Problem 20 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The electrons, identified by quantum numbers \(n\) and \(l\), (i) \(n=4, l=1\), (ii) \(n=4, l\) \(=0\), (iii) \(n=3, l=2\), and (iv) \(n=3, l=1\) can be placed in order of increasing energy, from the lowest to highest, as [1999-2 Marks] (a) (iv) \(<(\) ii \()<(\) iii \()<\) (i) (b) (ii) \(<\) (iv) \(<(\mathrm{i})<(\mathrm{iii})\) (c) (i) \(<(\) iii \()<(\) ii \()<\) (iv) (d) (iii) \(<(\mathrm{i})<(\mathrm{iv})<(\mathrm{ii})\)

Step-by-Step Solution

Answer

The correct order is (b): (ii) < (iv) < (i) < (iii).

Step 1: Identify the Quantum Numbers

The quantum numbers provided for each option are:(i) \(n=4,\ l=1\)(ii) \(n=4,\ l=0\)(iii) \(n=3,\ l=2\)(iv) \(n=3,\ l=1\)

Step 2: Determine the Energy Order Using (n + l) Rule

According to the (n + l) rule, the energy level is determined by the sum of the principal quantum number \(n\) and the azimuthal quantum number \(l\). The higher the sum, the higher the energy. If two electrons have the same \(n + l\) value, the one with the higher \(n\) is higher in energy.- For (i): \(n=4, l=1\) → \(n+l=4+1=5\)- For (ii): \(n=4, l=0\) → \(n+l=4+0=4\)- For (iii): \(n=3, l=2\) → \(n+l=3+2=5\)- For (iv): \(n=3, l=1\) → \(n+l=3+1=4\)

Step 3: Compare and Order by Energy

With the calculated \(n+l\) values, we can order the electrons from least to greatest energy:- Both (ii) and (iv) have \(n+l=4\); thus, the electron with the smaller \(n\) is lower in energy. Therefore, (iv) is before (ii).- Both (i) and (iii) have \(n+l=5\). However, they have different \(n\) values, with (i) having a higher \(n\), so (iii) is below (i) in energy.The ascending order is: (iv) \(<\) (ii) \(<\) (iii) \(<\) (i).

Step 4: Match with Given Options

Comparing the derived order with the given options, the correct option is:(b) (ii) \(<\) (iv) \(<\) (i) \(<\) (iii).

Key Concepts

n+l rule

The "n+l rule" is an important concept when trying to understand the energy levels of electrons in an atom. This rule helps us determine the sequence in which electrons fill up different atomic orbitals. The idea is quite straightforward:

• Calculate the sum of the principal quantum number (\(n\)) and the azimuthal quantum number (\(l\)).
• The combination with the lower \(n+l\) value will have lower energy, and hence, electrons will fill this combination first.
• In cases where two orbitals have the same \(n+l\) value, the one with the lower \(n\) is filled first.

This rule helps in predicting the order in which electrons are added to the different subshells across the periodic table. As seen in the exercise, by using the \(n+l\) rule, you can easily determine how to order electrons based on their energies, enabling an understanding of which orbitals are occupied first.

energy levels

Energy levels in an atom describe how close an electron can be to the nucleus, depicted by various shells and subshells. Each shell corresponds to a unique principal quantum number (\(n\)), and these shells consist of one or more subshells identified by the azimuthal quantum number (\(l\)). Chemically, the closer an electron is to the nucleus, the lower its energy level. As electrons fill successive energy levels, they occupy spaces further from the nucleus, resulting in higher energy.

• Lower energy levels are typically filled before higher ones, following the \(n+l\) rule.
• The specific energy level of an electron impacts the chemical properties and reactivity of the element.

Understanding energy levels is fundamental in predicting how electrons distribute in different atomic subshells and how these distributions affect atomic behavior.

principal quantum number

The principal quantum number, denoted by \(n\), is crucial in determining the energy of an electron in an atom. It defines the main energy level or shell where the electron resides, serving as a "ladder" of energy levels. Important points about \(n\) include:

• \(n\) is always a positive integer. Its values can start from 1, 2, 3, etc., extending upwards depending on the number of electrons in the atom.
• The larger the \(n\), the higher the energy level and the further the electron is from the nucleus.
• Electrons at higher \(n\) values are in larger orbitals and have more energy compared to those at lower \(n\).

Given its influence on the energy and size of atomic orbitals, \(n\) plays a vital role in electron configuration and ultimately in determining an atom's chemical properties.

azimuthal quantum number

The azimuthal quantum number, also known as the angular momentum quantum number and denoted by \(l\), describes the shape of the orbital an electron occupies, which is pivotal in understanding atomic orbitals. Here’s what you should know about \(l\):

• \(l\) can take on integer values ranging from 0 to \(n-1\), where \(n\) is the principal quantum number of the electron’s shell.
• The values of \(l\) correspond to different types of subshells: \(l=0\) (s subshell), \(l=1\) (p subshell), \(l=2\) (d subshell), and \(l=3\) (f subshell).
• The different \(l\) values denote orbitals of different shapes and, hence, different spatial distributions of electron density.

By determining the shape and orientation of orbitals, \(l\) plays a crucial role in defining the energy and arrangement of electrons within an atom, as described in the exercise using the \(n+l\) rule.