Question

What is a valid set of quantum numbers for the outermost electron of an aluminium atom?

Answer 1

Hint: Quantum numbers are a set of four numbers that are used to describe the state of an electron in an atom. These four quantum numbers are principal quantum number, azimuthal quantum number, magnetic quantum number, and spin quantum number. According to the Pauli exclusion principle, no two electrons can share the same set of four quantum numbers.

Complete answer:
The values obtained from solving the Schrodinger wave equation that is acceptable by the wave equation for hydrogen atoms are known as quantum numbers. There are a total of four quantum numbers that describe the probable position and energy of an electron in an atom.
These four quantum numbers are briefly described below:
1 – Principal quantum number (n): It designates the principal shell and energy of an electron. It also describes the size of the orbital as it is the most probable distance of electrons from the nucleus. Its value can be any positive integer. For example, for an electron present in a 2p subshell, the value of n is 2.
\[\text{n}=1,2,3.....\]
2 – Azimuthal quantum number (l): It denotes the subshell of an orbital with a particular principal quantum number in which an electron is present and it also specifies the shape of the orbital. The subshell with n=3 and l=0 is the 3s subshell. A letter is used to identify each value of l.
\[\begin{align}
  & l\text{ 0 1 2 3 4 5 }.... \\
 & \text{Letter s p d f g h }.... \\
\end{align}\]
3 – Magnetic quantum number $\left( {{\text{m}}_{\text{l}}} \right)$ : It tells the spatial orientation of an orbital of particular energy (n) and shape (l). its value is related to the value of the azimuthal quantum number. Its value range is given as:
\[{{\text{m}}_{l}}=-l,\text{ }\left( -l+1 \right),....,-1,0,1,....\left( l-1 \right),+l\]
4 – Spin quantum number $\left( {{\text{m}}_{\text{s}}} \right)$ : It specifies the spin orientation of an electron. Since an electron can spin in only one of two directions (up and down), the value of this quantum number can be:
\[{{\text{m}}_{\text{s}}}=+\dfrac{1}{2}\text{ or }-\dfrac{1}{2}\]
Now, the electronic configuration of aluminium is:
\[\text{Al}(13)-1{{\text{s}}^{2}}2{{\text{s}}^{2}}2{{\text{p}}^{6}}3{{\text{s}}^{2}}3{{\text{p}}^{1}}\]
Here, the outermost electron is present in the 3p subshell. So the possible and valid values of quantum numbers corresponding to this electron are:
\[\begin{align}
  & n=3 \\
 & l=1 \\
 & {{m}_{l}}=-1 \\
 & {{m}_{s}}=+\dfrac{1}{2} \\
\end{align}\]

Note:
Two electrons of the same atom can't have exactly the same quantum state or exactly the same values of the quantum numbers because an orbital cannot occupy more than two electrons and also two electrons in an orbit must always have opposite spins to minimize repulsion.