Problem 34 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The maximum number of electrons that can have principal quantum number, \(n=\) 3, and spin quantum \(m_{s}=-\frac{1}{2}\), is

Step-by-Step Solution

Answer

9 electrons can have \( n = 3 \) and \( m_s = -\frac{1}{2} \).

Step 1: Understanding Quantum Numbers

Quantum numbers are used to describe the position and spin of electrons in an atom. The principal quantum number, \( n \), determines the shell, and the spin quantum number, \( m_s \), has values of \( +\frac{1}{2} \) or \( -\frac{1}{2} \). For this exercise, we're considering electrons where \( n = 3 \) and \( m_s = -\frac{1}{2} \).

Step 2: Determine Number of Electrons Per Shell

For \( n = 3 \), we need to determine the number of electrons that can fill all available orbitals for this shell. Electrons fill according to specific subshells: \(3s\), \(3p\), and \(3d\). These subshells have 1, 3, and 5 orbitals respectively.

Step 3: Calculate Electrons with Specific Spin

Each orbital can accommodate 2 electrons with opposite spins. Therefore, for \( m_s = -\frac{1}{2} \) spin:- \(3s\) contains 1 orbital so holds 1 electron with \( m_s = -\frac{1}{2} \).- \(3p\) contains 3 orbitals allowing 3 electrons with \( m_s = -\frac{1}{2} \).- \( 3d \) contains 5 orbitals, accommodating 5 electrons with \( m_s = -\frac{1}{2} \).

Step 4: Summing the Electrons

Add the electrons with spin \( m_s = -\frac{1}{2} \) from each subshell: 1 from \(3s\), 3 from \(3p\), and 5 from \(3d\). That makes a total of 1 + 3 + 5 = 9 electrons that meet the criteria.

Key Concepts

Principal Quantum Number

The principal quantum number, designated as \( n \), is one of the four quantum numbers which define the energy and position of an electron in an atom. It primarily determines the energy level or "shell" of an electron in an atom, represented by integers starting from 1, where higher numbers correspond to higher energy levels.

For an electron with \( n = 3 \), it means that this electron is present in the third energy level of an atom. This third energy level supports electrons in the following subshells: 3s, 3p, and 3d. Each of these subshells contains a specific number of orbitals, where orbitals are regions around the nucleus where electrons are most likely to be found.

• The 3s subshell has 1 orbital.
• The 3p subshell has 3 orbitals.
• The 3d subshell includes 5 orbitals.

Understanding the principal quantum number helps in identifying how electrons configure themselves around the nucleus and also how energy changes across these levels.

Spin Quantum Number

The spin quantum number, denoted as \( m_s \), describes the intrinsic angular momentum or "spin" of an electron within an orbital. Electrons can have a spin of either \( +\frac{1}{2} \) or \( -\frac{1}{2} \). These values indicate the two possible orientations of an electron's spin in a magnetic field.

When arranging electrons in an atom, each orbital can hold up to two electrons, and these electrons must have opposite spins. This means if one electron has a spin of \( +\frac{1}{2} \), the other must have a spin of \( -\frac{1}{2} \). Hence, understanding the spin is crucial for predicting magnetic properties of an atom and ensuring that electrons are correctly paired within their respective orbitals.

In the context of this exercise, we are examining electrons with \( m_s = -\frac{1}{2} \) in the \( n = 3 \) energy level. Every orbital from the subshell 3s, 3p, and 3d can host one electron with this specific spin.

Electron Configuration

Electron configuration explains the distribution of electrons in an atom’s orbitals. It helps define the arrangement of electrons in various shells, subshells, and orbitals. Typically, electrons fill orbitals starting from the lowest energy level to the higher ones, a principle known as the Aufbau principle.

For \( n = 3 \):

• 3s subshell: holds 2 electrons in 1 orbital.
• 3p subshell: holds 6 electrons across 3 orbitals.
• 3d subshell: can accommodate 10 electrons in 5 orbitals.

Each orbital can accommodate a maximum of 2 electrons, each with opposite spins. Hence, to determine the total number of electrons that can have the same specific spin of \( m_s = -\frac{1}{2} \), we calculate as follows:

• 3s can hold 1 electron with \( m_s = -\frac{1}{2} \).
• 3p can accommodate 3 electrons with \( m_s = -\frac{1}{2} \).
• 3d can host 5 electrons with \( m_s = -\frac{1}{2} \).

Summing these values gives the total of 9 electrons meeting the problem criteria. Understanding the electron configuration allows chemists to predict chemical properties and reactivities of elements.