The principal quantum number is related to the energy and size of the atomic orbits. It is denoted by ‘n’ and has integral values. As n increases, the electronic cloud (orbital) becomes larger and the electron spends more time further away from the nucleus. All orbitals that have the same value of principal quantum number ‘n*’ *are said to be in the same energy level or shell.

Prerequisites |

Bohr atomic model |

Quantum numbers |

When we solve the Schrodinger wave equation (H𝛹 = E𝛹), we find a number of wave functions that describe the probability of finding electrons at different energy levels around the nucleus. Each of these orbitals is characterized by a series of numbers called quantum numbers*. *

A quantum number is an integer that labels the state of a system. It also describes the different properties of orbitals. Every electron in an atom will have its unique set of four quantum numbers in the form of n, ℓ, m_{ℓ}, and m_{s}*.*

- Principle quantum number (n)
- Azimuthal quantum number (ℓ)
- Magnetic quantum number (m
_{ℓ}) - Spin quantum number (m
_{s})

## Principle quantum number (n)

Principal Quantum Number (n) specifies the energy of an electron and the size of the orbital in an atom. It has integral values i.e. n = 1, 2, 3,^{…}, n. The larger value of ‘n’ corresponds to the average distance of an electron in the orbital from the nucleus.

On the basis of principal quantum number, the main shells of electrons in the periodic table are labeled as

K (n=1)

L (n=2)

M (n=3)

N (n=4)

The energy of an electron in an orbital with a quantum number *n *is given by the equation:

E_{n} = Z^{2}μe^{4} / 32ε^{2}_{o}ћ^{2}n^{2}

E_{n} = -2.178 x 10^{-18} (Z^{2}/n^{2}) J

An increase in principle quantum number (n) also means higher energy, because an electron spends more time farther from the nucleus. It shows that the electrons are less tightly bound to the nucleus and their energy is relatively increased.

**Number of orbits and electrons in a shell**

The total number of orbitals for a given value of principal quantum number (n) is n^{2}. The maximum number of electrons possible in a given shell is 2n^{2}.

### Principle energy levels

From the discoveries of the Hydrogen emission spectrum and the photoelectric effect, Neil Bohr proposed a new model of the atom in 1915. According to his postulates, an electron orbits around the nucleus at fixed energy levels. Moving away from the nucleus increases the energy of electrons. This model is also referred to as the planetary model. These energy levels are denoted by principal quantum numbers.

In short, principal quantum numbers determine all of the quantum numbers for a particular set of orbits.

**Related Resources**

- The Hamiltonian operator
- Quantum mechanical operators
- Singlet vs. triplet state
- Time independent Schrodinger wave equation

## Azimuthal Quantum Number (ℓ)

In relation to the principal quantum number, the angular momentum quantum number or azimuthal quantum number (ℓ) is related to the shape of atomic orbitals. It has integral values ℓ** **= 0, 1, 2,^{…}, n – 1 for each value of n. This quantum number divides the shells into smaller groups of orbitals, called subshells. For a particular orbital, the value of ℓ** **is commonly assigned by a letter.

Value of ℓ | Letter |

0 | s |

1 | p |

2 | d |

3 | f |

4 | g* |

5 | h* |

g^{* }and h^{* }are not used in the ground state of any known element.

## Magnetic Quantum Number (m_{ℓ})

The magnetic quantum number (m_{ℓ}) is related to the orientation of the orbital in space relative to other orbitals in the atom. It has integral values of 0, ±1, ±2, ±3, and so on. The number of m_{ℓ} values is also called degeneracy (the number of orbitals in an atom that is at the same energy level). There are (2ℓ+1) magnetic quantum numbers containing orbitals present in each subshell.

## Spin Quantum Number (m_{s})

The spin quantum number (m_{s}) is related to the spin an electron occupies in an orbital. It has integral (or half-integral) values of +½ and -½. +½ designates the up spin and -½ shows the down spin.

According to Pauli’s exclusion principle, no two electrons in the same atom can have the values of all quantum numbers identical. It means two electrons in the same orbitals must have opposite spins.

- The principal quantum number ‘n’ is associated with the energy and size of orbitals and has integral values i.e. n = 1, 2, 3,…
- The azimuthal quantum number (ℓ) associates with the shape of orbitals and has the values of 0, 1, 2, and 3. The alphabets assigned for these azimuthal quantum numbers are s, p, d, and f respectively.
- The magnetic quantum number is associated with energy levels within a subshell and has different values related to the azimuthal quantum number by the rule (2ℓ+1).
- The spin quantum number is associated with the orientation of electrons in an orbital and only has two values, +½, and -½.

## Key takeaways

## Concepts Berg

**What are the possible subshells when n = 4? How many electrons are contained by each of these subshells?**

For a given value of n possible subshells are n². So, for n=4 there are 16 possible subshells. n=4 means ℓ=0, 1, 2, 3, it includes s, p, d, f which can contain a maximum of 2, 6, 10, and 14 electrons respectively. Another formula (2n²) shows a total of 32 electrons for n=4 as well.

**What are the principal energy levels?**

An electron’s principal energy levels refer to the shells or orbitals where these electrons reside around the nucleus. These levels are denoted by ‘n’ and known as principal quantum numbers.

**Which energy level has the least energy?**

The principal energy levels for electrons are described as K, L, M, N, and so on. Among these electronic levels, the K shell has the least energy.

**How do you find the principal quantum number?**

In the periodic table, the period number shows the principal quantum number. There are a total of 7 periods in the periodic table so the maximum value of the principal quantum numbers yet known is 7.

**What is quantum energy?**

Quantum energy is a discrete amount of energy possessed by any system. Max Planck postulated that the energy is quantized and could be emitted or absorbed in integral multiples of small units of energy known as quantum.

E = hƲ

The energy is proportional to the frequency of the radiation and (h) is Planck’s constant having a value of 6.62 × 10^{-34} Js.

**What are quantum numbers?**

Quantum numbers are integers that label the state of the system. It describes the different properties of orbitals. Every electron in an atom will have its unique set of quantum numbers in the form of n, ℓ, m_{ℓ}, or m_{s}.

**What is the spin of an electron?**

The spin of an electron is a property of an electron related to the spin quantum number. The spin quantum number is associated with the spin angular momentum of an electron. According to Pauli’s exclusion principle, no two electrons can reside at a single point with the same spins.

**What are the 4 quantum numbers?**

- Principle quantum number (n)
- Azimuthal quantum number (ℓ)
- Magnetic quantum number (m
_{ℓ}) - Spin quantum number (m
_{s})

**Who proposed the principal quantum number?**

Niel Bohr proposed the principal quantum number (n) in 1913 by explaining the spectrum of the hydrogen atom.

**What are the possible values of the principal quantum number (n)?**

The possible values for the principal quantum number are 1, 2, 3, 4, and so on but can never be zero.

**What does the azimuthal quantum number determine?**

The azimuthal quantum number determines the angular momentum of the orbit and describes the shape of the orbital.

ℓ = 0 means s subshell

ℓ = 1 means p subshell

ℓ = 2 means d subshell

ℓ = 3 means f subshell.

**Does the principal quantum number determine the energy of an orbital?**

According to the formula from hydrogen atoms derivation, it shows the quantitative value of energy associated with a principal quantum number (n).

E = -2.178 x 10^{-18} (Z^{2}/n^{2}) J

**What three quantum numbers are associated with the 4d orbital?**

For a 4d orbital, three quantum numbers are associated as:

The value of principal quantum number (n) = 4 because it says itself.

The value of azimuthal or orbital quantum number (ℓ) = 2 for d orbital.

By (2ℓ + 1) rule, the value of magnetic quantum number (m_{ℓ}) = 2 has the possible values of 0, ±1, ±2.

**What does the magnetic quantum number determine?**

The magnetic quantum number describes the available energy levels within a subshell and determines the orientation of orbital angular momentum along a specified axis.

**What are the principal energy levels?**

The principal energy levels of an electron refer to the orbit or shell in which the electrons reside. The levels are denoted by the principal quantum number ‘n’.