Calculate number of electron for which n = 4 and S = +1/2.

The correct answer is 16.

Step-by-Step Explanation:

1. Identify the Principal Quantum Number ($n$):
We are given the principal quantum number $n=4$.

2. Determine the possible values for the Azimuthal Quantum Number ($l$):
For a given value of $n$, the azimuthal quantum number $l$ can take any integer value from $0$ to $n-1$.
For $n=4$, the possible values of $l$ are:
$l=0$ (s-subshell)
$l=1$ (p-subshell)
$l=2$ (d-subshell)
$l=3$ (f-subshell)

3. Determine the number of orbitals in each subshell:
The number of orbitals in a subshell is given by $2l+1$:
For $l=0$ (4s): $2\left(0\right)+1=1$ orbital
For $l=1$ (4p): $2\left(1\right)+1=3$ orbitals
For $l=2$ (4d): $2\left(2\right)+1=5$ orbitals
For $l=3$ (4f): $2\left(3\right)+1=7$ orbitals
Total number of orbitals in the $n=4$ shell = $1+3+5+7=16$ orbitals. (Alternatively, the total number of orbitals in a shell is $n^2=4^2=16$ orbitals).

4. Apply the Spin Quantum Number ($s$ or $m_s$) constraint:
Each orbital can hold a maximum of 2 electrons, one with spin up ($s=+1/2$) and one with spin down ($s=-1/2$).
Therefore, exactly one electron per orbital can have a spin of $s=+1/2$.
Since there are 16 orbitals in total for $n=4$, the number of electrons with $s=+1/2$ is exactly equal to the number of orbitals, which is 16.