Valid and Invalid Quantum Numbers

For an electron with angular momentum quantum number ℓ=0, the shape of the subshell electron cloud is spherical, which corresponds to the s orbital. Therefore, the correct answer is the s orbital.

The notation 4d indicates that the electrons are in the d subshell of the fourth energy level. Here, n=4 (the principal quantum number) and ℓ=2 (the azimuthal quantum number for d orbitals). Thus, the correct values are n=4 and ℓ=2.

The d subshell contains 5 orbitals, which can hold a total of 10 electrons. Each orbital can hold 2 electrons, and the five d orbitals are designated as dxy, dyz, dzx, dx2-y2, and dz2.

In quantum mechanics, the azimuthal quantum number "l" defines the shape of the orbital. For a p sublevel, "l" equals 1, indicating a dumbbell shape. Therefore, the correct answer is 1.

For quantum number l=2, the allowed m values range from -2 to +2. Since -1 falls within this range, an m value of -1 is permitted. Therefore, the correct answer is yes.

The total electron capacity for a principal quantum number n is given by the formula 2n². For n=2, the capacity is 2(2)² = 8. Therefore, the correct answer is 8.

In quantum numbers, ℓ (azimuthal quantum number) must be less than n. Since n=1, ℓ can only be 0. Therefore, ℓ=1 is invalid.

The quantum number m_s must be either +1/2 or -1/2. The value +3/4 is invalid, as it exceeds the allowed range for spin quantum numbers.

The principal quantum number n must be a positive integer (1, 2, 3, ...). Since n=0 is not valid, it is the incorrect part of the set of quantum numbers.

The set n=7, l=3, ml=-3,-2,-1,0,1,2,3 is not acceptable because for l=3, ml can only take values from -3 to 3, but the maximum value of l is n-1, which means n must be at least 4.