# What is a possible set of four quantum numbers (n,l,ml,ms) in order, for the highest energy electron in gallium?

Jun 18, 2016

Gallium ($\text{Ga}$) is atomic number $31$ and it is on column 13, row 4. So its electron configuration involves the $1 s$, $2 s$, $2 p$, $3 s$, $3 p$, $4 s$, $3 d$, and $4 p$ orbitals.

$\implies 1 {s}^{2} 2 {s}^{2} 2 {p}^{6} 3 {s}^{2} 3 {p}^{6} 3 {d}^{10} 4 {s}^{2} 4 {p}^{1}$

or

$\implies \textcolor{b l u e}{\left[A r\right] 3 {d}^{10} 4 {s}^{2} 4 {p}^{1}}$

The highest energy electron in $\text{Ga}$ is the single $4 p$ electron, which can either be in the $4 {p}_{x}$, $4 {p}_{y}$, or $4 {p}_{z}$ orbital, and it can either be spin up or spin down. So, there are $2 \times 3 = 6$ possible sets of quantum numbers.

For the $4 p$ orbital:

• $n = 1 , 2 , . . . , N \implies \textcolor{b l u e}{4}$ for the principal quantum number.
• $l = 0 , 1 , 2 , . . . , n - 1 \implies \textcolor{b l u e}{1}$ for the angular momentum quantum number.
• ${m}_{l} = \left\{0 , 1 , . . . , \pm l\right\} = \left\{0 , \pm 1\right\}$ for the magnetic quantum number, so there exist $2 l + 1 = 2 \left(1\right) + 1 = 3$ total $4 p$ orbitals.

For a $4 p$ electron:

• The quantum numbers $n$ and $l$ are fixed.
• ${m}_{l}$ will vary as $\textcolor{b l u e}{- 1}$, $\textcolor{b l u e}{0}$, or $\textcolor{b l u e}{+ 1}$ as mentioned above, and tells you that there are three $4 p$ orbitals.
• ${m}_{s}$, the spin quantum number, can be $\textcolor{b l u e}{\pm \frac{1}{2}}$.

Thus, the 6 possible sets of quantum numbers are:

1. $\left(n , l , {m}_{l} , {m}_{s}\right) = \textcolor{b l u e}{\text{("4,1,-1,+1/2")}}$
2. $\left(n , l , {m}_{l} , {m}_{s}\right) = \textcolor{b l u e}{\text{("4,1,0,+1/2")}}$
3. $\left(n , l , {m}_{l} , {m}_{s}\right) = \textcolor{b l u e}{\text{("4,1,+1,+1/2")}}$
4. $\left(n , l , {m}_{l} , {m}_{s}\right) = \textcolor{b l u e}{\text{("4,1,-1,-1/2")}}$
5. $\left(n , l , {m}_{l} , {m}_{s}\right) = \textcolor{b l u e}{\text{("4,1,0,-1/2")}}$
6. $\left(n , l , {m}_{l} , {m}_{s}\right) = \textcolor{b l u e}{\text{("4,1,+1,-1/2")}}$

Another way to represent this each one of these, respectively, is:

1. color(white)([(" ",color(black)(uarr),color(black)(ul(" ")),color(black)(ul(" "))), (color(black)("orbital": ), color(black)(4p_x),color(black)(4p_y),color(black)(4p_z)), (color(black)(m_l: ),color(black)(-1),color(black)(0),color(black)(+1))])

2. color(white)([(" ",color(black)(ul(" ")),color(black)(uarr),color(black)(ul(" "))), (color(black)("orbital": ), color(black)(4p_x),color(black)(4p_y),color(black)(4p_z)), (color(black)(m_l: ),color(black)(-1),color(black)(0),color(black)(+1))])

3. color(white)([(" ",color(black)(ul(" ")),color(black)(ul(" ")),color(black)(uarr)), (color(black)("orbital": ), color(black)(4p_x),color(black)(4p_y),color(black)(4p_z)), (color(black)(m_l: ),color(black)(-1),color(black)(0),color(black)(+1))])

4. color(white)([(" ",color(black)(darr),color(black)(ul(" ")),color(black)(ul(" "))), (color(black)("orbital": ), color(black)(4p_x),color(black)(4p_y),color(black)(4p_z)), (color(black)(m_l: ),color(black)(-1),color(black)(0),color(black)(+1))])

5. color(white)([(" ",color(black)(ul(" ")),color(black)(darr),color(black)(ul(" "))), (color(black)("orbital": ), color(black)(4p_x),color(black)(4p_y),color(black)(4p_z)), (color(black)(m_l: ),color(black)(-1),color(black)(0),color(black)(+1))])

6. color(white)([(" ",color(black)(ul(" ")),color(black)(ul(" ")),color(black)(darr)), (color(black)("orbital": ), color(black)(4p_x),color(black)(4p_y),color(black)(4p_z)), (color(black)(m_l: ),color(black)(-1),color(black)(0),color(black)(+1))])