Problem 121 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statement (s) is /are incorrect? (a) \(3 \mathrm{~s}\) orbital has 2 radical nodes. (b) Electronic configuration of \(\mathrm{Co}^{3+}\) is \(4 \mathrm{~s}^{2} 3 \mathrm{~d}^{4}\). (c) \(2 \mathrm{p}\) orbital has 1 radial node. (d) Node is a region around the nucleus of an atom where probability of finding electron is maximum.

Step-by-Step Solution

Answer

Statements (b), (c), and (d) are incorrect.

Step 1: Understanding Radical Nodes in Orbitals

A radial node is a region within an atom where the probability of finding an electron is zero. The number of radial nodes in an orbital is given by the formula: \( n - l - 1 \), where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number. For the \(3s\) orbital, \(n = 3\) and \(l = 0\). Thus, the number of radial nodes is \(3 - 0 - 1 = 2\). Therefore, statement (a) is correct.

Step 2: Determining the Electronic Configuration of Ions

The electronic configuration for cobalt (_{27}Co) in its neutral state is [Ar] \(3d^7 4s^2\). When it loses three electrons to form \( Co^{3+} \), electrons are removed first from the 4s and then the 3d orbital, resulting in [Ar] \(3d^6\). Thus, statement (b) given as \(4s^2 3d^4\) is incorrect.

Step 3: Identifying Radial Nodes for 2p Orbitals

Using the formula from step 1, for a \(2p\) orbital, \(n = 2\) and \(l = 1\). Therefore, the number of radial nodes is \(2 - 1 - 1 = 0\). Statement (c) is incorrect as it states there is 1 radial node.

Step 4: Understanding Nodes

A node is properly defined as a region where the probability of finding an electron is exactly zero, not maximum. Therefore, statement (d) is incorrect as it claims that a node is where the probability is maximum.

Key Concepts

Radial Nodes

In quantum chemistry, understanding the concept of radial nodes is fundamental when studying electron behavior in atoms. A radial node is a spherical region within an orbital where the probability of finding an electron is exactly zero. This is quite intriguing as it indicates specific regions within an atom that are completely void of electron presence.
To determine the number of radial nodes, we can use the formula: \( n - l - 1 \). Here, \( n \) represents the principal quantum number which indicates the size and energy level of the orbital, and \( l \), the azimuthal quantum number that represents the shape of the orbital.
Consider the \( 3s \) orbital: with \( n = 3 \) and \( l = 0 \), the number of radial nodes is calculated as \( 3 - 0 - 1 = 2 \). This confirms that the \( 3s \) orbital indeed has 2 radial nodes. In contrast, a \( 2p \) orbital has \( n = 2 \) and \( l = 1 \). Thus, \( 2 - 1 - 1 = 0 \), showing no radial nodes are present in \( 2p \) orbitals. These calculated nodes help us visualize where electrons can and cannot be within an atom.

Electronic Configuration

Electronic configuration is a representation of how electrons are distributed in an atom's orbitals. Each electron in an atom is characterized by its position in specific orbitals, and this arrangement influences the chemical behavior of the element. Knowing an element's electronic configuration allows us to predict how it will bond with others and its place in the periodic table.
For instance, cobalt (Co) has an atomic number of 27. Its neutral electronic configuration is \([\text{Ar}] 3d^7 4s^2\). However, when cobalt forms a \( \text{Co}^{3+} \) ion, it loses three electrons. Electrons are removed starting from the \( 4s \) orbital before the \( 3d \) orbital as \( 4s \) electrons are higher in energy despite being filled before \( 3d \). This results in the configuration \([\text{Ar}] 3d^6\). Remembering this rule is crucial for correctly writing electronic configurations of ions.

Quantum Numbers

Quantum numbers play a crucial role in describing the quantum state of an electron in an atom. They determine the electron's position and energy within an atom, akin to an address for the electron. There are four quantum numbers, each with a specific significance:

• Principal Quantum Number (\( n \)): It indicates the energy level or shell of the electron, essentially dictating the size of the orbital.
• Azimuthal Quantum Number (\( l \)): This number determines the shape of the orbital, with specific values representing s, p, d, f shapes, etc.
• Magnetic Quantum Number (\( m_l \)): Determines the orientation of the orbital in space. For a given \( l \), \( m_l \) can take values from \(-l \) to \(+l \).
• Spin Quantum Number (\( m_s \)): It specifies the direction of the spin of the electron, with a possible value of \( + \frac{1}{2} \) or \( - \frac{1}{2} \).

These numbers are essential to understand as they help scientists and students predict electron configurations and properties of atoms, which are crucial for further chemical reactions and bonding analysis.