Question

How many planar nodes are associated with each of the following atomic orbitals? (a) \(4 f\) (b) \(2 p\) (c) \(6 s\)

Short answer

(a) 3 planar nodes; (b) 1 planar node; (c) 0 planar nodes.

Step-by-step solution

Understanding the Concept

Planar nodes, also known as angular nodes, are regions where there is zero probability of finding an electron. The number of planar nodes is determined by the orbital angular momentum quantum number, denoted as \(l\). The value of \(l\) indicates the number of angular nodes in an orbital. Different sublevels correspond to different \(l\) values: \(s\) orbitals have \(l=0\), \(p\) orbitals have \(l=1\), \(d\) orbitals have \(l=2\), and \(f\) orbitals have \(l=3\).

Determining Planar Nodes for 4f

For the \(4f\) orbital, the angular momentum quantum number \(l\) is 3 because \(f\) orbitals correspond to \(l=3\). Thus, the \(4f\) orbital has 3 planar nodes.

Determining Planar Nodes for 2p

For the \(2p\) orbital, the angular momentum quantum number \(l\) is 1 because \(p\) orbitals correspond to \(l=1\). Thus, the \(2p\) orbital has 1 planar node.

Determining Planar Nodes for 6s

For the \(6s\) orbital, the angular momentum quantum number \(l\) is 0 because \(s\) orbitals correspond to \(l=0\). Thus, the \(6s\) orbital has 0 planar nodes.

Key concepts

Planar Nodes

Planar nodes, sometimes referred to as angular nodes, are fascinating regions within an atomic orbital. These are areas where the likelihood of finding an electron is exactly zero. Imagine a plane slicing through an orbital where there exists no electron probability. That's a planar node! The number of planar nodes in an orbital is intrinsically linked to the orbital's angular momentum quantum number, represented by the symbol \( l \).

• \( s \) orbitals: no planar nodes (\( l = 0 \))
• \( p \) orbitals: one planar node (\( l = 1 \))
• \( d \) orbitals: two planar nodes (\( l = 2 \))
• \( f \) orbitals: three planar nodes (\( l = 3 \))

Hence, the bigger the \( l \) value, the more planar nodes an orbital has. Understanding where planar nodes occur helps in visualizing the shape and orientation of each type of orbital, which is crucial for predicting chemical bonding and molecular geometry.

Quantum Number

The quantum number system is a set of four numbers that serves as the address of an electron within an atom. Here, we'll focus on the angular momentum quantum number, \( l \), since it's key to figuring out planar nodes.
The angular momentum quantum number tells us about the shape of an electron's orbital:

• It ranges from 0 to \( n-1 \), where \( n \) is the principal quantum number that signifies the energy level.
• The value of \( l \) is directly related to the type of orbital: \( s, p, d, \) or \( f \).
• Each different \( l \) leads to a respective orbital shape:
• \( l = 0 \) gives a spherical \( s \) orbital.
• \( l = 1 \) results in a dumbbell-shaped \( p \) orbital.
• \( l = 2 \) forms a cloverleaf \( d \) orbital.
• \( l = 3 \) leads to complex \( f \) orbitals.

By understanding \( l \), we can anticipate not just the number of planar nodes but also the overall geometry of electron occupation zones in an atom.

Angular Momentum

Angular momentum in atomic theory is more than just a concept from classical physics—it represents electron behavior in orbitals. The angular momentum quantum number \( l \) reflects the momentum of the electron's circular trajectory around the nucleus in quantized levels, which determines the shape of the orbital.
This quantum mechanical property helps in understanding why electrons do not just fall into the nucleus, instead, they occupy certain allowed orbits dictated by their angular momentum. The essence of angular momentum as it pertains to orbitals include:

• Orbitals with higher \( l \) values have more complex shapes.
• The total probability area for an electron increases with increasing \( l \).
• Higher \( l \) values contribute to the atom’s chemical properties, influencing characteristics like bonding and spatial orientation.

Thus, angular momentum helps form the basis of why atoms and molecules have the shapes and properties that they do. It's a fundamental driver of the architectural puzzle that is atomic structure.

Orbitals

Orbitals are essentially the homes of electrons within atoms—regions where an electron is most likely to be found. Each type of orbital (\( s, p, d, \) and \( f \)) has a distinct shape and set of nodes, based on its angular momentum quantum number \( l \) and principal quantum number \( n \).

• \( s \) orbitals are spherical and get larger as \( n \) increases.
• \( p \) orbitals look like dumbbells or figure eights, positioned along axes.
• \( d \) orbitals are more complex and can appear as cloverleaf shapes or with a unique donut shape.
• \( f \) orbitals are even more intricate, with complicated, multi-lobed structures.

Each orbital type is a part of an atom's electronic configuration, dictating how electrons fill in from lower to higher energy levels, known as the Aufbau Principle. Understanding orbitals is integral to analyzing how atoms interact and bond, forming the basis for all chemical reactions and the properties of matter.