Problem 9 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The quantum number of four electrons are given below: I. \(n=4, l=2, m_{l}=-2, m_{s}=-1 / 2\) II. \(n=3, l=2, m_{l}=1, m_{s}=+1 / 2\) III. \(n=4, l=1, m_{l}=0, m_{s}=+1 / 2\) IV. \(n=3, l=1, m_{l}=1, m_{\mathrm{s}}=-1 / 2\) The correct order of their increasing energies will be : (a) IV \(<\mathrm{III}<\mathrm{II}<\mathrm{I}\) (b) \(\mathrm{I}<\mathrm{II}<\mathrm{III}<\mathrm{IV}\) (c) \(\mathrm{IV}<\mathrm{II}<\mathrm{III}<\mathrm{I}\) (d) \(\mathrm{I}<\mathrm{III}<\mathrm{II}<\mathrm{IV}\)

Step-by-Step Solution

Answer

The order of increasing energies is IV < II < III < I.

Step 1: Understand the Quantum Numbers

Each electron is described by four quantum numbers: the principal quantum number \( n \), azimuthal quantum number \( l \), magnetic quantum number \( m_l \), and spin quantum number \( m_s \). The principal quantum number \( n \) indicates the shell, while the azimuthal quantum number \( l \) indicates the subshell (0 for \( s \), 1 for \( p \), 2 for \( d \), etc.). The energy of an electron in a multi-electron atom is primarily determined by the \( n \) and \( l \) values.

Step 2: Determine Energy Levels Based on \( n \) and \( l \)

The energy of the electron increases with increasing \( n + l \) value. If two electrons have the same \( n + l \) value, the one with the lower \( n \) value is lower in energy. This rule is known as the (n+l) rule or Madelung rule.

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

- For electron I: \( n=4, l=2 \) gives \( n+l=6 \).- For electron II: \( n=3, l=2 \) gives \( n+l=5 \).- For electron III: \( n=4, l=1 \) gives \( n+l=5 \).- For electron IV: \( n=3, l=1 \) gives \( n+l=4 \).

Step 4: Rank Electrons by Increasing Energy

Based on \( n+l \) values:- Electron IV (\( n+l=4 \)) is lowest in energy.- Electrons II and III both have \( n+l=5 \), but II has a lower \( n \) value than III, making II lower in energy than III.- Electron I, with \( n+l=6 \), is highest in energy.

Step 5: Write the Correct Order

Thus, the correct order of increasing energies is IV < II < III < I.

Key Concepts

Madelung Rule

The Madelung rule, also known as the (n+l) rule, is a guideline to predict the distribution of electrons among the orbitals of an atom in its ground state. This rule is instrumental when determining the order in which atomic orbitals are filled as electrons are added to an atom. The sum of the principal quantum number ( ") and the azimuthal quantum number ( $ " ) determines the order of orbital filling. Lower (n+l) values indicate lower energy levels and are filled first.

Here are the key points about the Madelung rule:

• For orbitals with the same (n+l) value, the one with the lower principal quantum number ( " ") is filled first because it's lower in energy.
• The rule applies to atoms in their ground state and helps determine electron configurations that correspond to the lowest energy.
• The sequence of filling based on the Madelung rule is reflected in the periodic table, explaining the general order: 1s, 2s, 2p, 3s, 3p, 4s, and so on.

Electron Configuration

Electron configuration is the arrangement of electrons in an atom's electron shells, subshells, and orbitals. It plays a crucial role in understanding chemical properties and reactivity. An electron configuration states how electrons are distributed among different orbitals, considering the energy levels and subshells.

When writing electron configurations:

• Identify the number of electrons in an atom, which equals its atomic number in a neutral atom.
• Use the Madelung rule and Pauli's exclusion principle to fill electrons in the appropriate orbitals.
• Represent the distribution using notation like 1s², 2s², 2p⁶, where 1s² refers to the s orbital in the first energy level holding two electrons.

Electron configuration influences chemical bonding and properties by dictating valence electrons' behavior and interactions. Configurations also explain periodic trends such as atomic size and ionization energy.

Energy Levels

Energy levels in atoms are specific zones where electrons are likely to be found. These levels are quantized, meaning electrons can only exist at specific energies. Each principal quantum number ( $ " ") corresponds to a different shell and a potential energy level for electrons in an atom.

Important aspects of energy levels include:

• The principal quantum number ( " ") determines the size and energy of the shell, with higher n values indicating higher energy and larger electron clouds.
• Subshells within each energy level, designated by the azimuthal quantum number ( $ " "), define the shape and specific energy of orbitals (s, p, d, f).
• Within these subshells, electrons can occupy specific orbitals with energies denoted by combinations like 3p or 4d, where the lowercase letter signifies the subshell and the number indicates the principal energy level.

Understanding energy levels is fundamental for explaining how electrons fill up in atoms and how this affects an atom's chemical properties and reactivity.