Question

What type of electron orbital (i.e., s, \(\mathrm{p}, \mathrm{d}\), or \(\mathrm{f}\) ) is designated by (a) \(\mathbf{n}=3, \ell=1, \mathbf{m}_{\ell}=1\) ? (b) \(\mathrm{n}=5, \ell=0, \mathrm{~m}_{\ell}=0 ?\) (c) \(\mathrm{n}=6, \ell=4, \mathrm{~m}_{\ell}=-4 ?\)

Short answer

Question: Based on the given quantum numbers, determine the type of electron orbitals. (a) n=3, l=1, ml=1 (b) n=5, l=0, ml=0 (c) n=6, l=4, ml=-4 Answer: (a) p orbital (b) s orbital (c) Invalid set of quantum numbers

Step-by-step solution

Review the quantum numbers and their meanings

The given quantum numbers are as follows: - n: The principal quantum number, which indicates the energy level of the electron and the size of the orbital. n can be any positive integer greater than or equal to 1. - l: The azimuthal quantum number, which indicates the shape of the orbital. l can be any integer from 0 to n-1. - ml: The magnetic quantum number, which indicates the orientation of the orbital. ml can be any integer from -l to l.

Determine the type of electron orbitals based on the value of l

Each value of the azimuthal quantum number l corresponds to a specific type of electron orbital: - l = 0: s orbital - l = 1: p orbital - l = 2: d orbital - l = 3: f orbital Now, let's determine the type of electron orbitals for each given set of quantum numbers:

(a) n=3, l=1, ml=1

For this set of quantum numbers, the value of l is 1. Therefore, it corresponds to a p orbital.

(b) n=5, l=0, ml=0

For this set of quantum numbers, the value of l is 0. Therefore, it corresponds to an s orbital.

(c) n=6, l=4, ml=-4

For this set of quantum numbers, the value of l is 4. Since there are no orbitals corresponding to l=4, this set of quantum numbers is not valid for any type of electron orbitals.

Key concepts

Understanding Quantum Numbers

Quantum numbers are essential for understanding the arrangement of electrons within an atom. Electrons aren't just randomly dispersed around the nucleus; they reside in specific energy states characterized by a unique set of quantum numbers. Four quantum numbers define the properties of these electrons:

• Principal Quantum Number (n): Indicates the energy level and size of the orbital. It can be any positive integer starting from 1. Higher 'n' means the electron is further from the nucleus and has more energy.
• Azimuthal Quantum Number (ℓ): Relates to the shape of the orbital and can have integer values from 0 to n-1. Each value corresponds to a specific shape, such as s (ℓ=0), p (ℓ=1), d (ℓ=2), and f (ℓ=3).
• Magnetic Quantum Number (mℓ): Describes the orbital's orientation in space, which affects how the orbital aligns in a magnetic field. It can range from -ℓ to +ℓ.
• Spin Quantum Number (ms): Describes the direction of the electron spin, which can either be +1/2 or -1/2, though this detail isn't directly involved in the current exercise.

Each electron in an atom has a unique set of quantum numbers, almost like an address that specifies exactly where it 'lives' in the atomic neighborhood. The exercise provided tests understanding of these quantum numbers and their implications for identifying electron orbitals.

Orbital Shapes and Their Significance

The shape of an electron orbital is critical to comprehending chemical bonding and reactions. The azimuthal quantum number is the key determinant of an orbital's shape:

• The s orbitals are spherical and centered around the nucleus.
• The p orbitals resemble dumbbells or figure-eights and are oriented along cartesian axes.
• The more complex d orbitals come in four clover-leaf shapes and one donut-shaped ring with a clover-leaf above and below it.
• The complex f orbitals are even more detailed, often visualized as flowers or intricate patterns with multiple lobes.

These differing shapes allow electrons to occupy specific regions in space, determining how atoms interact with each other. For example, the types of bonds formed between atoms and the geometry of molecules are influenced by the spatial distribution of these orbitals. Thus, when the exercise asks to identify the orbital type for given quantum numbers, it's asking which 'room' the electron resides in within the 'house' that is the atom.

The Significance of Energy Levels

Energy levels within an atom describe where electrons can be found relative to the nucleus. The principal quantum number, 'n', serves as the primary indicator of these energy levels. As 'n' increases, so does the electron's energy, as well as its average distance from the nucleus. It's like the floors of an atom's energy skyscraper - the higher the floor (energy level), the more energy the electron has.

Higher energy levels can hold more electrons because they have more orbitals. The first energy level has only one s orbital, the second has one s and three p orbitals, and this pattern expands as levels increase. The exercise solutions elucidate how electrons with certain 'n' and 'ℓ' values correspond to specific types of orbitals (s, p, d, f), each of which occupies a certain energy level. Incorrect quantum number combinations, such as those that don't exist in nature (for example, ℓ=4 for the current state of knowledge), challenge students to apply these rules correctly and understand the structured staircase of atom's energy levels.