State the total capacity for electrons in (a) \(\mathbf{n}=4\). (b) a 3s sublevel. (c) a d sublevel. (d) a p orbital.

To find the total electron capacity for each given case, we need to know the rules of electron configuration. There are three main rules we use: the Pauli Exclusion Principle, the Aufbau Principle, and Hund's Rule. The Pauli Exclusion Principle states that no two electrons in an atom can have the same four quantum numbers (n, l, m_l, and m_s). The Aufbau Principle informs us that the electrons will fill the lowest available energy level first. Hund's Rule states that when filling degenerate orbitals (those with the same energy level), electrons will fill each orbital with the same spin before pairing. Now, we will apply these rules to find the total electron capacity of each case.

When the principal quantum number \(n=4\), we have 4 possible sublevels: \(s, p, d, \text{ and } f\), with \(l=0, 1, 2, \text{ and } 3\) respectively. - For s sublevel, there is 1 orbital, and 2 electrons can fit in (1 orbital × 2 electrons per orbital) - For p sublevel, there are 3 orbitals, and each orbital can hold 2 electrons, so a total of 6 electrons. (3 orbitals × 2 electrons per orbital) - For d sublevel, there are 5 orbitals, and they can hold 2 electrons each, totaling 10 electrons. (5 orbitals × 2 electrons per orbital) - For f sublevel, there are 7 orbitals, each capable of holding 2 electrons, giving a total of 14 electrons. (7 orbitals × 2 electrons per orbital) The total electron capacity when \(n=4\) is the sum of all the electrons in all the sublevels: \(2+6+10+14=32\). So, the total electron capacity when \(n=4\) is 32 electrons.

In a 3s sublevel, the principal quantum number (n) is 3 and the angular quantum number (l) is 0. This means there is only 1 orbital in the s sublevel. Each orbital can occupy a maximum of 2 electrons, following the Pauli Exclusion Principle. Therefore, the total electron capacity for a 3s sublevel is 2 electrons.

In a d sublevel, the angular quantum number (l) is 2. The 5 orbitals can each hold a maximum of 2 electrons following the Pauli Exclusion Principle. Hence, the total electron capacity for a d sublevel is 5 orbitals × 2 electrons per orbital = 10 electrons.

Each p orbital can hold up to 2 electrons, abiding by the Pauli Exclusion Principle. So, the total electron capacity for a p orbital is 2 electrons. To summarize: (a) Total capacity for electrons when \(n=4\) is 32 electrons. (b) Total capacity for electrons in a 3s sublevel is 2 electrons. (c) Total capacity for electrons in a d sublevel is 10 electrons. (d) Total capacity for electrons in a p orbital is 2 electrons.