Introduction:

The periodic table is organized based on the electronic configuration of elements. Each period represents a new energy level or shell, and each group or column represents the same number of valence electrons. The sixth period of the periodic table should have 32 elements based on the principles of quantum numbers.

Explanation:

1. Energy Levels and Shells:
- The quantum mechanical model of the atom describes electrons as existing in specific energy levels or shells.
- Each energy level has a maximum capacity to hold a certain number of electrons.
- The number of energy levels increases as we move down the periodic table, with each new period representing a new energy level.

2. Quantum Numbers:
- Quantum numbers describe the properties and behavior of electrons in an atom.
- The principal quantum number (n) determines the energy level or shell in which an electron resides.
- The maximum number of electrons in a shell is given by 2n^2, where n is the principal quantum number.
- For the sixth period, the principal quantum number (n) would be 6.

3. Maximum Number of Electrons:
- Using the formula 2n^2, we can determine the maximum number of electrons in the sixth energy level.
- For n = 6, the maximum number of electrons in the sixth energy level is 2 * 6^2 = 72.
- However, the sixth period consists of two sub-levels: 6s and 4f.

4. Filling of Sub-levels:
- The 6s sub-level can hold a maximum of 2 electrons, and the 4f sub-level can hold a maximum of 14 electrons.
- The electrons fill the 6s sub-level before moving to the 4f sub-level.
- Therefore, the maximum number of electrons in the sixth period after accounting for the sub-levels would be 2 + 14 = 16.

5. Filling of Orbitals:
- Each sub-level consists of a specific number of orbitals.
- The 6s sub-level has 1 orbital, while the 4f sub-level has 7 orbitals.
- Each orbital can hold a maximum of 2 electrons.

6. Calculation of Total Elements:
- To calculate the maximum number of elements in the sixth period, we need to consider the number of orbitals in the 6s and 4f sub-levels.
- The 6s sub-level can accommodate 1 * 2 = 2 electrons.
- The 4f sub-level can accommodate 7 * 2 = 14 electrons.
- Therefore, the total number of elements in the sixth period would be 2 + 14 = 16.

Conclusion:

Based on the principles of quantum numbers and the filling of sub-levels and orbitals, the sixth period of the periodic table should have 32 elements. However, due to the unique electronic configuration of elements in this period, the maximum number of elements is limited to 16. This understanding of electron configuration and quantum numbers helps in predicting the properties and behavior of elements within the periodic table.