Problem 76 - A Complete Resource Book in Chemistry for JEE Main

Question

Radial nodes present in \(3 \mathrm{~s}\) and \(2 \mathrm{p}\) orbitals are respectively (a) 0,2 (b) 2,0 (c) 2,1 (d) 1,2

Step-by-Step Solution

Answer

Option (b) 2,0 is correct. 3s has 2 and 2p has 0 radial nodes.

Step 1: Understanding Radial Nodes

Radial nodes occur where the probability density of finding an electron is zero. They are determined for atomic orbitals using the formula: \( n - l - 1 \), where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number.

Step 2: Calculating Radial Nodes in the 3s Orbital

For the 3s orbital: the principal quantum number \( n = 3 \) and azimuthal quantum number \( l = 0 \) (since it's an s-orbital). Therefore, the number of radial nodes is: \( 3 - 0 - 1 = 2 \).

Step 3: Calculating Radial Nodes in the 2p Orbital

For the 2p orbital: the principal quantum number \( n = 2 \) and azimuthal quantum number \( l = 1 \) (since it's a p-orbital). Thus, the number of radial nodes is: \( 2 - 1 - 1 = 0 \).

Step 4: Identifying the Correct Option

By comparing the calculated radial nodes, we find that the 3s orbital has 2 radial nodes and the 2p orbital has 0 radial nodes. Hence, the correct option is (b) 2,0.

Key Concepts

Atomic Orbitals

Atomic orbitals are fundamental concepts in quantum mechanics that describe regions in an atom where there is a high probability of finding electrons. These orbitals are wave functions that are solutions to the Schrödinger equation, and each orbital is associated with particular energy levels.

Orbitals are categorized primarily into s, p, d, and f types:

• s-orbitals: Spherical in shape and can hold up to 2 electrons.
• p-orbitals: Dumbbell-shaped and can accommodate up to 6 electrons.
• d-orbitals and f-orbitals: More complex shapes, with d holding up to 10 electrons and f holding up to 14 electrons.

Understanding these orbitals is crucial because they determine how atoms bond and interact with each other. Each type of orbital offers specific insights into the arrangement and distribution of electrons around the nucleus.

Principal Quantum Number

The principal quantum number is a fundamental part of quantum mechanics that indicates the primary energy level of an electron in an atom. It is denoted by the symbol \( n \) and assumes positive integer values: 1, 2, 3, etc.

This number gives us insight into several aspects of an electron's position:

• Indicates the overall energy level and size of the electron cloud. Higher \( n \) values mean electrons have higher energy and are found further from the nucleus.
• Also dictates the count of sublevels within that energy level, as each \( n \) allows for n sublevels (e.g., n=3 includes s, p, and d sublevels).

By understanding \( n \), we can better anticipate electron arrangements, which affects reactivity and bonding characteristics of different elements.

Azimuthal Quantum Number

The azimuthal quantum number, often referred to as the angular momentum quantum number, identifies the shape and type of the orbital an electron occupies. It is denoted by the letter \( l \).

This quantum number can take on integer values ranging from 0 to \( n-1 \), where \( n \) is the principal quantum number. Each \( l \) value corresponds to a different orbital shape:

• \( l = 0 \): s-orbital, which is spherical.
• \( l = 1 \): p-orbital, which has a dumbbell shape.
• \( l = 2 \): d-orbital, which exhibits more complex shapes.
• \( l = 3 \): f-orbital, noted for even more intricate shapes.

Understanding \( l \) not only tells us about the orbital types but also helps in predicting the radial nodes and, consequently, the probability distribution of electrons in an atom.

3s Orbital

The 3s orbital belongs to the third principal energy level of an atom and is an example of an s-type orbital, noted for its spherical shape. It has the following characteristics:

• The principal quantum number, \( n \), is 3, indicating this orbital is in the third energy level, located further away from the nucleus compared to \( 1s \) or \( 2s \) orbitals.
• The azimuthal quantum number, \( l \), is 0, which confirms it is an s-orbital.
• Number of radial nodes is calculated as \( n - l - 1 = 3 - 0 - 1 = 2 \).

These radial nodes imply that within a 3s orbital, there are two spherical layers where the probability of finding an electron is zero. Such insights are crucial for understanding an atom's electron configuration and predicting its chemical behavior.

2p Orbital

The 2p orbital is part of the second principal energy level and is categorized as a p-type orbital. Its defining features include:

• Principal quantum number \( n \) is 2, showing it is part of the second energy level, relatively closer to the nucleus than any third-level orbitals.
• Azimuthal quantum number \( l \) is 1, indicating it is a p-orbital with a distinct dumbbell shape, typically oriented along the x, y, or z axes in three-dimensional space.
• Number of radial nodes is \( n - l - 1 = 2 - 1 - 1 = 0 \).

With no radial nodes, the 2p orbital's probability density does not cross zero along any radial direction from the nucleus. Comprehending these properties helps in visualizing electron distribution, which influences chemical bonding and reactivity.