**Electron Cloud -**We know that before the introduction of the electron cloud, de Broglie justified electron dual nature. He developed the relationship between wave nature as well as the particle nature of an atom.

Heisenberg has stated that it is not possible to measure simultaneously the position and momentum (velocity) of the electron with absolute accuracy.

In 1926, Erwin Schrodinger developed an atomic model taking into account both the wave and particle nature of the electron. This is known as the quantum (or wave) mechanical model of an atom. This is also known as quantum mechanics.

## Electron Cloud

- The probable electron cloud density corresponds to ψ (psi)
^{2}.

- Electron density value is high near the nucleus of the atom and it decreases as the distance of the electron from the nucleus increases.

- The probability does not become zero even at large distances from the nucleus although it may become negligible.

- Therefore, it is not possible to draw a boundary that will enclose the region of 100% probability.

However, for the sake of simplicity, one may draw an arbitrary boundary that may enclose the region in which the probability of finding the electron density is the maximum (about 90 to 95%).

**What is an Electron Cloud**The region in space around the nucleus where there is the maximum probability of finding the electron density is called orbital and the association of electrons is an electron Cloud. Thus, an orbital may be defined as :

Electron Cloud |

The region in the space around the nucleus where the probability of finding the electron is the maximum. The intensity of dotting in the electron cloud corresponds to probable positions of the electron. It may be noted that the spherical picture of the electron orbital is only a general representation. The electron cloud shape differs in their shapes and sizes depending upon the energy associated with them.

**Electron Cloud Definition**The probability of finding an electron at a point within an atom is proportional to the square of the wave function ψ (psi)^2. It is also called probability density and is positive. From the value of at different points within an atom, one can predict the region around the nucleus where the electron will be most probably found or located.

The electron cloud can be more clearly explained with Schrodinger's justification. Schrodinger's wave equation gives a better picture of the atom than the Bohr's Model.

### Electron Cloud Model

This model atom popularly known as Quantum Mechanical Model of Atom has the following important consequences.

- The energy of electrons in an atom is quantized i.e., it can have only certain specific values.

- The quantized energy levels in which electrons can be present are obtained from the solution of the Schrodinger wave equation.

- As pointed by Heisenberg's uncertainty principle, both the exact position and the exact velocity (or momentum) of an electron cannot be determined simultaneously. The path of the electron is only probable and not exact. This has ultimately led to the concept of atomic orbitals.

- The atomic orbital is represented by the wave function ψ (psi). which is also known as an orbital wave function. Since a number of such waves functions are possible for an electron, the corresponding atomic orbitals are also possible. In each orbital, the electron has a definite energy and it cannot have more than two electrons.

#### Cloud Model

The Schrodinger wave equation defined with the help of four constants. These constants are called quantum numbers. These are the set of four numbers that gives the complete information(address, energy, etc.) of the electron in an atom. These are designated as principal quantum number(n), azimuthal or secondary or angular momentum(l)and the magnetic quant. number(ml).In addition to these, the fourth Quan. number called spin quantum which represents the spin of the electron.Electron Cloud Shapes |

The four Index numbers which characterize the probability of location and energy of each electron in an atom.

These are:

- Principal quantum number (n).
- Azimuthal or angular momentum or subsidiary or orbital quant. number.
- Magnetic quant. number (m).
- Spin quant. number(s)

With the help of these quantum numbers(Quan.no), we can specify the position and energy of an electron, size shape and orientation of the orbital to which a particular electron belongs.

**Principal Quantum Number**

This number gives an idea of the major energy level in which the electron is present, it also gives the average distance of the electron from the nucleus, i.e., it determines the size of the orbital. In terms of wave mechanics, it gives the effective volume of the electron cloud. It is denoted by letter n which can have integral values excluding zero such as 1,2,3,4,…etc.

- If n = 1 it means the electron is present in first energy level, i.e., K-shell.
- n = 2 it means the electron is present in the second energy level, i.e., L-shell.
- n = 3 it means the electron is present in third energy level, i.e., M-shell and so on.

Sl. No. |
Energy level or Orbit (shell) |
Principal number ‘n’ |
Maximum Number of electrons (2n^{^2}) |

1 |
K |
1 |
2×1^{2}=2 |

2 |
L |
2 |
2×2^{2}=8 |

3 |
M |
3 |
2×3^{2}=18 |

4 |
N |
4 |
2×4^{2}=32 |

This quantum number gives the following information.

- It gives the average distance of the electron from the nucleus. Thus it determines the size of the orbital.
- The maximum number of electrons in any principal shell is given by 2n
^{2}, where n is the principal quantum number. - All the electrons having the same value of principal quantum numbers, however, do not have exactly the same energy.
- It helps to determine the energy of single-electron atoms such as hydrogen. For multi-electron species, the energy of an electron is determined by using the relation.

**Azimuthal or Angular Momentum Quantum Number(l)**This quantum number gives the following information.

- It tells the sub-shell or sub energy level or orbital in which the electron is present.
- It gives the angular kinetic energy associated due to the angular momentum of the electron.
- It gives the shape of the orbital in which the electron is located.

This quantum number is represented by letter l. For a given value of “n”, I can have values starting from l= 0 to l = n – 1 so that the total value of l is equal to the principal quantum number (=n).

- If n = 1, l can have only one value, i.e., l = 0.
- n = 2, I can have two values, i.e., l =0 and l = 1.
- n = 3, I can have three values, i.e., l = 0, 1 = l and l = 2.
- n = 4, I can have four values, i.e., l = 0, l = 1, 1 = 2 and l = 3.

Different values of l correspond to different subshells. The various subshells are designated as s, p, d, and f as shown below:

Value of l |
Subshell of orbital |

0 |
s (sharp) |

1 |
p (principal) |

2 |
d (diffuse) |

3 |
f (fundamental) |

- 1st energy level (n = 1) contains one subshell corresponding to l = 0, i.e., s-subshell.
- 2nd energy level (n = 2) contains two subshells corresponding to l = 0, (s-subshell) and l = 1 (p-subshell).
- 3rd energy level (n = 3) contains three subshells corresponding to l = 0, (s-subshell) and I = 1 (p-subshell) l = 2 (d-subshell).
- 4th energy level (n = 4) contains four subshells corresponding to l = 0, (s-subshell) ,l = 1 (p-subshell), l = 2 (d-subshell), l = 3 (f-subshell) and so on.

value of n |
l = n – 1 |
subshell (orbital shape) |
No. orbitals = 2l + 1 |
---|---|---|---|

1 |
0 |
s subshell |
1 (1 x s orbitals) |

2 |
1 |
p subshell |
3 (3 x p orbitals) |

3 |
2 |
d subshell |
5 (5 x d orbitals) |

4 |
3 |
f subshell |
7 (7 x f orbitals) |

For a given principal quantum number, the energies of the various subshells are in the order s < p <d< f. Thus an electron in s-subshell having the same value of n.

**Magnetic Quantum Number**An electron due to its angular motion around the nucleus generates magnetic fields that can interact with the external magnetic field. Under the influence of the external magnetic field, the electrons of a subshell can orient themselves in a certain specified region of space around the nucleus called orbital.

Thus magnetic Quan. number gives the following information:

- Magnetism generated due to the angular motion of the electron.
- This represents the number of orbitals in any subshell.
- This Quan. number is represented by m. It can have all the values from -l to +l through zero so that for each value of l, m has (2l+1)values.

- If l=0, m can have only one value, i.e., m=0 it means (s-subshell) has only one orientation of the electron in space, i.e., s-subshell has one orbital.
- l=1, m can have three values, i.e., -1, 0 and +1. It means p-subshell has three orbitals.
- L =2m can have five values, i.e. -2,-1, 0, +1,+2.

(d-subshell) means d-subshell has five orientations of the electron in space, i.e., d subshell has the orbitals.

Subshell |
Orbital or Azimuthal quantum number (l) |
Number of Orbital 3l + 1 |
Magnetic number (m or m_{l}) |

s |
0 |
1 |
0 |

p |
1 |
3( p_{x},p_{y},p_{z}) |
-1, 0, + 1 |

d |
2 |
5 (d_{x2-y2},d_{z2} ,d_{xy},d_{xz},d_{yz}) |
– 2, -1, 0, + 1, + 2 |

f |
3 |
7(f_{z3} ,f_{xz2},f_{xyz},f_{x(x2-3y2)},f_{yz2}, f_{z(x2-y2)},f_{y(3x2-y2)}) |

**Spin Quantum Number**The fourth quantum number arises from the spectral evidence that an electron in its motion around the nucleus also rotates or spins about its own axis. The spinning of an electron produces a magnetic field. Thus it behaves like a tiny bar magnet and consequently, it has spin angular momentum. This Quan. number gives the following information:

- Contribution of spin angular momentum to the total angular momentum of the electron.
- Spin orientation of the electron around its own axis.
- This Quan.number is represented by s. As an electron can spin either clockwise or anticlockwise, it can have only two values.
- Since the Quan. number can differ from one another by integers, s can have either +1/2 and -1/2. values depending upon the direction of spin. These values are chosen so as to be equal and opposite in sign and differ by unity. The two values are sometimes indicated by putting an arrowhead pointing upwards (↑) or downward (↓).

It should be noted that an electron can spin either clockwise but not in both directions at the same time.

Thus the four Quan. numbers characterize completely the address of an electron in a given atom. They give its position in the major energy level (n), the sub-energy level (l), the orbital (m) and the directions of its spin (s).

**Quantum Number Rules**- The maximum number of electrons in each principal shell(n) is given by 2n
^{(square)2}. - The Maximum number of orbitals in each principal shell is n
^{2}(square). - s-subshell has only one orbital with a maximum of two electrons.
- p-subshell has three orbital with a maximum of six electrons.
- d-subshell has five orbital with a maximum of 10 electrons.
- f-subshell has seven orbital with a maximum of 14 electrons.

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