Problem 3 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Consider the hypothetical situation where the azimuthal quantum number, \(l\), takes values \(0,1,2, \ldots . n+1\), where \(n\) is the principal quantum number. Then, the element with atomic number : [Main Sep. 03, 2020 (II)] (a) 9 is the first alkali metal (b) 13 has a half-filled valence subshell (c) 8 is the first noble gas (d) 6 has a \(2 p\)-valence subshell

Step-by-Step Solution

Answer

Option (d) is correct; atomic number 6 has a \( 2 p \)-valence subshell.

Step 1: Understanding Quantum Numbers

The azimuthal quantum number, denoted as \( l \), determines the shape of the orbital and typically ranges from \( 0 \) to \( n-1 \). In this hypothetical situation, it ranges from \( 0 \) to \( n+1 \). This means for the principal quantum number \( n \), \( l \) can take \( n+2 \) different values.

Step 2: Determine the Electron Configuration

The question implies that the periodic table format is altered due to the new range of \( l \). We need to understand the electron filling order influenced by \( n+1 \) values. Start by filling \( n = 1 \) to \( n = 3 \) shells with \( l = 0 \) to \( n+1 \), then map the atomic numbers to these configurations.

Step 3: Analyze Option (a): Atomic Number 9 as First Alkali Metal

Alkali metals have a single electron in the outermost shell. With the given rules, atoms \( n = 1 \) have \( l = 0, 1, 2 \); atoms at \( n = 2 \) are filled until \( Z=9 \). Therefore, \( Z=9 \) should have a full \[ s \] subshell. Check if remaining electrons fill as given, resulting in a single electron in a \[ p \] subshell.

Step 4: Analyze Option (b): Atomic Number 13 Half-Filled Valence Subshell

Half-filled subshell indicates subshell with \( (l+1)/2 \) electrons. Check configuration for \( Z=13 \), evaluating the distribution to see if it results in a half-filled \( p \) or \( d \) subshell.

Step 5: Analyze Option (c): Atomic Number 8 as First Noble Gas

Noble gases have full outer electron shells. With this periodic table, compute the configuration for \( Z=8 \). Examine if it has a closed outer shell indicating nobility.

Step 6: Analyze Option (d): Atomic Number 6 with \( 2p \)-Valence Subshell

For \( n=2 \), \( l\) allows possibilities like \( 0 \), \( 1 \), \( 2 \). Find the electron configuration for \( Z=6 \) to confirm if it includes a \( p \) orbital in the valence shell.

Step 7: Evaluate and Compare the Options

Compare each statement against the hypothetical situation. Accurate determination depends on confirming which condition matches the new filling order of electrons in each scenario described.

Key Concepts

Azimuthal Quantum Number

The azimuthal quantum number, commonly denoted as \( l \), plays a crucial role in determining the shape of atomic orbitals. Normally, \( l \) ranges from \( 0 \) to \( n-1 \), where \( n \) is the principal quantum number. This means that for any given value of \( n \), the azimuthal quantum number allows us to depict the subshells associated with an electron configuration. Each \( l \) value corresponds to a different type of orbital:

• \( l = 0 \): s-orbital (spherical shape)
• \( l = 1 \): p-orbital (dumbbell shape)
• \( l = 2 \): d-orbital (cloverleaf shape)
• \( l = 3 \): f-orbital (complex shapes)

However, in the hypothetical scenario we are examining, \( l \) can take values from \( 0 \) all the way to \( n+1 \). This adjustment drastically changes the electron configuration scaffolding by adding new possible orbitals, thereby affecting how electrons populate these orbitals. A better understanding of this modified range illustrates the fundamental adaptability of electron configurations under different theoretical conditions.

Principal Quantum Number

The principal quantum number, symbolized by \( n \), defines the principal energy level or shell of an electron. The value of \( n \) is a positive integer: 1, 2, 3, etc.

This number not only designates the energy level of the electron but also indicates the average distance of the electron from the nucleus:

• The larger the value of \( n \), the higher the energy level and, generally, the farther the electron is from the nucleus.
• Each energy level can hold a fixed number of electrons, determined by the formula \( 2n^2 \).

In our exercise, adjusting the values of \( l \) to range up to \( n+1 \) requires a re-evaluation of how these principal quantum numbers function within theoretical atom models. This modification impacts not only the potential electron subshells but also the energy hierarchy within each shell, leading to novel interpretations of electron configurations.

Electron Configuration

Electron configuration describes the distribution of electrons among the various orbitals of an atom. Typically, electrons fill orbitals in a specific order, known as Aufbau principle: from lower to higher energy orbitals. This filling order follows the sequence seen in a standard periodic table.

In our modified scenario, with \( l \) ranging up to \( n+1 \), the periodic table layout is dramatically different. Here's how this affects electron configurations:

• Each principal energy level will potentially host more subshells than conventionally allowed, altering the traditional Aufbau principle.
• As a consequence, elements acquire different electron configurations, resulting in unexpected properties and new placements in a hypothetical periodic table.

This allows us to explore theoretical scenarios where the elemental characteristics based on electron configuration deviate from traditional ones, leading to engaging insights into atomic structure complexity and variability.