Problem 125 - A Complete Resource Book in Chemistry for JEE Main

Question

In which of the orbital/orbitals radial node and angular nodes are same? (a) \(3 \mathrm{p}\) (b) \(4 \mathrm{p}\) (c) \(6 \mathrm{f}\) (d) \(5 \mathrm{~d}\)

Step-by-Step Solution

Answer

The orbitals where radial and angular nodes are the same are 3p and 5d.

Step 1: Understanding Nodes

Nodes are regions where the probability density of finding an electron is zero. There are two types of nodes: radial nodes and angular nodes. Radial nodes depend on the principal quantum number \(n\) and angular momentum quantum number \(l\). Angular nodes are determined by \(l\).

Step 2: Calculate Radial Nodes

Radial nodes are calculated using the formula: \( \text{Radial Nodes} = n - l - 1 \).

Step 3: Calculate Angular Nodes

Angular nodes are equal to the angular momentum quantum number \( l \).

Step 4: Analyze the Orbital 3p

For the 3p orbital: \( n = 3 \) and \( l = 1 \). Radial nodes: \( 3 - 1 - 1 = 1 \). Angular nodes: \( 1 \). Radial and angular nodes equal.

Step 5: Analyze the Orbital 4p

For the 4p orbital: \( n = 4 \) and \( l = 1 \). Radial nodes: \( 4 - 1 - 1 = 2 \). Angular nodes: \( 1 \). Radial and angular nodes differ.

Step 6: Analyze the Orbital 6f

For the 6f orbital: \( n = 6 \) and \( l = 3 \). Radial nodes: \( 6 - 3 - 1 = 2 \). Angular nodes: \( 3 \). Radial and angular nodes differ.

Step 7: Analyze the Orbital 5d

For the 5d orbital: \( n = 5 \) and \( l = 2 \). Radial nodes: \( 5 - 2 - 1 = 2 \). Angular nodes: \( 2 \). Radial and angular nodes equal.

Key Concepts

Radial Nodes

In quantum chemistry, radial nodes are specific regions around the nucleus of an atom where the probability of finding an electron is zero. These nodes occur at certain radial distances from the nucleus. The concept of radial nodes is crucial for understanding the shape and size of atomic orbitals.

The formula to calculate radial nodes is:

• Radial Nodes = \( n - l - 1 \)

Here, \( n \) is the principal quantum number, and \( l \) is the angular momentum quantum number.A higher number of radial nodes indicates a larger extent of the orbital from the nucleus. For instance, a 3p orbital has 1 radial node, indicating one region where the probability of finding an electron is zero, apart from the nucleus itself.

Angular Nodes

Angular nodes are another type of node where the probability density of finding an electron drops to zero. Unlike radial nodes, angular nodes are related to the shape of the orbital. These nodes result from the angular part of the wave function becoming zero at certain angles.

To determine the number of angular nodes, you simply look at the angular momentum quantum number \( l \):

• Angular Nodes = \( l \)

Each type of orbital has a unique number of angular nodes.

• s-orbitals (\( l = 0 \)) have no angular nodes.
• p-orbitals (\( l = 1 \)) have one angular node.
• d-orbitals (\( l = 2 \)) have two angular nodes.
• f-orbitals (\( l = 3 \)) have three angular nodes.

The presence of angular nodes influences the shape of the orbital, with each additional node giving orbitals a more complex geometry.

Principal Quantum Number

The principal quantum number, denoted by \( n \), is one of the four quantum numbers used to describe the unique quantum state of an electron in an atom. It plays a pivotal role in defining the electron's energy level and distance from the nucleus.

Larger values of \( n \) correspond to higher energy shells and indicate that electrons are farther from the nucleus. The principal quantum number is always a positive integer:

• \( n = 1, 2, 3, 4, \)...

This number helps determine the size of the orbital and the energy of the electron within an atom. For instance, a 3p orbital has a principal quantum number of \( n = 3 \), which signifies it is in the third energy level of an atom. The principal quantum number also dictates the maximum number of electrons possible in a shell, calculated as \( 2n^2 \). Thus, for \( n = 3 \), a maximum of 18 electrons can occupy this shell.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted by \( l \), is integral to understanding the shape of electron orbitals. It defines the subshell or orbital type within a principal energy level and determines the angular shape and number of nodal surfaces in an orbital.

The value of \( l \) ranges from 0 to \( n-1 \), where \( n \) is the principal quantum number:

• \( l = 0 \) corresponds to an s-orbital
• \( l = 1 \) corresponds to a p-orbital
• \( l = 2 \) corresponds to a d-orbital
• \( l = 3 \) corresponds to an f-orbital

Each type of orbital has a distinct shape owing to the angular momentum quantum number. For example, the angular momentum quantum number \( l = 1 \), indicates a p-orbital which is dumbbell-shaped. Understanding \( l \) helps in visualizing how electrons are distributed around the nucleus and how they influence the chemical characteristics of an atom.