Question

The electronic configuration of $Eu$ (atomic number 63) , $Gd$ (atomic number 64), $Tb$ (atomic number 65) are:
A. $\left[ {Xe} \right]4{f^7}6{s^2},\left[ {Xe} \right]4{f^8}6{s^2}$ and $\left[ {Xe} \right]4{f^8}5{d^1}6{s^2}$
B. $\left[ {Xe} \right]4{f^7}5{d^1}6{s^2},\left[ {Xe} \right]4{f^7}5{d^1}6{s^2}$ and $\left[ {Xe} \right]4{f^9}6{s^2}$
C. $\left[ {Xe} \right]4{f^6}5{d^1}6{s^2},\left[ {Xe} \right]4{f^7}5{d^1}6{s^2}$ and $\left[ {Xe} \right]4{f^8}5{d^1}6{s^2}$
D. $\left[ {Xe} \right]4{f^7}6{s^2},\left[ {Xe} \right]4{f^7}5{d^1}6{s^2}$ and $\left[ {Xe} \right]4{f^9}6{s^2}$

Answer 1

Hund’s rule: It is defined as every orbital in a subshell is singly occupied with one electron before any one orbital is doubly occupied i.e. completely filled. And all the orbitals have singly occupied electrons with the same spin.
Aufbau principle: In this principle, atomic orbitals which have lowest energy are filled first after that the orbitals with high energies are filled.
Pauli’s exclusion principle: It states that for any electrons the values of all the four quantum numbers cannot be the same. They differ in at least one quantum number.

Complete step by step solution:
Let us first talk about quantum numbers.
Quantum number: It is defined as the set of numbers which describes the position and energy of electrons in an atom. There are four quantum numbers: principal, azimuthal, magnetic and spin quantum numbers.
Principle quantum number: It is defined as the quantum number which describes the electron’s state. It is represented by $n$. It’s value starts from $1$.
Azimuthal quantum number: It is defined as a quantum number which describes the shape of the orbital and its orbital angular momentum. It is represented by $l$. It’s value is from $0$ to $(n - 1)$.For $s$, $l = 0$ for $p{\text{ l}} = 1$ and so on.
Magnetic quantum number: It is defined as the quantum number which describes the orientation in shape of orbitals. It is represented by $m$. Its value is from $- l$ to $l$. They generally represent the subshell of the orbitals. For example: For s shell $l = 0$. So the value of $m = 0$. Hence there is only a subshell for s-shell. Similarly for p shell $l = 1$. So the value of $m$ can be $- 1,0,1$. Hence there will be three subshells for the p-shell. In general the number of subshells is equal to $2l + 1$.
Spin quantum number: It describes the angular momentum of the electron. Spin quantum numbers have two values $+ \dfrac{1}{2}$ or $- \dfrac{1}{2}$. At a time electrons can have one spin value.
Degenerate orbitals: Those orbitals of the same subshell which have the same energies, are known as degenerate orbitals. For example: In $2p$ shell there are three subshells as $2{p_x}, 2{p_y}, 2{p_z}$. They have the same energy. So we can say that degenerate orbitals have the same principal quantum number and azimuthal quantum number.
The rules to fill the electrons in the orbits are as:
Hund’s rule: It is defined as every orbital in a subshell is singly occupied with one electron before any one orbital is doubly occupied i.e. completely filled. And all the orbitals have singly occupied electrons with the same spin.
Aufbau principle: In this principle, atomic orbitals which have lowest energy are filled first after that the orbitals with high energies are filled.
Pauli’s exclusion principle: It states that for any electrons the values of all the four quantum numbers cannot be the same. They differ in at least one quantum number.
Now here we are given with the atom having atomic numbers as $63,64,65$. Now the atomic number of noble gas which comes before these atoms is xenon having atomic number $54$.
Now, first $54$ electrons of all these atoms will be filled as xenon. So we write [Xe ] in the beginning of each electronic configuration. The elements in the given question are lanthanides. So there will be electrons in the f-orbitals. After that for atomic number $63$ we are left with $9$ electrons. So the next two electrons will go into the 6s-orbit and then the next seven electrons will go into 4d-orbits. Hence the electronic configuration of an element having atomic number $63$ will be $\left[ {Xe} \right]4{f^7}6{s^2}$ because half filled orbitals are more stable. And similarly the element having atomic number $64$ will be as $\left[ {Xe} \right]4{f^7}5{d^1}6{s^2}$ and the lastly the element having atomic number $65$ will be like $\left[ {Xe} \right]4{f^9}6{s^2}$.

Hence, option (D) is correct.

Note: For d-shell there are five subshells as the value of azimuthal quantum number $l$ is $2$. So total number of magnetic quantum numbers i.e. $m = 2l + 1 = 5$. They are as: ${d_{xy}}, {d_{yz}}, {d_{xz}}, {d_{{x^2} - {y^2}}}, {d_{{z^2}}}$. They all have the same principal and azimuthal quantum number but have different magnetic and spin quantum numbers.