As girlfriend know, we usage four quantum numbers to define the position and also spin of one electron in one atom.

Each electron has its unique set that quantum numbers, which means that 2 electrons can share one, two, or even three quantum numbers, but never all four.

Now, friend are offered a #color(red)(4)d# orbital and asked to find how numerous sets that quantum numbers can define an electron located in together an orbital, or, in other words, how countless 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 located on the fourth energy level

The subshell in i beg your pardon the electron is situated is defined by the angular magnetic quantum number, #l#, which for the fourth energy level bring away the following values

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

Since you"re in search of the d-subshell, friend will require #l=2#.

The specific orbital in i beg your pardon the electron is located is offered 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 that these five values explains one of the 5 d-orbitals obtainable in a d-subshell.

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

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Now, since each orbital can hold a maximum of two electrons, one through spin-up and also one v spin-down, it follows that the d-obitals deserve to hold a total of

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

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

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

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

five electrons will share the turn 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 that quantum numbers that deserve to be provided to define an electron situated in among the 5 d-orbitals