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) \(<(\mathrm{iii})<(\mathrm{ii})<\) (iv) (d) (iii) < (i) < (iv) < (ii)

Step-by-Step Solution

Answer

The order is (ii) < (iv) < (i) < (iii), which is option (b).

Step 1: Understanding Quantum Numbers

The quantum numbers given are \(n\) and \(l\), where \(n\) is the principal quantum number indicating the energy level and \(l\) is the azimuthal quantum number indicating the subshell: \(l=0\) is \(s\), \(l=1\) is \(p\), \(l=2\) is \(d\).

Step 2: Calculating Energy Level using n and l

To determine the energy order of orbitals, we use the rule: the lower the sum of \(n + l\), the lower the energy. For the same \(n + l\) value, the orbital with the smaller \(n\) has lower energy.

Step 3: Calculate for Each Orbital

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

Step 4: Ordering by Increasing Energy

From the \(n + l\) values, order the orbitals from lowest to highest energy:1. (ii) \(n + l = 4\)2. (iv) \(n + l = 4\), but \(n = 3\) is lower than \(n = 4\)3. (i) \(n + l = 5\)4. (iii) \(n + l = 5\), but \(n = 3\) is lower than \(n = 4\)

Step 5: Selecting the Correct Option

Translate the order from Step 4 into the format given by the possible answer choices: The correct order based on increasing energy is (ii), (iv), (i), (iii), matching option (b).

Key Concepts

Principal Quantum Number

The principal quantum number is symbolized by \( n \). It plays an essential role in the atomic structure by **defining the energy level** of an electron within an atom. The value of \( n \) can be any positive integer, starting from 1. Each increase in \( n \) represents a higher energy level. Consequently, electrons in higher principal quantum levels are found further from the nucleus.

Important characteristics of \( n \):

• It describes the overall **size and energy** of an electron cloud.
• Higher \( n \) values correspond to higher energy levels, meaning the electron cloud is larger and the electron has more energy.

Therefore, the principal quantum number is often considered the most important of the quantum numbers when assessing electron energy. This includes the energy hierarchy noted in the exercise where electrons with lower \( n \) have lower energy when all other factors are equal.

Azimuthal Quantum Number

The azimuthal quantum number, denoted by \( l \), is crucial for understanding the **shape and type of subshell** an electron occupies within a given energy level. Also termed the orbital (or angular momentum) quantum number, \( l \) can take on integer values ranging from 0 to \( n-1 \). These include:

• \( l = 0 \): s subshell (spherical shape)
• \( l = 1 \): p subshell (dumbbell shape)
• \( l = 2 \): d subshell (more complex shape with four lobes)

Each increase in \( l \) represents a different shape of the electron cloud around the nucleus. The particular \( l \) value not only helps identify the type of orbital but also influences the electron's energy.

In the context of electron configuration, the azimuthal quantum number helps determine orbital shape and energy level. As noted in the exercise, when assessing which electron configuration has lower energy, \( l \) combined with \( n \) determines the electron's energy priority.

Orbital Energy Order

The order of electron energy in orbitals is determined using the **\( n + l \) rule**. This guideline ranks the energy of electrons by evaluating the sum of their respective quantum numbers \( n \) and \( l \).

• The lower the \( n + l \) value, the lower the energy of the electron.
• If two orbitals share the same \( n + l \) value, the one with the smaller \( n \) is lower in energy.

This method helps scientists systematically determine how electrons fill an atom's orbitals. In the exercise, this rule allowed us to rank the given set of orbitals by \( n + l \) values, helping to choose the correct electron arrangement. It's essential for predicting electron configuration and understanding chemical properties in elements.