Determine the maximum number of electrons that can have each of the following designations: \(2 f, 2 d_{x y}, 3 p, 5 d_{y z},\) and 4\(p .\)

For each designation, let's identify the principal quantum number (n), angular momentum quantum number (l), magnetic quantum number (m_l), and spin quantum number (m_s). 1. 2f: n = 2, l = 3 (since f orbitals correspond to l = 3) 2. 2d_{xy}: n = 2, l = 2 (since d orbitals correspond to l = 2), m_l = -2 (xy refers to the 2nd d orbital) 3. 3p: n = 3, l = 1 (since p orbitals correspond to l = 1) 4. 5d_{yz}: n = 5, l = 2 (since d orbitals correspond to l = 2), m_l = -1 (yz refers to the 3rd d orbital) 5. 4p: n = 4, l = 1 (since p orbitals correspond to l = 1)

According to the Pauli Exclusion Principle, no two electrons in an atom can have the same set of quantum numbers. Since the spin quantum number (m_s) can have only two values (+1/2 and -1/2), at most, two electrons can occupy the same orbital with the same values of n, l, and m_l. To find the maximum number of electrons for each designation, we need to calculate the number of possible m_l values.

For each designation, let's calculate the number of possible m_l values. Note that for each l value, the range of m_l values is -l, -(l-1), ..., 0, ...., (l-1), l. 1. 2f: For l = 3, m_l values range from -3 to 3. There are a total of 7 possible m_l values. 2. 2d_{xy}: For l = 2 and m_l = -2, there's only one possible m_l value. 3. 3p: For l = 1, m_l values range from -1 to 1. There are a total of 3 possible m_l values. 4. 5d_{yz}: For l = 2 and m_l = -1, there's only one possible m_l value. 5. 4p: For l = 1, m_l values range from -1 to 1. There are a total of 3 possible m_l values.

According to Hund's Rule, electrons will fill the orbitals in such a way that they maintain the maximum possible multiplicity (maximum number of unpaired electrons). So, the maximum number of electrons for each designation can be determined by multiplying the number of m_l values by 2 (for the two possible spin quantum numbers, +1/2 and -1/2). 1. 2f: Maximum electrons = 7 m_l values * 2 spins = 14 2. 2d_{xy}: Maximum electrons = 1 m_l value * 2 spins = 2 3. 3p: Maximum electrons = 3 m_l values * 2 spins = 6 4. 5d_{yz}: Maximum electrons = 1 m_l value * 2 spins = 2 5. 4p: Maximum electrons = 3 m_l values * 2 spins = 6 In conclusion, the maximum number of electrons for each designation is as follows: - 2f: 14 electrons - 2d_{xy}: 2 electrons - 3p: 6 electrons - 5d_{yz}: 2 electrons - 4p: 6 electrons