Problem 170 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following sets of quantum numbers represents the highest energy of an atom? \(\quad\) (a) \(\mathrm{n}=3, l=2, m=1, \mathrm{~s}=+1 / 2\) (b) \(\mathrm{n}=4, l=0, \mathrm{~m}=0, \mathrm{~s}=+1 / 2\) (c) \(\mathrm{n}=3, l=0, \mathrm{~m}=0, \mathrm{~s}=+1 / 2\) (d) \(\mathrm{n}=3, l=1, \mathrm{~m}=1, \mathrm{~s}=+1 / 2\)

Step-by-Step Solution

Answer

Option (b) \( n=4, l=0, m=0, s=+1/2 \) has the highest energy.

Step 1: Understanding Quantum Numbers

Quantum numbers are used to describe the electron configuration in an atom. The principal quantum number \( n \) indicates the energy level of an electron and its relative distance from the nucleus. The azimuthal quantum number \( l \) refers to the shape of the electron's orbital, with values ranging from 0 to \( n-1 \). The magnetic quantum number \( m \) indicates the orientation of the orbital, ranging from \( -l \) to \( +l \). Lastly, the spin quantum number \( s \) can be either \( +\frac{1}{2} \) or \( -\frac{1}{2} \).

Step 2: Assessing Energy Levels

The energy of an electron in an atom is primarily determined by the principal quantum number \( n \), with a higher \( n \) indicating higher energy levels. Additionally, for the same \( n \), a higher azimuthal quantum number \( l \) indicates higher energy. Therefore, to find the highest energy, compare both \( n \) and \( l \).

Step 3: Compare Options Based on \( n \) and \( l \)

For each option: - (a) \( n=3, l=2 \) - (b) \( n=4, l=0 \) - (c) \( n=3, l=0 \) - (d) \( n=3, l=1 \) Option (b) has the highest \( n \), but the lowest \( l \). Option (a) has the highest \( l \) among \( n=3 \).

Step 4: Determine the Highest Energy Set

Option (b) has the highest principal quantum number \( n=4 \), generally indicating higher energy despite having the lowest \( l \). Hence, the set \( n=4, l=0, m=0, s=+1/2 \) represents the highest energy state among the given options.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is a fundamental concept in quantum mechanics. It signifies the main energy level of an electron within an atom and is a positive integer value starting from 1. The higher the \( n \), the further the electron's energy level is from the nucleus, resulting in higher energy.

• A lower \( n \) indicates that the electron is closer to the nucleus and has lower energy.
• For example, an electron with \( n=4 \) is at a higher energy level than an electron with \( n=3 \).

Understanding \( n \) is crucial because it determines the size of the orbital and influences the electron’s energy level, playing a key role in determining the overall behavior of the atom.

Azimuthal Quantum Number

The azimuthal quantum number, \( l \), also known as the angular momentum quantum number, corresponds to the shape of the orbital occupied by an electron. It ranges from 0 to \( n-1 \). This value is essential in defining the subshells within a principal energy level:

• For \( l=0 \), the shape is spherically symmetric, and this is known as an \( s \) orbital.
• When \( l=1 \), the shape is more complex, like the double lobes of a \( p \) orbital.

The azimuthal quantum number provides deeper insight into the electron's spatial distribution within an atom, contributing to its overall energy configuration. At the same \( n \), a higher \( l \) value means a higher energy level because the electrons are moving in more complex paths and shapes.

Electron Configuration

Electron configuration is a method of describing the distribution of electrons in an atom's orbitals. Understanding this is crucial, as it tells us where electrons are, and helps in predicting chemical behavior.

• Electrons fill lower energy orbitals before they fill higher energy ones, complying with the Aufbau principle.
• The configuration can be represented using notation that includes the principal quantum number, the orbital letter, and the number of electrons in those orbitals.
• For example, a carbon atom with six electrons has the electron configuration: \( 1s^2 2s^2 2p^2 \).

Through electron configuration, one can understand the chemical bonding, magnetism, and reactivity of an element by observing how its electrons are arranged in different orbitals.

Orbital Energy Levels

Orbital energy levels describe the varying energies associated with the electrons in their respective orbitals. These energies are determined by the electron’s position relative to the nucleus and the orbital type they occupy.

• Energy levels increase with increasing principal quantum number \( n \).
• Within the same energy level, energy increases with a higher azimuthal quantum number \( l \).

For instance, in general, a \( 3p \) orbital has higher energy compared to a \( 3s \) orbital. Orbital energy levels are essential for understanding the order in which electrons occupy orbitals and how atoms engage in chemical bonding. Using these energy principles, one can predict which atomic or molecular configurations are more or less stable, based on the potential energy of the electrons involved.