Problem 138 - A Complete Resource Book in Chemistry for JEE Main

Question

Match the following Column-I (a) \(2 \mathrm{~s}, 3 \mathrm{~s}, 4 \mathrm{~s}, 4 \mathrm{~d}\) (order of increasing energy) (b) Quantum numbers of \(2 \mathrm{~s}^{2}\) electrons \(\begin{array}{cccc}\mathrm{n} & l & \mathrm{~m} & \mathrm{~s} \\ 2 & 0 & 0 & +1 / 2 \\ 2 & 0 & 0 & -1 / 2\end{array}\) Column-II (p) Bohr (q) Hund's rule (r) Pauli's exclusion principle (s) Aufbau principle (t) Size of orbit

Step-by-Step Solution

Answer

(a) matches with (s), (b) matches with (r).

Step 1: Analyzing Energy Levels

We need to order the given energy levels in Column-I part (a) by their increasing energy. The order is determined by the principle of Aufbau, where lower n and l values fill first. The order is: 2s < 3s < 4s < 4d. This matches with Aufbau principle, Column-II '(s)'.

Step 2: Understanding Quantum Numbers

Column-I part (b) lists the quantum numbers of the 2s electrons. The principle quantum number n=2 indicates the second shell. The azimuthal quantum number l=0 corresponds to the s orbital. Magnetic quantum number m=0 suggests only one position, and spin quantum number ±1/2 means paired electrons. These quantum numbers obey Pauli's exclusion principle, corresponding to Column-II '(r)'.

Key Concepts

Quantum Numbers

Quantum numbers are the unique set of numerical values that describe the specific energies and positions of electrons in an atom. There are four quantum numbers that together determine the arrangement of electrons:

• Principal Quantum Number ( ext{n} ): Indicates the main energy level or shell of an electron. It takes positive integers (1, 2, 3,...), with higher values corresponding to higher energy levels.
• Azimuthal Quantum Number ( ext{l} ): Defines the shape of the orbital. It takes integral values from 0 to ( ext{n} - 1 ). For example, for a principal quantum number ( ext{n} ) of 2, ( ext{l} ) can be 0 or 1.
• Magnetic Quantum Number ( ext{m} ): Describes the orientation of the orbital in space. It includes integers between -( ext{l} ) and +( ext{l} ).
• Spin Quantum Number ( ext{s} ): Refers to the spin of the electron, which can be either +1/2 or -1/2. This reflects the two possible spin orientations of an electron.

Each electron in an atom has a unique set of quantum numbers, making it possible to map all electrons in an element.

Aufbau Principle

The Aufbau Principle is a fundamental guideline in understanding how electrons populate the different energy levels in an atom. According to this principle, electrons fill orbitals starting from the lowest available energy levels before moving to higher ones. This sequential filling assumes that electrons will fill orbital levels in such a manner as to produce the lowest possible energy configuration for the atom. When organizing the order of filling, the principle depends on two quantum numbers:

• Principal Quantum Number (n)
• Azimuthal Quantum Number (l)

The energy of an orbital increases with higher values of ( ext{n} ) and ( ext{l} ). For instance, the sequence 2s, 3s, 4s, 4d reflects increasing energy, as determined by the rules of Aufbau.

Pauli's Exclusion Principle

Pauli's Exclusion Principle is pivotal in explaining the arrangement of electrons within individual orbitals. It states that no two electrons in a single atom can have the same set of four quantum numbers. This principle forms the basis for electron pairing and the unique electron configuration of elements. In simpler terms, in any given atom, an orbital can hold a maximum of two electrons, and these must have opposite spins. This is indicated by the spin quantum number, ext{s} , which can be +1/2 or -1/2. Because of Pauli’s exclusion principle, each electron in an atom has a unique identifier due to its unique combination of quantum numbers.

Energy Levels

Energy levels, or electron shells, are the fixed orbits around the nucleus of an atom where electrons are likely to be found. Each level corresponds to a different amount of energy that an electron can have. The higher the principal quantum number ( ext{n} ), the higher the energy level and the farther the electron is likely to be from the nucleus. In relation to the Aufbau Principle, electrons will naturally fill from lower to higher energy levels. This helps in understanding the electron configuration and the sequential occupation of orbitals by electrons, following patterns dictated by principles like Pauli’s exclusion and Hund's rule.

Hund's Rule

Hund's Rule is useful in determining how electrons are distributed among orbitals of the same sublevel (degenerate orbitals). The rule states that electrons will singly occupy each orbital of the same energy before pairing up. This concept minimizes electron repulsion within a sublevel and results in potential energy that is lower.

• Each orbital at a given energy level is filled singly first.
• Paired occupancy happens only when all orbitals in a sublevel are filled with one electron each.

This arrangement leads to a maximum multiplicity, meaning it ensures the electrons are as spread out as possible within a sublevel to reduce repulsion and stabilize the atom.