Problem 104 - A Complete Resource Book in Chemistry for JEE Main

Question

The electrons, identified by quantum numbers \(\mathrm{n}\) and \(l\) (i) \(\mathrm{n}=4 . l=1\) (ii) \(\mathrm{n}=4, l=0\) (iii) \(\mathrm{n}=3, l=2\) (iv) \(\mathrm{n}\) \(=3, l=1\) can be placed in order of increasing energy from the lowest to highest as (a) (iv) \(<(i i)<(i i i)<(i)\) (b) (ii) \(<(\mathrm{iv})<(\mathrm{i})<(\mathrm{iii})\) (c) (i) \(<\) (iii) \(<(i i)<\) (iv) (d) (iii) < (i) < (iv) < (ii)

Step-by-Step Solution

Answer

(b) (ii) < (iv) < (iii) < (i)

Step 1: Understanding Quantum Numbers

The quantum numbers use the principal quantum number \( n \) and the azimuthal quantum number \( l \) to describe the energy and shape of atomic orbitals. \( n \) indicates the shell, or distance from the nucleus, and \( l \) indicates the subshell type (0 = s, 1 = p, 2 = d, 3 = f).

Step 2: Determine Energy Order Rule

Generally, electron orbitals fill in increasing order of \( n + l \). If two orbitals have the same \( n + l \) value, the one with the lower \( n \) fills first, as per the Aufbau principle.

Step 3: Calculate \( n + l \) Values

Calculate the \( n + l \) values for each set of quantum numbers:- \((i): n = 4, l = 1\), so \( n + l = 5 \)- \((ii): n = 4, l = 0\), so \( n + l = 4 \)- \((iii): n = 3, l = 2\), so \( n + l = 5 \)- \((iv): n = 3, l = 1\), so \( n + l = 4 \)

Step 4: Compare \( n \) If \( n + l \) Is Equal

For (i) and (iii), both have \( n + l = 5 \), so compare \( n \): - (iii) has \( n = 3 \) and (i) has \( n = 4 \).Thus, (iii) fills before (i).

Step 5: Order the Configurations From Lowest to Highest Energy

The order based on \( n+l \) values and \( n \) comparisons is:- (ii): \( n + l = 4 \)- (iv): \( n + l = 4 \)- (iii): \( n + l = 5, n = 3 \)- (i): \( n + l = 5, n = 4 \)Thus, the correct order is (ii) < (iv) < (iii) < (i).

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by the symbol \( n \), is a fundamental component in the quantum mechanical model of atomic structure. This number is crucial as it describes the main shell or energy level of an electron within an atom. Simply put, think of \( n \) as representing the distance of an electron's orbit from the nucleus.

• A larger \( n \) corresponds to electrons that are further from the nucleus.
• These electrons have higher energy and are part of larger atomic orbitals.
• Commonly, \( n \) values range from 1 to 7 in ground state atoms, though theoretically, they can be any positive integer.

Understanding \( n \) is vital because it lays the groundwork for determining other quantum numbers and ultimately the electron configuration.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \( l \), defines the shape of the electron's orbital and is sometimes called the angular momentum quantum number. The value of \( l \) is intrinsically linked to the principal quantum number \( n \).

• \( l \) can range from 0 to \( n-1 \) for each principal quantum number \( n \).
• Each distinct value of \( l \) corresponds to a particular type of subshell or orbital type, which are:
  • \( l = 0 \): s-orbital (spherical)
  • \( l = 1 \): p-orbital (dumbbell shaped)
  • \( l = 2 \): d-orbital (more complex shapes)
  • \( l = 3 \): f-orbital (even more complex)

By determining \( l \), we understand more about the shape and energy associated with the electron's position, which is crucial for detailing electron configurations in multi-electron systems.

Aufbau Principle

The Aufbau Principle is a fundamental rule in chemistry that governs the sequence in which electrons populate available atomic orbitals. As atoms build in complexity, following this principle ensures that electrons fill orbitals in a specific, orderly fashion based on energy levels.

• The principle dictates that electrons occupy orbitals starting from the lowest energy level before moving to higher ones.
• A useful rule derived from the Aufbau principle is calculating \( n + l \) value to determine the order of filling.
• If two orbitals share an \( n + l \) value, the orbital with the lower \( n \) fills first.

The careful adherence to the Aufbau principle helps predict the electron configuration of atoms accurately, highlighting their chemical and physical properties, and plays a vital role in understanding the periodic table layout.