As friend know, we use four quantum numbers to describe the position and also spin of one electron in an atom. Each electron has actually its unique set of quantum numbers, which way that 2 electrons deserve to share one, two, or also three quantum numbers, but never every four.

Now, girlfriend are given a #color(red)(4)d# orbital and asked to uncover how plenty of sets that quantum numbers can explain an electron located in together an orbital, or, in various other words, how many electrons have the right to occupy a #color(red)(4)d# orbital.

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

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

The subshell in which the electron is located is described by the angular magnetic quantum number, #l#, which for the fourth power level take 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 looking for the d-subshell, friend will require #l=2#.

The specific orbital in i beg your pardon the electron is situated is provided by the magnetic quantum number, #m_l#. Because that any d-subshell, the magnetic quantum number deserve to take the values

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

Each of these 5 values describes one the the 5 d-orbitals accessible in a d-subshell. Finally ,the spin quantum number, #m_s#, can only take 2 values, #-1/2# because that an electron that has actually spin-down and also #+1/2# because that an electron that has spin-up.

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Now, because each orbital can hold a preferably of two electrons, one with spin-up and also one through spin-down, it follows that the d-obitals deserve to hold a full of

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

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

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

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

five electrons will certainly share the rotate quantum number

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

two electrons will 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 certainly thus have #10# sets of quantum numbers that deserve to be provided to describe an electron situated in one of the five d-orbitals