Question

How many electrons in an atom have the following quantum numbers?
n=4, $m_3=-1/2$

• A. 32
• B. 18
• C. 8
• D. 16

Answer 1

The correct option is D 16

For $n^{th}$ orbital possible values of the azimuthal quantum number, l are from 0 to (n-1). Total of n values.

In $n=4$ and $l=0,1,2,3$

Magnetic quantum number $m_l$ have values from -l to +l and total of $2l+1$ values.

For n=4, possible values of l and $m_l$ are:

$m_l=0$ for l=0; total $m_l$ values =1

$m_l=-1,0,1$ for l=1; total $m_l$ values =3

$m_l=-2,-1,0,1,2$ for l=2; total $m_l$ values =5

$m_l=-3,-2,-1,0,1,2,3$ for l=3; total $m_l$ values =7

Total number of orbitals = total values of $m_l$

for n=4,

$\therefore 1+3+5+7=16$ orbitals

Each orbital can occupy a maximum of two-electron

Number of electrons $=2\times 16=32$

spin quantum number for one electron in an orbital is $m_S=+\frac{1}{2}$ and other electron has $m_S=-\frac{1}{2}$

Thus electrons with $m_S=-\frac{1}{2}$ will be $\frac{1}{2}\times 32=16$

Hence, the correct option is $\text{D}$