half-filled orbital contains one electron .
- ( ↑↓ ) full-filled orbital contains 2 pairing electrons that have opposite spins and called spin paired state .
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 :
- Consists of a number of sublevels equals its number ( n = no. of l values ) .
- Consists of a number of orbitals equals the number of the level ( n² ) .
- The number of electrons occupying it , equals two times the square of the level number ( 2n² ) .
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
There are some important rules , which must be considered in distributing electrons in the atom , These rules are :
- Pauli’s exclusion principle .
- Aufbau ( building-up ) principle .
- Hund’s rule .
Pauli’s exclusion principle
When ( np ) sublevel contains one electron , it is not necessary to be :
( ml = − l ) , but it may be equal 0 or + l
( ms = + ½ ) , but it may be equal − ½
Aufbau ( building-up ) principle
Aufbau ( building-up ) principle : Electrons occupy the sub-levels in the order of increasing their energy , the lowest energy sub-levels are filled firstly .
The sub-levels of the same principal energy level differ slightly from each other in energy .
Arrangement of sub-levels according to their energy depends on :
Sum of ( n + l ) , Ex : Energy of 4s sub-level is lower than that of 3d sub-level , because sum of ( n + l ) of 4s sub-level is less that of 3d sub-level .
Order of the principal energy level , in case of equality of sum ( n + l ) values for two sub-levels , Ex : Energy of 3p sub-level is lower than that of 4s sub-level , Because ( n ) value of 3p sub-level is lower than that of 4s sub-level .
The sequence of energy sub-levels is arranged ascendingly according to their energy follows
The order : 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5d < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p
Filling the energy sublevels : s up to 2 electrons , p up to 6 electrons , d up to 10 electrons , f up to 14 electrons .
The electron configuration for one of the energy levels can be expressed as the following : 2p6
Where , 2 is the principal energy level ( n ) , p is the energy sub-level , 6 is number of electrons in the orbitals of sub-level .
Example : write the electron configuration for the following elements , according to building-up principle : 11Na , 20Ca , 30Zn , 19K , 26Fe .
11Na : 1s² , 2s² , 2p6 , 3s1
20Ca : 1s² , 2s² , 2p6 , 3s² , 3p6 , 4s²
30Zn : 1s² , 2s² , 2p6 , 3s² , 3p6 , 4s² , 3d10
19K : 1s² , 2s² , 2p6 , 3s² , 3p6 , 4s1
26Fe : 1s² , 2s² , 2p6 , 3s² , 3p6 , 4s² , 3d6
Hund’s rule : No electron pairing takes place in a gives sub-level until each orbital contains one electron .
Rules of filling the energy sub-levels with electrons , according to Hund’s rule :
- The orbitals of the same sub-levels are equal in their energy .
- The orbitals of the same sub-levels are filled by the unpaired electrons firstly , The spin of single electron in the same direction gives the atom stability .
- When the two electrons are paired in one orbital , they have opposite spins , to decrease the repulsive force between them , then the two electrons are in a spin-paired .
- Electrons pairing takes place in the orbitals of the same sub-level after occupying all orbitals by unpaired electrons .
- The electrons prefers to be paired with another electron in one orbital of the same sub-level rather than being transferred to a higher energy sub-level .
The spin of single electrons in the same sub-levels orbitals is in the same direction , because this state gives the atom more stability .
The electron prefers to occupy an orbital alone in the same sub-level rather than pairing with another one in the same orbital , Because when two electrons are paired in one orbital , ( in spite of their opposite spins ) , there must be a repulsive force that decreases the stability of the atom ( increasing in its energy ) .
The electron prefers pairing with another one in orbital in the same sub-level rather than transferring to the higher energy sub-level , because the required energy to overcome the repulsive force between the two paired electrons is less than that is required for transferring the electron to a higher sub-level .
Example : Predict the atomic number for each of the following elements :
An element whose electronic configuration is 1s² , 2s² , 2p3 → 7
An element whose principal energy level ( n = 3 ) contains 3 electrons → The electronic configuration : 1s² , 2s² , 2p6 , 3s² , 3p1 , So , the atomic number = 13
An element whose last 3s-sublevel is half-filled with electrons → The electronic configuration : 1s² , 2s² , 2p6 , 3s1 , So , the atomic number = 11 .