Problem 9 - A Complete Resource Book in Chemistry for JEE Main

Question

In which orbital is the angular momentum of an electron zero? (a) \(2 \mathrm{p}\) (b) \(2 \mathrm{~s}\) (c) \(3 \mathrm{~d}\) (d) \(4 \mathrm{f}\)

Step-by-Step Solution

Answer

The angular momentum of an electron is zero in the \(2\mathrm{~s}\) orbital.

Step 1: Understanding Angular Momentum in Orbitals

The angular momentum (denoted by the quantum number \(l\)) of an electron in an atomic orbital is related to the shape of the orbital. It is calculated using the formula \( L = \sqrt{l(l+1)}\hbar \), where \( L \) is the angular momentum and \( \hbar \) is the reduced Planck's constant. An electron in an orbital with \( l = 0 \) will have zero angular momentum.

Step 2: Identifying the Quantum Numbers

Different orbitals are associated with specific values of the azimuthal quantum number \( l \):- \( s \)-orbital: \( l = 0 \)- \( p \)-orbital: \( l = 1 \)- \( d \)-orbital: \( l = 2 \)- \( f \)-orbital: \( l = 3 \)Thus, the \( s \)-orbital is the only orbital where \( l = 0 \).

Step 3: Determining the Correct Orbital

Based on the values of \( l \) for different orbitals, the \( 2 \mathrm{s} \) orbital has an \( l \) value of 0. Therefore, it has zero angular momentum due to its quantum number \( l = 0 \). The \( 2 \mathrm{p} \), \( 3 \mathrm{d} \), and \( 4 \mathrm{f} \) orbitals have non-zero angular momenta corresponding to their respective \( l \) values of 1, 2, and 3.

Key Concepts

Quantum Numbers

Quantum numbers are essential for understanding the structure of electrons in an atom. They give us valuable information about the location and energy of electrons. There are four main quantum numbers, each offering specific details about the electron's state:

• Principal Quantum Number (\(n\)): This number signifies the main energy level of the electron. It's always a positive integer (e.g., 1, 2, 3), determining the size and energy of the orbital.


• Azimuthal Quantum Number (\(l\)): Also known as the angular momentum quantum number, it describes the shape of the orbital. Its value ranges from 0 to \(n-1\), corresponding to different orbital types.


• Magnetic Quantum Number (\(m_l\)): This number describes the orientation of the orbital in space. It can have integer values from \(-l\) to \(+l\).


• Spin Quantum Number (\(m_s\)): It describes the electron's spin direction. Its value can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Together, these numbers provide a comprehensive description of an electron's position within an atom, allowing us to predict its behavior and interactions.

S Orbitals

S orbitals are the simplest type of atomic orbitals, characterized by their spherical shape. This is what makes them distinct compared to other orbital types. Every energy level has one s orbital.

• Spherical Symmetry: The s orbitals have a perfectly spherical shape, which means the probability of finding an electron is the same at every angle at a fixed distance from the nucleus.


• Occurrence: Each principal energy level (e.g., 1, 2, 3...) starts with an s orbital, denoted as 1s, 2s, 3s, and so on.


• Quantum Number (\(l\) Value): For s orbitals, the azimuthal quantum number \(l = 0\). This is why they have zero angular momentum.

The simplicity and symmetry of s orbitals mean they are often filled first in an atom. This is because they have the lowest energy within an energy level.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \(l\), plays a crucial role in defining the shape and energy of orbitals. It informs us about the angular momentum of the electron within an orbital. Let's delve into what this number signifies:

• Determination of Orbital Shape: The value of \(l\) determines the shape of the orbital:
  • \(l = 0\): s orbitals (spherical)
  
  
  • \(l = 1\): p orbitals (dumbbell-shaped)
  
  
  • \(l = 2\): d orbitals (cloverleaf-shaped)
  
  
  • \(l = 3\): f orbitals (complex shapes)


• Connection to Angular Momentum: The azimuthal quantum number directly affects the angular momentum of an electron. For instance, an electron in an orbital where \(l = 0\) has zero angular momentum.


• Range of Values: The possible values of \(l\) range from 0 to \(n-1\), where \(n\) is the principal quantum number.

Understanding \(l\) is essential for predicting an electron's behavior, its bonding capabilities, and its place within an atom. It provides a clear indication of an electron's angular properties and orbital shape.