# Quantum Number

**Origin of Quantum Number:** In 1926, Erwin Schrodinger developed an atomic model taking into account both the wave and particle nature of the electron. This is known as quantum (or wave) mechanical model of an atom. The wave mechanics is also called Quantum Mechanics. It describes the electron as a three-dimensional wave in the electronic field of the positively charged nucleus. The wave motion of the electron in this field is described with the help of an equation known as **Schrodinger wave equation.**

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## What is Quantum Number?

The Schrodinger wave equation defined with the help four constants. **These constants are called as quantum numbers.** These are the set of four numbers which 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(m_{l}).In addition to these, the fourth Quan. number called spin quantum which represents the spin of the electron.

## Quantum Number Definition

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 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 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) |

Thus we can say that:

- 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 which 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 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)}) |
-3, – 2, – 1, 0, + 1, + 2, + 3 |

## 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. 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
^{2}. - The Maximum number of orbitals in each principal shell is n
^{2}. - s-subshell has only one orbital with the maximum of two electrons.
- p-subshell has three orbital with maximum of six electrons.
- d-subshell has five orbital with the maximum of 10 electrons.
- f-subshell has seven orbital with maximum of 14 electrons.

## Principal Quantum Number Chart

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