Problem 98 - A Complete Resource Book in Chemistry for JEE Main

Question

Probability of finding the electron \(\psi^{2}\) of \(\mathrm{s}\) orbital doesn't depend upon (a) azimuthal quantum number. (b) energy of s orbital (c) principal quantum number (d) distance from nucleus (r)

Step-by-Step Solution

Answer

The probability density \( \psi^2 \) of an \( s \)-orbital doesn't depend on the azimuthal quantum number.

Step 1: Understanding the Concept

The probability density function for an electron, denoted as \( \psi^2 \), represents the likelihood of finding an electron in a region of space. For an \( s \)-orbital, this function primarily depends on the distance from the nucleus (\( r \)), as \( s \)-orbitals are spherically symmetrical around the nucleus.

Step 2: Analyzing the Impact of the Principal Quantum Number

The principal quantum number (\( n \)) influences the size and energy level of the orbital. It affects the probability density function by altering the radial distribution of the orbital. Thus, \( \psi^2 \) is dependent on the principal quantum number.

Step 3: Considering the Azimuthal Quantum Number

The azimuthal quantum number (\( l \)) defines the shape of the orbital. However, for \( s \)-orbitals, this is always zero (\( l=0 \)). Hence, \( \psi^2 \) does not depend on the azimuthal quantum number for an \( s \)-orbital.

Step 4: Examining Energy of the s Orbital

The energy of an orbital includes contributions from both the principal quantum number and the effect of electron shielding. Thus, indirectly, \( \psi^2 \) is related to the energy of the \( s \)-orbital because the radial distribution can change with energy levels.

Step 5: Evaluating Distance from the Nucleus

For \( s \)-orbitals, the probability density \( \psi^2 \) changes with the distance \( r \) from the nucleus. As you move further from the nucleus, the value of \( \psi^2 \) varies, confirming that it depends on \( r \).

Key Concepts

Azimuthal Quantum Number

The azimuthal quantum number, commonly represented by the symbol \( l \), plays a crucial role in defining the shape of an atomic orbital.
The value of this quantum number determines the subshells within a main energy level and is based on the principal quantum number \( (n) \). However, for \( s \)-orbitals specifically, the azimuthal quantum number is always zero \( (l = 0) \).
This particularity means that the shape of the \( s \)-orbital is spherical, and consequently, the azimuthal quantum number does not affect the electron probability function \( \psi^2 \). This is because the symmetry of the \( s \)-orbital around the nucleus makes \( l \) irrelevant in its probability distribution.
In summary:

• Determines the subshell (i.e., shape and type of orbital).
• It's zero for all \( s \)-orbitals, hence doesn't affect \( \psi^2 \) for \( s \)-orbitals.
• Shapes other subshells like \( p \), \( d \), and \( f \).

Although \( l \) is a vital component for most orbitals, its role in \( s \)-orbitals is minimal because it doesn't alter the spherical nature associated with them.

Principal Quantum Number

The principal quantum number, denoted as \( n \), is foundational in establishing the size and energy of an atomic orbital.
This number determines the primary shell or energy level that an electron occupies. It ranges from 1 to infinity, with higher numbers indicating higher energy levels and larger orbitals.
In terms of the probability distribution of electrons, \( n \) critically influences the function \( \psi^2 \) as it dictates the average distance and energy associated with an electron in an orbital.
Larger values of \( n \) lead to:

• Increased energy levels.
• Larger orbitals, meaning electrons are likely found farther from the nucleus.
• Influences the radial distribution function, adapting the spread of electron density.

Therefore, while the principal quantum number directly impacts the orbital size and energy, it also indirectly affects the likelihood of finding an electron at various distances from the nucleus.

Radial Distribution Function

The radial distribution function is a fundamental concept that describes how electron probability density varies with distance from the nucleus.
Specifically for \( s \)-orbitals, this function indicates the likelihood of finding an electron at a certain radius \( r \). Reflecting spherical symmetry, the radial distribution function is pivotal for understanding the spatial distribution of electrons in an \( s \)-orbital.
The function is derived from two factors:

• The wave function \( \psi \), which is spherically symmetrical for \( s \)-orbitals.
• The square of the radial part of the wave function, weighted by \( r^2 \), making it dependent on the radial distance.

The radial distribution peaks at specific distances, known as nodes, where beyond this, the probability decreases again.
These nodes help identify structures and properties of atoms, making it clear how distance impacts electron distribution throughout the atomic structure." Therefore, the radial distribution function plays a significant role in visualizing electron positions and is crucial for interpreting the probability density for \( s \)-orbitals.