Problem 75 - A Complete Resource Book in Chemistry for JEE Main

Question

If the nitrogen atom has electronic configuration \(1 \mathrm{~s}^{7}\), it would have energy lower than that of the normal ground state configuration \(1 \mathrm{~s}^{2} 2 \mathrm{~s}^{2} 2 \mathrm{p}^{3}\), because the electrons would be closer to the nucleus. Yet \(1 \mathrm{~s}^{7}\) is not observed because it violates (a) Heisenberg uncertainty principle (b) Hund's rule (c) Pauli's exclusion principle (d) Bohr postulates of stationary orbits

Step-by-Step Solution

Answer

The configuration \(1\,\mathrm{s}^{7}\) is not observed because it violates the Pauli's exclusion principle (option c).

Step 1: Understanding Electron Configuration

The electron configuration tells us how electrons are distributed around the nucleus of an atom. Normally, electrons fill the lowest energy orbitals first, following a specific pattern or rules.

Step 2: Analyzing the Given Configuration

The suggested configuration for nitrogen is \(1\,\mathrm{s}^{7}\). This states that the 1s orbital contains 7 electrons, which is unusual because the 1s orbital can hold a maximum of 2 electrons.

Step 3: Applying Pauli's Exclusion Principle

Pauli's exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers. In a single orbital (like 1s), this means it can only hold 2 electrons—one with a spin up and one with spin down. Therefore, filling the 1s orbital with more than 2 electrons violates this principle.

Step 4: Conclusion on the Configuration's Validity

Given that \(1\,\mathrm{s}^{7}\) violates the Pauli's exclusion principle, it is not a feasible configuration. The correct distribution avoids filling any orbital beyond its allowed capacity according to these principles.

Key Concepts

Pauli's Exclusion Principle

The Pauli's Exclusion Principle is a fundamental rule in quantum mechanics that plays a crucial role in determining the electronic configuration of atoms. It was proposed by Wolfgang Pauli in 1925. According to this principle, no two electrons in an atom can have the same set of four quantum numbers. Each electron in an atom is described by four distinct quantum numbers: the principal quantum number (), the azimuthal quantum number (l), the magnetic quantum number (m), and the spin quantum number (s). This uniqueness ensures that electrons within the same orbital must differ by at least one quantum number, specifically the spin quantum number.

• The spin quantum number can either be +\(\frac{1}{2}\) or -\(\frac{1}{2}\), which means an orbital can only hold a maximum of two electrons with opposite spins.
• If more electrons are forced into an orbital than it can hold, the Pauli's Exclusion Principle is violated, as would occur in the suggested configuration of \(1s^7\) for nitrogen.
• This principle helps maintain the integrity of atomic structures, ensuring that electrons are distributed in a way that builds up the stability of the atom's energy state.

This rule is fundamental in explaining why electrons occupy different energy levels and why the periodic table has its specific structure.

Electron Orbitals

Electron orbitals are regions around the nucleus of an atom where electrons are likely to be found. They are defined by wave functions and are essential components of the quantum mechanical model of the atom. Each orbital can host a certain number of electrons and is characterized by its shape and energy.

• The variety of orbitals includes s, p, d, and f, each with unique shapes and capacities.
• The s orbitals are spherical and are the simplest, with the ability to hold up to 2 electrons.
• P orbitals are dumbbell-shaped and can hold up to 6 electrons in total, spread over three p orbitals in a shell.

Understanding orbitals is paramount as they define how electrons build up inside an atom, which subsequently determines the chemical properties of an element. Electrons fill these orbitals starting from the lowest energy, following principles like the Aufbau principle and, crucially, the Pauli's Exclusion Principle. This hierarchy explains the standard configuration of atoms, such as nitrogen's proper ground state configuration \(1s^2 2s^2 2p^3\), instead of anomalous, non-viable configurations.

Quantum Numbers

Quantum numbers are a set of numerical values that provide solutions to the Schrödinger equation for electrons in atoms. They describe the properties of atomic orbitals and the properties of electrons in those orbitals, giving a complete address for each electron in an atom. There are four quantum numbers:

• Principal quantum number \(n\): Indicates the energy level and relative distance of the orbital from the nucleus, with values like 1, 2, 3, etc.
• Azimuthal quantum number \(l\): Connected to the shape of the orbital, ranging from 0 to \(n-1\). For example, for \(n=3\), \(l\) could be 0 (s orbital), 1 (p orbital), or 2 (d orbital).
• Magnetic quantum number \(m\): Defines the orientation of the orbital with values between \(-l\) and \(+l\).
• Spin quantum number \(s\): Represents the electron's spin direction, either +\(\frac{1}{2}\) or -\(\frac{1}{2}\).

Together, these numbers offer a comprehensive description of an electron's position and movement within an atom, ensuring that each one is uniquely identifiable according to the Pauli's Exclusion Principle. This sequential filling of quantum states underlies the periodic table's order and explains the configuration of elements like nitrogen.