Question

What feature of an orbital is related to each of the following? (a) Principal quantum number \((n)\) (b) Angular momentum quantum number \((l)\) (c) Magnetic quantum number \(\left(m_{i}\right)\)

Short answer

(n) Size and energy level, (l) Shape, (m_i) Orientation in space

Step-by-step solution

- Principal Quantum Number ()

The principal quantum number, denoted by , determines the size and energy level of an orbital. It represents the shell or energy level in which an electron resides and can take on any positive integer value (1, 2, 3, ...). The larger the value of , the higher the energy level and the larger the orbital.

- Angular Momentum Quantum Number ()

The angular momentum quantum number, denoted by , determines the shape of the orbital. It can take on integer values from 0 to ( - 1). Each value of corresponds to a different type of orbital: 0 (s-orbital), 1 (p-orbital), 2 (d-orbital), and so on.

- Magnetic Quantum Number ()

The magnetic quantum number, denoted by , determines the orientation of the orbital in space. It can take on integer values ranging from - to +, including zero. This quantum number specifies the exact orbital within a subshell where an electron is likely to be found.

Key concepts

Principal Quantum Number (n)

The principal quantum number, denoted by , plays a crucial role in the structure of atomic orbitals. It's a fundamental concept in quantum mechanics and helps to describe the specific energy levels in which electrons can be found.

Here are some key points to understand about :

• represents the energy level of an orbital.
• It can take on any positive integer value, such as 1, 2, 3, and so on.
• The larger the value of , the higher the energy level and the farther the orbital is from the nucleus.

For example, an electron in an orbital with = 1 is in the first energy level, which is closest to the nucleus, while an electron in an orbital with = 3 is in a higher energy level and farther from the nucleus. The principal quantum number is essential for understanding the electron configuration and the way electrons are distributed within an atom.

Angular Momentum Quantum Number (l)

The angular momentum quantum number, denoted by , defines the shape of an orbital. It provides more detail about the electron's state within a given energy level.

Here are the main points about :

• can take integer values from 0 up to -1.
• Each value of corresponds to different orbital shapes:
• = 0: s-orbital (spherical shape)
• = 1: p-orbital (dumbbell shape)
• = 2: d-orbital (cloverleaf shape)
• = 3: f-orbital (complex shapes)

This quantum number helps in defining the orbital type within a particular energy level. For instance, if =2, there might be s, p, and d orbitals in that energy level, each with a unique shape and properties. By understanding , one can predict the types and number of orbitals present in each energy level of an atom.

Magnetic Quantum Number (ml)

The magnetic quantum number, denoted by , specifies the orientation of an orbital in space. It's another layer of detail in the quantum mechanical model of the atom, provided within each subshell.

Important aspects of include:

• can take on integer values from to , including zero.
• This defines the number of orbitals and their spatial orientations within a subshell.

For instance, in the p-orbital (=1), can be -1, 0, or 1. This means there are three p-orbitals within the p-subshell, each aligned differently in space (along the x, y, or z axes).

This quantum number is essential for understanding the magnetic properties of atoms and how electrons are spatially distributed around the nucleus, ultimately affecting the behavior of the atom in magnetic fields.

These three quantum numbers (,, and ) collectively provide a comprehensive description of the unique quantum state of an electron within an atom. By understanding these concepts, students can grasp the foundational principles of atomic structure and electron configuration in greater depth.