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 \(\begin{array}{cccc}\mathrm{n} & l & \mathrm{~m} & \mathrm{~s} \\ 2 & 0 & 0 & +1 / 2 \\ 2 & 0 & 0 & -1 / 2\end{array}\) (c) (d) Principal quantum no. \(n\) 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), (d) matches with (t).

Step 1: Understanding Column-I Elements

First, we need to understand each component of Column-I: (a) represents orbitals in the order of increasing energy. (b) pertains to quantum numbers for the electrons in the 2s2 configuration. (c) is indeterminate, assuming it's an error. (d) relates to a principal quantum number, typically denoted by 'n'.

Step 2: Analyzing Column-II Elements

In Column-II, each item relates to principles and terms in quantum mechanics: (p) Bohr model describes hydrogen-like atom's electron orbits. (q) Hund's rule suggests maximizing spin multiplicity. (r) Pauli's exclusion principle states no two electrons can have the same set of quantum numbers. (s) Aufbau principle provides guidance on electron filling order of orbitals. (t) Size of orbit relates to the principal quantum number determining orbital size.

Step 3: Matching Elements from Column-I to Column-II

Match each element as per its description:(a) corresponds to \(s\): the Aufbau principle, as it describes the order of filling electrons in orbitals based on increasing energy.(b) relates to \(r\): Pauli's exclusion principle, as it deals with quantum numbers, ensuring each electron is unique.(d) matches with \(t\): Size of orbit, since the principal quantum number 'n' is directly linked to the size of the orbit.

Key Concepts

Aufbau principle

The Aufbau principle is crucial in understanding how electrons are filled into orbitals when building an electron configuration. Essentially, this principle outlines the order in which electrons should fill an atom’s electron shells, starting from the lowest energy level to higher ones. It derives its name from the German word "Aufbau", meaning "building up" or "construction".

The basic idea is:

• Electrons occupy the lowest energy orbitals first.
• Once an orbital is filled, electrons fill into the next higher energy orbital.
• Following the order of increasing energy levels, this keeps the system stable and prevents unnecessary repulsion.

For example, consider the electron configuration of carbon. Using the Aufbau principle, we fill the 1s, then 2s, and then 2p orbitals sequentially. This ensures the electron arrangement is in its lowest energy state. When analyzing electronic configurations using the Aufbau principle, it helps us predict the chemical properties and behaviors of elements, as it defines how they interact and bond with other atoms.

Pauli's exclusion principle

Pauli's exclusion principle is a fundamental rule in quantum mechanics introduced by the physicist Wolfgang Pauli. This principle asserts that no two electrons within an atom can possess the identical set of four quantum numbers.

Understanding quantum numbers is essential here:

• The principal quantum number ( ) indicates the energy level and size of the orbital.
• The azimuthal quantum number (l) describes the shape of the orbital.
• The magnetic quantum number (m) specifies the orientation of the orbital in space.
• The spin quantum number (s) determines the direction of the electron's spin, either +1/2 or -1/2.

This principle is pivotal in assigning electrons to orbitals in an atom. It effectively allows a maximum of two electrons per orbital, and only if these electrons have opposite spins. For instance, in a 2s orbital configuration for electrons, one will have a spin of +1/2 and the other -1/2, ensuring compliance with Pauli's exclusion principle. This ultimately contributes to the unique electronic structure of each element.

Principal quantum number

The principal quantum number, denoted by 'n', refers to the primary factor determining an electron's energy level and the size of the orbital in which it resides. It is a positive integer (n = 1, 2, 3,...) and acts as the primary indicator for the distance between the nucleus and the electron itself.

Here's how it impacts the electron's placement:

• Higher values of 'n' indicate electrons located further from the nucleus.
• As 'n' increases, the orbital's size and energy levels also increase.
• The principal quantum number is integral in calculating the atom's total potential energy and plays a key role in determining its reactivity and bonding capabilities.

It's important to note that 'n' works in tandem with other quantum numbers to give a precise mapping of an electron's properties. For example, when discussing the principal quantum number in relation to the size of an orbit, we understand that greater 'n' translates to larger orbitals, helping us conceptualize electron placement in multi-electron systems.