Final answer:
The maximum number of electrons with n=4 and m=+1 in an atom is 6, considering the possible values of l and the two possible electron spins.
Step-by-step explanation:
The maximum number of electrons in an atom that can have the quantum numbers n=4, m=+1 is determined by the possible values of the other quantum numbers. When n=4, the angular momentum quantum number l can have values of 0, 1, 2, and 3. Given that m equals +1 in this scenario, l must be at least 1 because m can range from -l to +l. This rules out l=0. If l=1, m can be -1, 0, +1, if l=2, m can be -2, -1, 0, +1, +2, and if l=3, m can be -3, -2, -1, 0, +1, +2, +3. Therefore, the l values compatible with m=+1 are 1, 2, and 3.
The spin quantum number ms could be either +1/2 or -1/2 regardless of the l value. As a result, for each l value that allows for an m value of +1, there can be two electrons (one with ms=+1/2 and one with ms=-1/2). Thus, with l values of 1, 2, and 3, the total number of electrons that could have m=+1 would be 2 for each l value, so 2(l=1) + 2(l=2) + 2(l=3) gives us a total of 6 electrons.