Question

How many planar nodes are associated with each of the following orbitals? (a) \(2 s\) (b) \(5 d\) (c) \(5 f\)

Short answer

(a) 0, (b) 2, (c) 3 planar nodes.

Step-by-step solution

Understanding Planar Nodes

A planar or angular node is a region within an atomic orbital where the probability of finding an electron is zero. The number of planar nodes is determined by the azimuthal quantum number, \( l \).

Quantum Numbers for Orbitals

For a given orbital, identify the principal quantum number \( n \) and the azimuthal quantum number \( l \):(a) For \(2s\), \(n = 2\) and \(l = 0\).(b) For \(5d\), \(n = 5\) and \(l = 2\).(c) For \(5f\), \(n = 5\) and \(l = 3\).

Calculate Angular Nodes

The number of angular nodes in an orbital is given by the formula: \( l \).(a) For \(2s\), angular nodes = 0(b) For \(5d\), angular nodes = 2(c) For \(5f\), angular nodes = 3

Key concepts

Quantum Numbers

Quantum numbers are crucial in describing the properties and behaviors of electrons in atoms. They are like unique addresses for electrons, defining their positions, energy levels, and other characteristics within an atom. Quantum numbers can be compared to an elaborate coding system that differs for every electron.

There are four types of quantum numbers:

• Principal quantum number (\( n \)): Indicates the main energy level or shell where the electron is located. It is a positive integer (1, 2, 3,...).
• Azimuthal quantum number (\( l \)): Provides information on the shape of the orbital. It ranges from 0 to \( n-1 \).
• Magnetic quantum number (\( m_l \)): Describes the orientation of the orbital in space. It ranges from \( -l \) to \( +l \).
• Spin quantum number (\( m_s \)): Indicates the spin direction of the electron, either \( +\frac{1}{2} \) or \( -\frac{1}{2} \).

Every electron in an atom has a unique set of these four quantum numbers, which together define its position and energy state within the atom.

Planar Nodes

Planar nodes, also known as angular nodes, are areas within an atomic orbital where the probability of finding an electron is zero. These nodes are essential for understanding electron distribution and the shape of orbitals. Planar nodes occur where the wave function of an electron is zero, which means electrons cannot exist at these points.

The number of planar nodes in an orbital is directly related to the azimuthal quantum number (\( l \)).

• For \( s \) orbitals, \( l = 0 \), there are no planar nodes.
• For \( p \) orbitals, \( l = 1 \), there is one planar node.
• For \( d \) orbitals, \( l = 2 \), there are two planar nodes.
• For \( f \) orbitals, \( l = 3 \), there are three planar nodes.

Understanding planar nodes helps in visualizing the different shapes and orientations of orbitals, aiding in the prediction of chemical properties and reactivities of elements.

Angular Nodes

Angular nodes, synonymous with planar nodes, also depict regions in an atomic orbital where there is zero probability of discovering an electron, providing a crucial insight into the familiar shapes seen in graphical depictions of atomic orbitals.

Angular nodes are differentiated from radial nodes, where the term 'radial node' represents a spherical region (concentric circles around the nucleus) of zero electron probability.

The formula to calculate the number of angular nodes is simply the azimuthal quantum number, \( l \). Here's how it applies:

• Angular nodes = \( l \)
• If \( l = 0 \), there are 0 angular nodes (like \( s \) orbitals).
• If \( l = 1 \), there is 1 angular node (as in \( p \) orbitals).
• If \( l = 2 \), there are 2 angular nodes (as seen in \( d \) orbitals).
• If \( l = 3 \), there are 3 angular nodes (such as in \( f \) orbitals).

Angular nodes are key to understanding how electrons orient themselves spatially around a nucleus, helping predict the geometry and bonding patterns in molecules.