Problem 139 - A Complete Resource Book in Chemistry for JEE Main

Question

Match the following Column-I (a) \(2 \mathrm{~s}\) (b) \(2 \mathrm{p}\) (c) \(3 \mathrm{~s}\) (d) \(3 \mathrm{p}\) Column-II (p) sum of \((\mathrm{n}+1)\) is 3 (q) total number of nodes are two (r) Only one node (s) No radial node (t) No angular node

Step-by-Step Solution

Answer

(a) matches (r) and (t), (b) matches (p) and (s), (c) matches (q) and (t), (d) matches uncertain based on given properties.

Step 1: Identify Principles for Quantum Numbers

To match the given orbitals with the properties, we need to understand quantum numbers. The principal quantum number (n) indicates the shell, and the azimuthal quantum number (l) defines the subshell. For s orbitals, l = 0, and for p orbitals, l = 1. Nodes include radial nodes which depend on (n - l - 1) and angular nodes, which equal l.

Step 2: Consider Each Option in Column-I

For each option in Column-I, calculate the corresponding properties. - (a) **2s**: n = 2, l = 0. - (b) **2p**: n = 2, l = 1. - (c) **3s**: n = 3, l = 0. - (d) **3p**: n = 3, l = 1.

Step 3: Match (a) 2s with Column-II

The 2s orbital (n = 2, l = 0) has 1 radial node since n - l - 1 = 1, and 0 angular nodes because l = 0. Thus, (a) 2s matches (r) Only one node and (t) No angular node.

Step 4: Match (b) 2p with Column-II

The 2p orbital (n = 2, l = 1) has 0 radial nodes and 1 angular node because l = 1. The matching is with (p) sum (n+l) = 3 and (s) No radial node.

Step 5: Match (c) 3s with Column-II

The 3s orbital (n = 3, l = 0) has 2 radial nodes (n - l - 1 = 2) and 0 angular nodes. This matches (q) total number of nodes are two and (t) No angular node.

Step 6: Match (d) 3p with Column-II

The 3p orbital (n = 3, l = 1) has 1 radial node and 1 angular node (l = 1). Match this with (p) sum (n+l) = 4 if available properties in the column may allow matching to sum calculations; however, flexibility will depend confirmed alignments in x or y.

Key Concepts

Orbital Nodes

In quantum mechanics, nodes are regions where the probability of finding an electron is zero. There are two main types of nodes: radial and angular. Radial nodes are spherical surfaces within an atom where there is no probability of finding an electron, and they depend on both the principal quantum number (n) and the azimuthal quantum number (l). Specifically, the number of radial nodes can be calculated using the formula \[ \text{Radial nodes} = n - l - 1 \].
On the other hand, angular nodes are planes where the electron probability drops to zero, and they are determined solely by the azimuthal quantum number (l).
To calculate the total number of nodes, add the radial and angular nodes together. Recognizing these concepts is crucial to predict electron distributions in atoms and their corresponding quantum characteristics.

s and p Orbitals

In atomic structure, orbitals describe where electrons are likely to be found. The simplest type of orbital is the s orbital, which is spherical with no angular nodes. This means the probability of locating the electron is the same at all angles from the nucleus.
For s orbitals, since the azimuthal quantum number \( l = 0 \), they do not have angular nodes, only radial nodes.
In contrast, p orbitals, which have \( l = 1 \), exhibit one angular node. This results in their characteristic dumbbell shape, as the probability of finding an electron is zero along the plane that the node lies in. Moreover, since p orbitals differ in orientation, they contribute to the shape and magnetic properties of the atom.
Understanding these fundamental shapes helps in illustrating how atoms bond and interact in complex ways.

Matching Quantum Characteristics

To match orbitals with their respective quantum characteristics, understanding the principal \( n \) and azimuthal \( l \) quantum numbers is essential. The principal quantum number (n) dictates the energy level or shell that an electron occupies, while the azimuthal number (l) designates the shape of the orbital.
For example, consider a 3p orbital: here, \( n = 3 \) and \( l = 1 \). The calculation for total nodes, which includes both radial and angular nodes, can be instrumental in matching quantum characteristics. In this case:

• 1 radial node: since \( n - l - 1 = 1 \)
• 1 angular node: because \( l = 1 \)

This results in two total nodes and a sum of quantum numbers \( (n+l) = 4 \).
Such mathematical technique enables precise pairing of orbitals to given properties, ensuring that scientists and students alike can understand and predict the electronic structure of atoms.