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

The electron capacities for s, p, and d sublevels are 3, 9, and 15, respectively. The total number of electrons that can fit in the n=3 level is 27. The electron configurations for atomic numbers 8 and 17 are 1s³ 2s³ 2p² and 1s³ 2s³ 2p⁹ 3s¹, respectively.

Step-by-step solution

(a) Electron capacity of s, p, and d sublevels

The electron capacity of a sublevel depends on the number of available orbitals in that sublevel and the number of electrons that can fit in each orbital. In this case, the number of electrons per orbital has changed due to the alteration of the spin quantum number. Instead of 2 electrons per orbital, we now have 3 electrons per orbital. For an s sublevel, there is only 1 orbital. Therefore, the electron capacity of an s sublevel would be: $$ 1 \cdot 3 = 3 $$ For a p sublevel, there are 3 orbitals. Therefore, the electron capacity of a p sublevel would be: $$ 3 \cdot 3 = 9 $$ For a d sublevel, there are 5 orbitals. Therefore, the electron capacity of a d sublevel would be: $$ 5 \cdot 3 = 15 $$

(b) Number of electrons in \(\mathbf{n}=3\) level

To find the number of electrons that can fit in the \(n=3\) level, we need to consider the electron capacities of all the sublevels in that energy level. For \(n=3\), we have the following sublevels: 3s, 3p, and 3d. - The electron capacity of the 3s sublevel can hold 3 electrons, as determined in part (a). - The electron capacity of the 3p sublevel can hold 9 electrons, as determined in part (a). - The electron capacity of the 3d sublevel can hold 15 electrons, as determined in part (a). Therefore, the total number of electrons that can fit in the \(n=3\) level is: $$ 3 + 9 + 15 = 27 $$

(c) Electron configurations for atomic numbers 8 and 17

We can now determine the electron configurations for elements with atomic numbers 8 and 17 by following the order of filling sublevels: 1s\(\to\)2s\(\to\)2p\(\to\)3s\(\to\)3p\(\to\)4s\(\to\)3d\(\to\)... For atomic number 8, we distribute 8 electrons to sublevels: 1. 1s: 3 electrons 2. 2s: 3 electrons 3. 2p: 2 electrons (remaining) Thus, the electron configuration for atomic number 8 is: $$ 1s^3 2s^3 2p^2 $$ For atomic number 17, we distribute 17 electrons to sublevels: 1. 1s: 3 electrons 2. 2s: 3 electrons 3. 2p: 9 electrons 4. 3s: 1 electron (remaining) Thus, the electron configuration for atomic number 17 is: $$ 1s^3 2s^3 2p^9 3s^1 $$

Key concepts

Understanding Electron Configuration

Electron configuration is a form of notation which depicts the distribution of electrons across the orbitals of an atom or molecule. Imagine it like organizing students in a classroom where there are rules on where everyone can sit. Similarly, electrons follow specific rules when filing into orbitals.

Here's the quick guide: Electrons fill into orbitals starting from the lowest energy level moving up. They follow a predetermined sequence based on increasing energy levels and sublevels, designated as 1s, 2s, 2p, 3s, 3p, and so on.

Furthermore, the concept obeys the Pauli Exclusion Principle, which, in simple terms, states that no two electrons in the same atom can have identical sets of quantum numbers. Typically, an orbital can fit two electrons with opposing spins—a characteristic dictated by the spin quantum number. However, the exercise posed a unique scenario where the spin quantum number allowed for three possible values, leading to each orbital being able to hold three electrons, not just two.

For example, if we were to look at the element with atomic number 8 under normal rules, its electron configuration would be 1s² 2s² 2p⁴. However, with the altered rules from the exercise, it's 1s³ 2s³ 2p² because each orbital can hold one extra electron.

Spin Quantum Number and its Impact

The spin quantum number is an intrinsic property of electrons, almost like each electron has its own little compass needle, pointing up or down. This value is vital in determining the magnetic properties of an atom and is critical in the Pauli Exclusion Principle.

Typically, the spin quantum number can have only two possible values: +1/2 or -1/2, representing the two possible spin states of an electron. As mentioned earlier, this factor limits the number of electrons in an orbital to two. Each electron in an orbital must have an opposite spin to adhere to the Pauli principle.

However, in our exercise's hypothetical scenario, a spin quantum number of zero is introduced, bumping up the potential occupancy of each orbital from two to three electrons. This alteration has profound effects on the arrangement of electrons in an atom and thus on chemical properties and bonding behaviors. In reality, this possibility does not exist — it's strictly a thought experiment to help us understand the importance of spin numbers.

Sublevel Electron Capacity Explained

Sublevels, designated as s, p, d, and f, each have a characteristic shape and a related capacity for electrons. The 's' sublevel has one orbital, 'p' has three, 'd' has five, and 'f' has seven. In normal circumstances, each orbital houses two electrons.

Here's how it usually breaks down:

• An s sublevel can hold up to 2 electrons (1 orbital x 2).
• A p sublevel can hold up to 6 electrons (3 orbitals x 2).
• A d sublevel can hold up to 10 electrons (5 orbitals x 2).
• An f sublevel can hold up to 14 electrons (7 orbitals x 2).

The proposed hypothetical scenario changes these capacities significantly. Treating each orbital like a room with an extra bed, the s sublevel now holds 3, the p sublevel 9, and the d sublevel 15. Thus, students can visualize the increased 'accommodation' for electrons in each type of sublevel. But remember, this inflated capacity is purely theoretical and would dramatically alter atomic and molecular structures if it were true.