Question

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

Answer 1

Gallium (`"Ga"`) is atomic number `31` and it is on column 13, row 4.

So its electron configuration involves the `1s`, `2s`, `2p`, `3s`, `3p`, `4s`, `3d`, and `4p` orbitals.

`=> 1s^2 2s^2 2p^6 3s^2 3p^6 3d^10 4s^2 4p^1`

or

`=> color(blue)([Ar] 3d^10 4s^2 4p^1)`

The highest energy electron in `"Ga"` is the single `4p` electron, which can either be in the `4p_x`, `4p_y`, or `4p_z` orbital, and it can either be spin up or spin down. So, there are `2xx3 = 6` possible sets of quantum numbers.

For the `4p` orbital:

• `n = 1,2, . . . , N => color(blue)(4)` for the principal quantum number.
• `l = 0,1,2, . . . , n-1 => color(blue)(1)` for the angular momentum quantum number.
• `m_l = {0, 1, . . . , pml} = {0, pm1}` for the magnetic quantum number, so there exist `2l+1 = 2(1) + 1 = 3` total `4p` orbitals.

For a `4p` electron:

• The quantum numbers `n` and `l` are fixed.
• `m_l` will vary as `color(blue)(-1)`, `color(blue)(0)`, or `color(blue)(+1)` as mentioned above, and tells you that there are three `4p` orbitals.
• `m_s`, the spin quantum number, can be `color(blue)(pm1/2)`.

Thus, the 6 possible sets of quantum numbers are:

1. `(n,l,m_l,m_s) = color(blue)("("4,1,-1,+1/2")")`
2. `(n,l,m_l,m_s) = color(blue)("("4,1,0,+1/2")")`
3. `(n,l,m_l,m_s) = color(blue)("("4,1,+1,+1/2")")`
4. `(n,l,m_l,m_s) = color(blue)("("4,1,-1,-1/2")")`
5. `(n,l,m_l,m_s) = color(blue)("("4,1,0,-1/2")")`
6. `(n,l,m_l,m_s) = color(blue)("("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))])`