As friend know, we usage four quantum numbers to explain the position and spin of one electron in an atom.

Each electron has actually its unique set the quantum numbers, which method that two electrons have the right to share one, two, or even three quantum numbers, but never every four.

Now, you are provided a #color(red)(4)d# orbital and also asked to find how plenty of sets the quantum numbers can define an electron situated in together an orbital, or, in other words, how numerous electrons have the right to occupy a #color(red)(4)d# orbital.

So, the principal quantum number, #n#, describes the energy level on which the electron is located. In this case, you have actually

#n = color(red)(4) -># the electron is situated on the fourth energy level

The subshell in i m sorry the electron is located is defined by the angular magnetic quantum number, #l#, which because that the fourth energy level bring away the adhering to values

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

Since you"re searching for the d-subshell, you will need #l=2#.

The specific orbital in i m sorry the electron is situated is offered by the magnetic quantum number, #m_l#. Because that any d-subshell, the magnetic quantum number have the right to take the values

#m_l = -2, -1, color(white)(-)0, +1, +2#

Each the these five values describes one that the 5 d-orbitals obtainable in a d-subshell.

Finally ,the spin quantum number, #m_s#, deserve to only take two values, #-1/2# because that an electron that has actually spin-down and #+1/2# because that an electron that has spin-up.

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Now, since each orbital deserve to hold a maximum of two electrons, one v spin-up and also one with spin-down, it complies with that the d-obitals deserve to hold a complete of

#"2 e"^(-)"/ orbital" xx "5 orbitals" = "10 e"^(-)#

Each of this ten electron will have actually its unique set of 4 quantum numbers.

all the ten electrons will certainly share the principal and also angular momentum quantum numbers

#n= color(red)(4)" "# and #" "l=2#

five electrons will share the rotate quantum number

#m_s = -1/2" "# or #" "m_s = +1/2#

two electrons will certainly share the magnetic quantum number

#m_l = -2" "# or #" "m_l = -1" "# or #" "m_l = color(white)(-)0" "# or #" "m_l = +1" "# or #" "m_l = +2#

You will thus have #10# sets of quantum number that have the right to be supplied to explain an electron situated in one of the 5 d-orbitals