Problem 126 - A Complete Resource Book in Chemistry for JEE Main

Question

Four different set of quantum numbers for 4 electrons are given below : \(\mathrm{e}_{1}=4,0,0-1 / 2 ; \mathrm{e}_{2}=3,1,1-1 / 2\) \(\mathrm{e}_{3}=3,2,2+1 / 2 ; \mathrm{e}_{4}=3,0,0,+1 / 2\) Then which of the following order of energies of \(\mathrm{e}_{1}, \mathrm{e}_{2}\), \(\mathrm{e}_{3}\) and \(\mathrm{e}_{4}\) are against the quantum rules ? (a) \(\mathrm{e}_{1}>\mathrm{e}_{2}>\mathrm{e}_{3}>\mathrm{e}_{4}\) (b) \(\mathrm{e}_{4}>\mathrm{e}_{3}>\mathrm{e}_{2}>\mathrm{e}_{1}\) (c) \(\mathrm{e}_{3}>\mathrm{e}_{1}>\mathrm{e}_{2}>\mathrm{e}_{4}\) (d) \(e_{2}>e_{3}>e_{4}>e_{1}\)

Step-by-Step Solution

Answer

Option (a) is against quantum rules.

Step 1: Review Quantum Numbers

Each electron is described by four quantum numbers: \(n\), \(l\), \(m_l\), and \(m_s\). The principal quantum number \(n\) defines the energy level. The azimuthal quantum number \(l\) defines the subshell, with values ranging from 0 to \(n-1\). The magnetic quantum number \(m_l\) can take values between \(-l\) to \(+l\). Finally, the spin quantum number \(m_s\) can be either \(+1/2\) or \(-1/2\).

Step 2: Determine Energy Levels Based on n and l

The general rule for determining the energy of an electron is to use \(n+l\) values. If two electrons have the same \(n+l\), the one with the lower \(n\) is lower in energy. Use this to calculate for each electron: - \(e_1: n+l = 4+0 = 4\) - \(e_2: n+l = 3+1 = 4\) - \(e_3: n+l = 3+2 = 5\) - \(e_4: n+l = 3+0 = 3\)

Step 3: Rank Energies of Electrons

Based on the \(n+l\) rule: - \(e_4\) has the lowest energy (\(n+l = 3\)).- \(e_1\) and \(e_2\) are next, both have an \(n+l\) value of 4, requiring further judgment using \(n\) (both have \(n=3\), so they are equal in energy level).- \(e_3\) has the highest energy with \(n+l = 5\).

Step 4: Identify Which Option Violates Quantum Rules

Review each energy order given in the options:(a) \(e_1 > e_2 > e_3 > e_4\),(b) \(e_4 > e_3 > e_2 > e_1\),(c) \(e_3 > e_1 > e_2 > e_4\),(d) \(e_2 > e_3 > e_4 > e_1\).Comparing each option with actual calculated energy levels (\(e_4 < e_1 = e_2 < e_3\)),Option (a) contradicts it the most as it has \(e_1\) as highest even though \(e_4\) should be lower than others.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), is a fundamental part of the quantum mechanical description of atoms. It determines the overall size and energy level of an electron's orbital.

• \( n \) can take positive integer values: 1, 2, 3, and so on.
• The larger the value of \( n \), the higher the energy level and the further the orbital is from the nucleus.
• Electrons in higher energy levels with larger \( n \) values are generally less tightly bound to the nucleus and are said to have higher potential energy.

The principal quantum number is crucial in the ranking of electron energies when combined with other quantum numbers to evaluate the electron's placement within an atom.

Azimuthal Quantum Number

Also known as the angular momentum quantum number, the azimuthal quantum number, represented by \( l \), characterizes the subshell of an electron within an energy level.

• For a given principal quantum number \( n \), \( l \) can take integer values from 0 to \( n-1 \).
• This quantum number determines the shape of the orbital and influences angular momentum.
• The subshells corresponding to different \( l \) values are denoted using letters: \( l = 0, 1, 2, 3 \) are referred to as s, p, d, and f, respectively.

The azimuthal quantum number helps to distinguish between different types of orbitals within the same energy level, affecting their energy due to variations in electron cloud shapes.

Electron Energy Levels

Electron energy levels in atoms are dictated by the combined values of quantum numbers. Primarily, energy is influenced by the \( n+l \) rule, a useful guideline to rank their energies.

• The sum \( n+l \) provides a general gauge of the energy an electron in an orbital has.
• If electrons share the same \( n+l \) value, the one with the lower \( n \) has lower energy, as it is closer to the nucleus.
• This rule helps predict which electron configurations will be more stable. Higher \( n+l \) values indicate higher energy and therefore less stability.

Using this information, we can predict how electrons are arranged in orbitals and their respective energy levels.

Magnetic Quantum Number

The magnetic quantum number, symbolized as \( m_l \), assists in detailing an electron's orientation within a subshell. This quantum number is important for understanding more about individual electron properties.

• For a given azimuthal quantum number \( l \), \( m_l \) can vary from \(-l\) to \(+l\), including zero.
• This range allows \( 2l+1 \) possible orientations or orbitals within a subshell.
• The magnetic quantum number is vital when considering how electrons fill these orbitals, especially when addressing magnetic fields or electron placement in multi-electron atoms.

Though it doesn't affect energy levels directly like \( n \) or \( n+l \), \( m_l \) is useful in complex atomic settings, where electron distribution needs to be precisely understood.