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

1 Answer
Nov 3, 2015

Answer:

Here's what I got.

Explanation:

Your starting point here will be gallium's electron configuration.

Gallium, #"Ga"#, is located in period 4, group 13 of the periodic table and has an atomic number equal to #31#. This means that a neutral gallium atom will have a total of #31# electrons surrounding its nucleus.

The electron configuration of gallium looks like this - I'll use the noble gas shorthand notation

#"Ga: " ["Ar"]3d^10 4s^2 4p^1#

Now, you're interested in finding the possible sets of quantum numbers that describe the highest-energy electron that belongs to a gallium atom.

As you know, the quantum numbers are defined

figures.boundless.com

So, the highest-energy electron found in gallium is located in a 4p-orbital, which means that right from the start you know that the value of its principal quantum number, #n#, will be #4#.

Now for the angular momentum quantum number, #l#, which describes the subshell in which the electron resides.

Notice that the fourth energy level has total of #4# subshells, each corresponding to a different value of #l#

  • #l =0 -># the s-subhell
  • #l = 1 -># the p-subshell
  • #l=2 -># the d-subshell
  • #l=3 -># the f-subshell

SInce your electron is located in the p-subshell, it follows that its #l# value wil be #1#.

The magnetic quantum number, #m_l#, tells you exactly in which orbital you can expect to find the electron.

For the p-subshell, #l=1#, the magnetic quantum number can take the values

  • #m_l = -1 -># the #p_x# orbital
  • #m_l = color(white)(-)0 -># the #p_y# orbital
  • #m_l = color(white)(-)1 -># the #p_z# orbital

Since the p-subshell only contains one electron, you can place it in the first available p-orbital, which is #p_x#, for which #m_l = -1#.

Finally, the spin quantum number, #m_s#, can only take one of two possible values

  • #m_s = -1/2 -># a spin-down electron
  • #m_s = +1/2 -># a spin-up electron

Since the orbital only contains one electron, it follows that it could be either spin-up or spin-down, so you get two possible sets of quantum numbers

#n=4 -> l=1 -> m_; = -1 -> m_2 = -1/2#

A spin-down electron located in the #4p_x# orbital

#n=4 -> l=1 -> m_; = -1 -> m_2 = +1/2#

A spin-up electron located in the #4p_x# orbital