Question

Suppose that the spin quantum number could have the values \(\frac{1}{2}, 0,\) and \(-\frac{1}{2}\). Assuming that the rules governing the values of the other quantum numbers and the order of filling sublevels were unchanged, (a) what would be the electron capacity of an s sublevel? a p sublevel? a d sublevel? (b) how many electrons could fit in the \(\mathbf{n}=3\) level? (c) what would be the electron configuration of the element with atomic number \(8 ? 17 ?\)

Short answer

Question: In an alternate version of the quantum numbers, the spin quantum number has three values instead of the usual two: +1/2, 0, and -1/2. Determine the electron capacity of s, p, and d sublevels, the number of electrons that can fit in the n=3 energy level, and the electron configurations of elements with atomic numbers 8 and 17 in this alternate version. Answer: In this alternate version with three possible spin quantum number values, the electron capacities of s, p, and d sublevels are 3, 9, and 15 electrons, respectively. In the n=3 energy level, 27 electrons can fit. The electron configurations of elements with atomic numbers 8 and 17 are 1s² 2s² 2p⁴ and 1s² 2s² 2p⁹ 3s² 3p², respectively.

Step-by-step solution

Recall the quantum numbers and their allowed values

In the standard model, there are four quantum numbers for an electron in an atom: 1. Principal quantum number (n): It determines the energy level and can have positive integer values (1, 2, 3, ...). 2. Azimuthal quantum number (l): It determines the shape of the orbital and can have integer values ranging from 0 to n-1. 3. Magnetic quantum number (m_l): It determines the orientation of the orbital and can have integer values ranging from -l to l. 4. Spin quantum number (m_s): It represents the intrinsic angular momentum of the electron, and can have values +1/2 or -1/2. In this exercise, the spin quantum number is allowed to have three values: 1/2, 0, and -1/2.

Determine the electron capacity of s, p, and d sublevels

Recall the relation between l and sublevels (s, p, d, f): l = 0 for s-sublevel, l = 1 for p-sublevel, l = 2 for d-sublevel, and so on. The electron capacity for a sublevel is given by the formula: Capacity = Number of orbitals × Allowed spins per orbital Now, let's find the electron capacity for each sublevel. a) s-sublevel: l = 0, so the allowed value for m_l is 0. This means there is only one orbital in an s-sublevel. The allowed m_s values are +1/2, 0, and -1/2, so there are 3 allowed spins per orbital. Hence, the electron capacity of an s-sublevel = 1 × 3 = 3 electrons. b) p-sublevel: l = 1, so the allowed values for m_l are -1, 0, and 1. This means there are three orbitals in a p-sublevel. The allowed m_s values are +1/2, 0, and -1/2, so there are 3 allowed spins per orbital. Hence, the electron capacity of a p-sublevel = 3 × 3 = 9 electrons. c) d-sublevel: l = 2, so the allowed values for m_l are -2, -1, 0, 1, and 2. This means there are five orbitals in a d-sublevel. The allowed m_s values are +1/2, 0, and -1/2, so there are 3 allowed spins per orbital. Hence, the electron capacity of a d-sublevel = 5 × 3 = 15 electrons.

Find the number of electrons that can fit in the n=3 energy level

In the n=3 level, we have three possible sublevels (s, p, and d), with l values of 0, 1, and 2, respectively. The electron capacity of each sublevel is: - s-sublevel: 3 electrons (from Step 2a) - p-sublevel: 9 electrons (from Step 2b) - d-sublevel: 15 electrons (from Step 2c) So, in the n=3 level, we can fit the sum of these capacities: 3 + 9 + 15 = 27 electrons.

Determine the electron configuration of elements with atomic numbers 8 and 17

First, recall the order of filling sublevels: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, and so on. (a) For an element with atomic number 8, we have 8 electrons to fill: 1s², 2s², 2p⁴ (b) For an element with atomic number 17, we have 17 electrons to fill: 1s², 2s², 2p⁹, 3s², 3p² The electron configurations in this alternate model for elements with atomic numbers 8 and 17 are 1s² 2s² 2p⁴ and 1s² 2s² 2p⁹ 3s² 3p², respectively.

Key concepts

Spin Quantum Number

The spin quantum number, denoted by m_s, is a fundamental property of electrons that describes their intrinsic angular momentum, or 'spin'. In the standard atomic model, this quantum number can take on one of two possible values: +1/2 or -1/2, which represent the two possible orientations of an electron's spin – 'spin-up' or 'spin-down' respectively.

The concept of the spin quantum number is crucial for understanding how electrons fill available atomic orbitals in an atom. The Pauli exclusion principle asserts that no two electrons in an atom can have the same set of four quantum numbers, and since there are only two possible spin states, each orbital can accommodate up to two electrons, one 'spin-up' and one 'spin-down'.

If we were to entertain a hypothetical scenario where the spin quantum number could adopt an additional value of 0, this would drastically alter the capacity of atomic orbitals and therefore the entire electron configuration of atoms.

Electron Configuration

The arrangement of electrons in an atom is referred to as its electron configuration. It follows a specific order of 'building up' from lower to higher energy levels and orbitals, a principle known as the Aufbau principle. Each electron is placed into the lowest energy orbital available, following the rules dictated by the quantum numbers.

The commonly taught sequence for filling up the orbitals is 1s, 2s, 2p, 3s, 3p, and so forth, with the numbers representing the principal quantum number n, and the letters s, p, d, and f indicating the shape of the orbital. With the alteration of the spin quantum number to include a zero value, you would find unconventional electron capacities, leading to unusual electron configurations. This would particularly impact the chemical properties of elements, as electron configuration is key to understanding an element's reactivity, bonding, and placement in the periodic table.

Atomic Orbital Capacity

The atomic orbital capacity denotes the maximum number of electrons that an orbital can hold. According to the conventional quantum mechanical model, an s orbital can hold 2 electrons, a p orbital can hold 6, a d orbital 10, and an f orbital 14. These capacities can be calculated using the formula 2(2l + 1), where l is the azimuthal quantum number corresponding to the shape of the orbital.

However, if we introduce the hypothetical scenario where the spin quantum number includes an additional value, we must adjust our calculations accordingly. This would triple the capacity of each atomic orbital, dramatically increasing the number of electrons it could hold and redefining many fundamental concepts in chemistry, such as the periodic table structure and the nature of chemical bonding. For example, an s orbital's capacity would increase from 2 to 3, and a p orbital's capacity from 6 to 9, impacting the entire architecture of electron allocation in atoms.

Principal Quantum Number

The principal quantum number, symbolized as n, is integral to the quantum mechanical model of the atom. It specifies the energy level of an electron in an atom and determines the distance of an electron from the nucleus; higher values of n mean higher energy levels and a greater radius from the nucleus. The principal quantum number starts at 1 and increases in positive integer increments (2, 3, 4, ...).

Each energy level can hold a distinct number of sublevels, and thus, electrons, with the maximum capacity determined by the formula 2n². Altering the spin quantum number would not directly affect the principal quantum number but would significantly increase the number of electrons that each main energy level could accommodate by changing the electron capacity of each sublevel. For instance, with the principal quantum number of 3 (n = 3), the typical electron holding capacity is 18. However, in our hypothetical scenario with the expanded spin values, it would rise to 27, leading to profound implications for the organization of electrons in each energy level and requiring a reevaluation of electron distribution across periods in the periodic table.