Problem 71 - Chemistry The Central Science

Question

What is the maximum number of electrons that can occupy each of the following subshells? (a) 3p, (b) \(5 d,(\mathbf{c}) 2 s,(\mathbf{d}) 4 f .\)

Step-by-Step Solution

Answer

The maximum number of electrons that can occupy the given subshells are: (a) 3p: \(6\) electrons, (b) 5d: \(10\) electrons, (c) 2s: \(2\) electrons, and (d) 4f: \(14\) electrons.

Step 1: Determine the angular quantum number (ℓ) for each subshell

For each subshell given, determine the angular quantum number (ℓ) corresponding to the subshell type (s=0, p=1, d=2, f=3): (a) 3p - ℓ=1 (p subshell) (b) 5d - ℓ=2 (d subshell) (c) 2s - ℓ=0 (s subshell) (d) 4f - ℓ=3 (f subshell)

Step 2: Apply the 2(2ℓ + 1) formula for each subshell

Now that we have the angular quantum numbers for each subshell, we will use the formula 2(2ℓ+1) to find the maximum number of electrons that can occupy each given subshell. (a) For 3p: Maximum number of electrons = 2(2(1) + 1) = 2(2+1) = 2(3) = 6 (b) For 5d: Maximum number of electrons = 2(2(2) + 1) = 2(4+1) = 2(5) = 10 (c) For 2s: Maximum number of electrons = 2(2(0) + 1) = 2(0+1) = 2(1) = 2 (d) For 4f: Maximum number of electrons = 2(2(3) + 1) = 2(6+1) = 2(7) = 14

Step 3: Write the final answer

The maximum number of electrons that can occupy each of the given subshells are: (a) 3p: 6 electrons (b) 5d: 10 electrons (c) 2s: 2 electrons (d) 4f: 14 electrons

Key Concepts

Quantum Numbers

Quantum numbers are like an address system for electrons within an atom, helping us understand their position and energy. There are four quantum numbers, but we'll focus on the angular quantum number for now.

• The **principal quantum number (n)**: This indicates the main energy level or shell. For example, in 3p, the principal quantum number is 3.
• The **angular (or azimuthal) quantum number (ℓ)**: This determines the shape of the orbital. It's key to understanding what type of subshell we have, whether it’s s, p, d, or f. The angular quantum number can be 0 (s), 1 (p), 2 (d), or 3 (f).
• Other quantum numbers can specify orientations and spins, but let's keep it simple for now.

These numbers together help predict where electrons are and how many fit into certain areas around an atom. They play a crucial role in determining the chemical behavior of elements.

Subshell

A subshell is a subdivision of electron shells separated by electron orbitals. It provides a more detailed structure than the principal energy level alone.

• Each shell can have one or more subshells, identified by the value of the angular quantum number (ℓ).
• Subshells are usually labeled s, p, d, and f, corresponding to the values of ℓ: 0, 1, 2, and 3, respectively.
• For instance, in the 3p subshell, 'p' indicates the shape and type of the orbitals, while 3 is the main energy level.

Subshells are crucial in predicting electron arrangements and calculating how many electrons can fit, using specific formulas based on quantum numbers. Methods like the 2(2ℓ + 1) formula help determine these capacities by calculating possible orientations.

Angular Quantum Number

The angular quantum number (ℓ) plays a vital role in determining the shape and type of an electron's orbital. It helps us visualize what the space around an atom looks like where electrons might be found.

• This number varies based on the type of subshell:
  • s (ℓ=0),
  • p (ℓ=1),
  • d (ℓ=2),
  • f (ℓ=3).
• Recognizing the angular quantum number allows us to use the formula for maximum electron capacity in a subshell: 2(2ℓ + 1).
• Applying these values shows us potential numbers like in a 2s subshell (ℓ=0) fitting 2 electrons, or a 4f subshell (ℓ=3) accommodating 14 electrons.

Understanding the angular quantum number enables us to map out where electrons are likely to be and how atoms interact, impacting everything from chemical reactions to the properties of substances.