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 orbital where radial and angular nodes are the same is the \(5d\) orbital.

Step 1: Understanding Nodes

To find out in which orbitals radial nodes and angular nodes are the same, we need to understand both types of nodes. Radial nodes are regions where the probability of finding an electron is zero and occur due to the radial part of the wave function. Angular nodes are related to the shape of the orbital and occur due to the angular part of the wave function.

Step 2: Expression for Nodes

The number of radial nodes is given by the formula: \(n - ext{l} - 1\) and the number of angular nodes is equal to \( ext{l}\), where \(n\) is the principal quantum number and \(l\) is the azimuthal quantum number.

Step 3: Calculating for Each Orbital

- For \(3p\): \(n = 3, l = 1\). Thus, radial nodes \(= 3 - 1 - 1 = 1\), angular nodes \(= 1\).- For \(4p\): \(n = 4, l = 1\). Thus, radial nodes \(= 4 - 1 - 1 = 2\), angular nodes \(= 1\).- For \(6f\): \(n = 6, l = 3\). Thus, radial nodes \(= 6 - 3 - 1 = 2\), angular nodes \(= 3\).- For \(5d\): \(n = 5, l = 2\). Thus, radial nodes \(= 5 - 2 - 1 = 2\), angular nodes \(= 2\).

Step 4: Identifying Matching Nodes

We are looking for the orbital where the radial nodes equal the angular nodes. Here, only the \(5d\) orbital has the same number of radial nodes (2) and angular nodes (2).

Key Concepts

Radial Nodes

Radial nodes are fascinating in quantum chemistry and are closely linked to the structure of an atom's electron cloud. These nodes occur in spherically symmetric shells and represent regions where the probability of finding an electron drops to zero. In other words, these are points within the radial distribution of the electron cloud where electrons simply can't exist.

The number of radial nodes in an orbital is determined by:

• The principal quantum number () which describes the size of the orbital
• And the azimuthal quantum number (ℓ) which indicates the shape of the orbital

To find out the number of radial nodes, you can use the formula: \(n - \ell - 1\). For example, in a 5d orbital, we have a principal quantum number of 5 and an azimuthal quantum number of 2. Calculating, we get: \(5 - 2 - 1 = 2\) radial nodes. Simple, yet incredibly insightful when understanding electron probabilities across different orbitals.

Angular Nodes

Angular nodes are regions tied to the angular part of the wave function of an electron. These nodes are responsible for the specific shapes of orbitals and define where an electron cannot be found due to angular orientation, rather than radial distance.

These nodes are directly related to the azimuthal quantum number (ℓ). The number of angular nodes is simply equivalent to ℓ. Here’s a quick breakdown:

• P orbitals (ℓ = 1) have one angular node
• D orbitals (ℓ = 2) have two angular nodes
• F orbitals (ℓ = 3) have three angular nodes

A clear example is the 5d orbital, where ℓ equals 2, resulting in 2 angular nodes. These nodes are not concentric spheres, but rather, they form planar or conical shapes depending on the type of orbital. They're crucial for defining the geometric structure of orbitals and understanding how electrons fill these regions.

Quantum Numbers

Quantum numbers are key to unlocking the mysteries of atomic structure. They describe specific characteristics of electrons within atoms and help us determine things such as an electron's energy level and the shape of its path around the nucleus.

Four important quantum numbers include:

• The principal quantum number ( ): It tells us the shell or energy level of an electron.
• The azimuthal quantum number (ℓ): It defines the shape of the orbital (s, p, d, f).
• The magnetic quantum number (m): It details the specific orbital within a subshell that an electron occupies.
• The spin quantum number (s): It describes the spin and magnetic orientation of electrons.

Putting these together, quantum numbers help explain the probable location of electrons and the arrangement of atoms in periodic tables. They are fundamental for understanding electron configurations and behaviors inherent to elements. In exercises, like the one with orbitals and nodes, these numbers guide us to predictively solve problems about electron positioning.