n the basis of quantum numbers, justify that the sixth period of the periodic tableshould have 32 element

The sixth period of the periodic table corresponds to the filling of the 6th energy level of an atom.

1. The principal quantum number (n) for this period is 6.

2. According to the quantum mechanical model of the atom, each energy level can hold a maximum of 2n^2 electrons, where n is the principal quantum number.

3. So, for the 6th energy level, it can hold a maximum of 2*(6^2) = 72 electrons.

4. However, not all of these 72 electron spaces are filled in the 6th period. This is because the filling of electron spaces also depends on the other quantum numbers - the azimuthal quantum number (l), the magnetic quantum number (m), and the spin quantum number (s).

5. In the 6th period, the following sublevels are filled: 6s, 4f, 5d, and 6p. The 6s, 5d, and 6p sublevels can each hold a maximum of 2 electrons (for s) and 10 electrons (for d and p), and the 4f sublevel can hold a maximum of 14 electrons.

6. Therefore, the total number of electrons that can be accommodated in the 6th period is 2 (for 6s) + 14 (for 4f) + 10 (for 5d) + 6 (for 6p) = 32 electrons.

7. Since each element is defined by its atomic number, which is the number of protons in its nucleus, and since in a neutral atom the number of protons is equal to the number of electrons, the 6th period of the periodic table should have 32 elements.

This is how quantum numbers justify that the sixth period of the periodic table should have 32 elements.