Question

Number of electrons with$+ \dfrac{1}{2}$ spin in $n = 3$ is:
A. 18
B. 9
C. 3
D. 1

Answer 1

The set of numbers used to describe the position and energy of the electron in an atom are called quantum numbers. There are four quantum numbers, namely, principal, azimuthal, magnetic and spin quantum numbers. The values of the conserved quantities of a quantum system are given by quantum numbers.

Complete step by step answer:
In quantum mechanics, the principal quantum number ($n$ ) is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable. Apart from the principal quantum number, the other quantum numbers for bound electrons are the azimuthal quantum number ($l$ ), the magnetic quantum number ($m$ ), and the spin quantum number ($s$ ).
The principal quantum number describes the electron shell, or energy level, of an electron. The value of $n$ ranges from 1 to the outermost shell containing the valence electron of that atom. Thus, for a value of $n = 3$, the shell consists of $2{n^2}$ electron (according to the Bohr’s principle). The sub shells that are present in the shell in which $n = 3$are: $3s,3p$ and $3d$. In these sub-shells, the $3s$ subshell consists of two electrons out of which one must be in $+ \dfrac{1}{2}$ spin and the other must have $- \dfrac{1}{2}$ spin. Similarly, in the $3p$ sub-shell, there are six electrons out of which there are 3 electrons with $+ \dfrac{1}{2}$ spin and three electrons with $- \dfrac{1}{2}$ spin. In the same way, there are ten electrons in the $3d$ sub-shell out of which five are in $+ \dfrac{1}{2}$ spin and the rest five are in $- \dfrac{1}{2}$ spin.
Thus, the total number of electrons with $+ \dfrac{1}{2}$ spin in $n = 3$ is 9.

Therefore, the correct option is B

Note:
An electron in the orbital of an atom can only have two spins: $+ \dfrac{1}{2}$ and $- \dfrac{1}{2}$ . The spin quantum number is a quantum number that describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle. The spin quantum number is designated by the letter s, and is the fourth of a set of quantum numbers (the principal quantum number, the azimuthal quantum number, the magnetic quantum number, and the spin quantum number), which completely describe the quantum state of an electron.