Question

Maximum number of electrons in a subshell or an atom is determined by the formula:
A.$2{n^2}$
B.$4l + 2$
C.$2l + 2$
D.$4l$

Answer 1

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.

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.
Now we know that the maximum number of subshells is equal to $2l + 1$ and the number of electrons in one subshell is equal to $2$. So the maximum of electrons in a subshell or an atom is $2(2l + 1) = 4l + 2$.

Hence option B is correct.

Note:
For d-shell there are five subshells as 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.