Question

1. provide the complete set of quantum numbers for all the electrons that could populate the 3d subshell of an atom.

Explanation:

Step1: Determine the principal quantum number ($n$)

The principal quantum number $n$ is related to the shell. For the $3d$ sub - shell, $n = 3$.

Step2: Determine the azimuthal quantum number ($l$)

The azimuthal quantum number $l$ for $d$ sub - shell is $l=2$.

Step3: Determine the magnetic quantum number ($m_l$)

The magnetic quantum number $m_l$ for a given $l$ has values $m_l=-l,-l + 1,\cdots,l$. For $l = 2$, $m_l=-2,-1,0,1,2$.

Step4: Determine the spin quantum number ($m_s$)

The spin quantum number $m_s$ can have two values, $m_s=+\frac{1}{2}$ or $m_s=-\frac{1}{2}$ for each combination of $n$, $l$, and $m_l$.

Answer:

The set of quantum numbers are:
$(n = 3,l = 2,m_l=-2,m_s=+\frac{1}{2})$, $(n = 3,l = 2,m_l=-2,m_s=-\frac{1}{2})$,
$(n = 3,l = 2,m_l=-1,m_s=+\frac{1}{2})$, $(n = 3,l = 2,m_l=-1,m_s=-\frac{1}{2})$,
$(n = 3,l = 2,m_l=0,m_s=+\frac{1}{2})$, $(n = 3,l = 2,m_l=0,m_s=-\frac{1}{2})$,
$(n = 3,l = 2,m_l=1,m_s=+\frac{1}{2})$, $(n = 3,l = 2,m_l=1,m_s=-\frac{1}{2})$,
$(n = 3,l = 2,m_l=2,m_s=+\frac{1}{2})$, $(n = 3,l = 2,m_l=2,m_s=-\frac{1}{2})$