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Principal Quantum Number

20th Jan 2020 @ 4 min read

Physical Chemistry

The principal quantum number is a set of positive integers that decide the size and energy of an orbital. It is denoted by n. Thus, n = 1, 2, 3, 4… It is one of the four quantum numbers that identify an electron in the atom; the others are the azimuthal quantum number, the magnetic quantum number, and the spin quantum number.

Bohr model

Since the principal quantum number was first originated in the Bohr model, let's try to understand it using the Bohr model.

In the Bohr model, an electron revolves the central nucleus in a circular path. These paths are called orbits. The orbits are fixed and discrete. Each orbit is assigned a positive number starting with one from the innermost orbit. These numbers are nothing but principal quantum numbers. For the first orbit, n = 1, for the second, n = 2, for the third, n = 3…

Principal quantum numbers represent shells.
The Bohr model

Each principal quantum number corresponds to a shell. Shells, unlike the quantum number, is represented alphabetically starting with K. Thus, n = 1 represents K shell, n = 2 represents L shell, n = 3 represents M shell, n = 4 represents N shell, and so forth.

In the beginning, we have mentioned the principal quantum number decides the size and energy of an electron in the atom. This can be seen from the equations below.

The radius of an atom according to the Bohr model

Here, rn is the radius of the atom with the atomic number Z. rn is directly proportional to the square of n; thus, the size of an atom increases with the principal quantum number.

In the equation below, En is the energy of an electron in an orbit. Although En is inversely proportional to the square of n, its value increases with n because of the negative sign.

The energy associated with the nth orbit

Both above formulae are applicable to the Bohr model, not the real-world quantum model, but the trend that the size and energy increase with the principal quantum number holds true.

Quantum model

In the quantum model, there are no orbits. Instead, we have orbitals, a probabilistic cloud of an electron. The value of n is still decisive in estimating the size and the energy of an electron. As n increases, the electron gets further from the nucleus.

Note: There are other factors also that affect the energy of an electron. However, the principal quantum number is to a great extent responsible. In hydrogen, the size and energy are solely determined by the value of n.

When an electron jumps from a higher principal quantum number to lower, it emits energy. And energy is absorbed when it ascends.

Relation with other quantities

No. of electrons

The maximum number of electrons in a shell is given by 2n2.

n = 1 can have utmost 2 electrons, n = 2 can have utmost 8 electrons, n = 3 can have utmost 18 electrons, and so on.

Azimuthal quantum numbers

The azimuthal quantum number l can have n values ranging from 0, 1, 2 … n − 1. Also, the azimuthal quantum number represents the number of subshells.

For n = 1, l = 0 and the number of subshells is 1. For n = 3, l = 0, 1, 2 and the number of subshells is 3.

Magnetic quantum numbers

The magnetic quantum number ml represents the number of orbitals. For a shell, ml can have n2 values, i.e. n2 orbitals.

Radial quantum numbers

The radial quantum number nr is related by the principal quantum number n by the equation below.

Radial quantum number

The radial quantum number represents the number of nodes, a point at which the probability of finding an electron is zero.

The above relations can be summarized in the table below.

Relation between the principal quantum number and other quantities
Principal quantum number (n) Shell Subshells (= n) Orbitals (= n2) Max electrons in a shell (= 2n2)
1 K 1 1 2
2 L 1 4 8
3 M 3 9 18
4 N 4 16 32
5 O 5 25 50

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Quantum Number Atomic Structure

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