Problem 37 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following is not possible? (a) \(\mathrm{n}=2, l=1, \mathrm{~m}=0\) (b) \(\mathrm{n}=2, l=0, \mathrm{~m}=-1\) (c) \(\mathrm{n}=3, l=0, \mathrm{~m}=0\) (d) \(\mathrm{n}=3, l=1, \mathrm{~m}=-1\)

Step-by-Step Solution

Answer

Option (b) is not possible because \(m = -1\) is invalid for \(l = 0\).

Step 1: Understand Quantum Numbers

A quantum mechanical model uses quantum numbers to describe the state of electrons in an atom. The four quantum numbers are principal quantum number (), azimuthal quantum number (\(l\)), magnetic quantum number (\(m\)), and spin quantum number. Here, we focus on , \(l\), and \(m\).

Step 2: Principal Quantum Number ( ) Analysis

The principal quantum number ( ) indicates the energy level and distance of the electron from the nucleus. It's a positive integer starting from 1. All the given options have valid values.

Step 3: Azimuthal Quantum Number ( \(l\)) Analysis

The azimuthal quantum number (\(l\)) describes the shape of the orbital. It ranges from 0 to -1. Check each option:- For (a) and (b), = 2, so \(l\) can be 0 or 1.- For (c) and (d), = 3, so \(l\) can be 0, 1, or 2.Thus, all \(l\) values are suitable based on given values.

Step 4: Magnetic Quantum Number ( \(m\)) Analysis

The magnetic quantum number (\(m\)) defines the orientation of the orbital and ranges from -\(l\) to \(l\). Check each option:- (a) \(m = 0\), which is valid for \(l = 1\)- (b) \(l = 0\), so \(m\) must be 0. But \(m = -1\) is invalid.- (c) \(m = 0\), which is valid for \(l = 0\)- (d) \(m = -1\), which is valid for \(l = 1\)Thus, option (b) is invalid because \(m = -1\) is not possible for \(l = 0\).

Key Concepts

Principal Quantum Number

The principal quantum number, symbolized as \( n \), is a fundamental concept in quantum mechanics that defines the energy level of an electron in an atom. It gives us insight into the size and energy of the electron's orbital. The principal quantum number is always a positive integer, for example, 1, 2, 3, and so on. This number indicates how far an electron is from the nucleus, with higher numbers suggesting a greater distance and therefore higher energy levels.

• It helps categorize electron shells or energy levels in an atom.
• Higher \( n \) values mean electrons are further from the nucleus.
• The principal quantum number determines the electron's potential energy within its shell.

Understanding \( n \) is crucial for predicting an atom's chemical behavior since it affects the atom's size and how it interacts with other atoms.

Azimuthal Quantum Number

The azimuthal quantum number, denoted by \( l \), describes the shape of the electron's orbital within a given energy level. This quantum number answers the question of what shape the electron cloud will have around the nucleus. The value of \( l \) ranges from 0 to \( n-1 \) for each principal quantum number. Each \( l \) value corresponds to a different orbital type:

• \( l = 0 \): An 's' orbital, spherical in shape.
• \( l = 1 \): A 'p' orbital, dumbbell-shaped.
• \( l = 2 \): A 'd' orbital, more complex forms.
• \( l = 3 \): An 'f' orbital, even more intricate shapes.

These different shapes of orbitals accommodate electrons differently and hence are vital in determining how an atom will bond with others. In the exercise example, correctly determining \( l \) was essential to find out which electron configurations were possible.

Magnetic Quantum Number

The magnetic quantum number, symbolized as \( m \), provides an understanding of the orientation of an electron's orbital in space relative to the other orbitals. For each orbital shape (denoted by \( l \)) there are specific orientations, defined by \( m \), which ranges from \(-l\) to \( l \). The values of \( m \) tell us how the orbitals are oriented in space, particularly in response to external magnetic fields.

• For \( l = 0 \), \( m \) can only be 0.
• For \( l = 1 \), \( m \) can be -1, 0, or 1.
• For \( l = 2 \), \( m \) ranges from -2 to 2.

In solving quantum problems, ensuring that \( m \) aligns with the possible values based on \( l \) is crucial. For instance, as per the problem's solution, if \( l = 0 \), then \( m \) must be 0, making any other value incorrect for that scenario. Understanding the magnetic quantum number helps predict how these orbitals will behave under external influences.