The maximum number of electrons that can have the principal quantum number 2 in a given atom is

8 electrons

Explanation:

to solve this problem it is necesarry to take into account the nume of the other quantum number l, ml, and ms:

l: angular quantum number = l=n-1

ml: magnetic quantum number = ml=-l, ...,0, ...,+l

It is also neccesary to take into account each n state.

for n=1

- l=1-1=0

- ml=0

each ml state has two electrons with antiparallel spins. Hence, for n=1 you have 2 electron.

forn n=2

- l=2-1=1

- ml=-1,0,1

each ml state has two electrons. hence, for n=2 you have 3*2=6 electrons.

Finally, the total of electrons is 6+2=8