Problem 81 - Chemistry The Science in Context

Question

What set of orbitals corresponds to each of the following sets of quantum numbers? How many electrons could occupy these orbitals? a. \(n=2, \ell=0\) b. \(n=3, \ell=1\) c. \(n=4, \ell=2\) d. \(n=1, \ell=0\)

Step-by-Step Solution

Answer

Question: Determine the type of orbitals and the number of electrons that can occupy these orbitals for the following sets of quantum numbers. a. \(n=2, \ell=0\) b. \(n=3, \ell=1\) c. \(n=4, \ell=2\) d. \(n=1, \ell=0\) Answer: a. The orbitals are 2s, and they can hold 2 electrons. b. The orbitals are 3p, and they can hold 6 electrons. c. The orbitals are 4d, and they can hold 10 electrons. d. The orbitals are 1s, and they can hold 2 electrons.

Step 1: a. \(n=2, \ell=0\)

Quantum numbers \(n\) and \(\ell\) signify the principal quantum number and the angular momentum quantum number, respectively. For \(n=2\) and \(\ell=0\), we have an s-type orbital (\(\ell=0\) corresponds to s orbitals). The set of orbitals corresponding to these quantum numbers is \(2s\). Since s orbitals hold a maximum of 2 electrons, there could be 2 electrons occupying these orbitals.

Step 2: b. \(n=3, \ell=1\)

For \(n=3\) and \(\ell=1\), the angular momentum quantum number (\(\ell=1\)) represents p-type orbitals. In this case, the corresponding set of orbitals is \(3p\). P orbitals can hold a maximum of 6 electrons, so 6 electrons could potentially occupy these orbitals.

Step 3: c. \(n=4, \ell=2\)

The quantum numbers \(n = 4\) and \(\ell = 2\) correspond to d-type orbitals (\(\ell=2\) signifies d orbitals). The set of orbitals corresponding to these values is \(4d\). Since d orbitals can hold a maximum of 10 electrons, there could be 10 electrons occupying these orbitals.

Step 4: d. \(n=1, \ell=0\)

The given quantum numbers, \(n = 1\) and \(\ell = 0\), represent an s-type orbital (\(\ell=0\) corresponds to s orbitals). The set of orbitals corresponding to these quantum numbers is \(1s\). S orbitals can hold a maximum of 2 electrons, so there could be 2 electrons occupying these orbitals.

Key Concepts

Orbitals

Orbitals are regions within an atom where electrons are most likely to be found. They represent the space around the nucleus that has a high probability of containing an electron. In essence, an orbital is a unique pattern that signifies where electrons are likely to be located.
There are different types of orbitals, each with a specific shape and capacity for housing electrons.

• S-type orbitals: These are spherical and can hold a maximum of 2 electrons.
• P-type orbitals: These have a dumbbell shape and can accommodate up to 6 electrons.
• D-type orbitals: More complex in shape, capable of holding up to 10 electrons.
• F-type orbitals: These are even more complex and can hold up to 14 electrons.

Each type of orbital is defined by a unique set of quantum numbers, which further delineate their energy levels and shapes.

Principal Quantum Number

The principal quantum number, denoted as \( n \), is a crucial element in quantum mechanics that determines the energy level of an electron within an atom. It essentially tells us the size of the orbital and the distance of the electron from the nucleus.
In simpler terms, it can be thought of as a "shell" surrounding the nucleus. The values of \( n \) are positive integers: 1, 2, 3, etc., and as \( n \) increases, the electron is located further from the nucleus.

• When \( n = 1 \): Electrons are closest to the nucleus and have the lowest energy.
• Higher \( n \) values: Electrons are found further out and have higher energy levels.

Thus, the principal quantum number not only helps in assessing the energy of an electron but also categorizes the atom's size.

Angular Momentum Quantum Number

The angular momentum quantum number, represented by \( \ell \), describes the shape of the orbital and is dependent on the principal quantum number \( n \).
This is a crucial aspect of defining the electron's environment and relates directly to the sub-levels within each principal energy level. For any given value of \( n \), \( \ell \) can take on integer values ranging from 0 to \( n-1 \).

• \( \ell = 0 \): Corresponds to s orbitals, which are spherical.
• \( \ell = 1 \): Corresponds to p orbitals, which are dumbbell-shaped.
• \( \ell = 2 \): Corresponds to d orbitals, which have a more complex shape.
• \( \ell = 3 \): Corresponds to f orbitals, even more complex in configuration.

Understanding \( \ell \) is key to grasping how electrons fill different spaces within the atom, as it directly influences the orbital type and electron distribution.

Electron Configuration

Electron configuration is a way of describing the arrangement of electrons in an atom's orbitals. It provides a concise method for representing which orbitals contain electrons and how these electrons are distributed across different orbitals of an atom.
This concept is vital in explaining an atom's chemical properties, bonding behavior, and reactivity.

• Notation: Electron configurations are typically notated by writing series of numbers and letters, such as 1s², 2s², 2p⁶.
• The "s, p, d, f" letters: Indicate the type of orbitals.
• Superscript number: Represents the number of electrons in those orbitals.

Understanding electron configuration helps predict and explain how atoms interact, how they bond in chemical reactions, and why certain elements behave the way they do.