Question

On the basis of quantum numbers, justify that the sixth period of the periodic table should have 32 elements.

Answer 1

Recall the significance of principle quantum number ($n$) and azimuthal quantum number ($l$). A period indicates the value of $n$. For a value of $n$, values of $l$ range from $0\;{\text{to}}\;n - 1$. For each subshell, orbitals are ($2l + 1$). So, find the total number of orbitals available in the sixth period. Each orbital can accommodate a maximum two electrons.

Complete step by step answer:
Principle quantum number ($n$) determines the shell in which an electron is present. Azimuthal quantum number ($l$) determines the subshell in a principal energy shell to which an electron belongs.
A period number in the periodic table of elements indicates the value of $n$ for the outermost shell. Therefore, the value of $n$ for the sixth period is 6. Since, we know for a given value of $n$, $l$ have values ranging from $0\;{\text{to}}\;n - 1$. Thus, for $n = 6$, $l$ can have values: $0,\;1,\;2,\;3,\;4$.
For each value of $l$, there is a corresponding subshell assigned and there are ($2l + 1$) orbitals in each subshell. Hence, the total number of orbitals in each subshell are as follows:

Value of $l$	0	1	2	3
Subshell notation	$s$	$p$	$d$	$f$
Number of orbitals ($2l + 1$)	1	3	5	7

According to Aufbau’s principle, electrons are filled in different orbitals in order of their increasing energies. For the sixth period, electrons can fill in only $6s,4f,5d{\text{ and }}6p$ sub-shells. Now, using the above table,$6s$ has one orbital, $4f$ has seven orbitals, $5d$ has five orbitals and $6p$ has 3 orbitals. Hence, the total number of orbitals available are: $1 + 7 + 5 + 3 = 16$.
According to Pauli’s exclusion principle, each orbital can accommodate a maximum of only 2 electrons. Thus, 16 orbitals can accommodate a maximum 32 electrons. Hence, we justified that the sixth period of the periodic table should have 32 elements.

Note: Electrons first occupy the lowest energy orbital available to them and then enter into the higher energy orbitals. The order in which the energies of orbitals increase and hence the order in which the orbitals are filled is as follows:
$1s,\;2s,\;2p,\;3s,\;3p,\;4s,\;3d,\;4p,\;5s,\;4d,\;5p,\;4f,\;5d,\;6p,\;7s...$