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

Question

Probability of finding the electron \(\psi^{2}\) of 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

It doesn't depend on (a) azimuthal quantum number.

Step 1: Understanding the Electron Probability for s Orbital

For an s orbital, the probability density of finding an electron can be described by \(\left| \psi \right|^2\), where \(\psi\) is the wave function. It mainly depends on \(r\), the distance from the nucleus, and is spherically symmetrical. Unlike other orbitals, the s orbital does not depend on the azimuthal (angular) placement, typically described by the magnetic quantum number.

Step 2: Identifying Dependencies of \\(\psi^2\\)

The probability density function \(\psi^2\) does not depend on (a) the azimuthal quantum number (which is 0 for s orbitals), (b) the energy of the s orbital directly, or (c) the principal quantum number directly. However, it is directly affected by (d) the distance \(r\) from the nucleus, as this dictates the radial distribution of electron density.

Step 3: Selecting the Correct Option based on Dependency Analysis

Since the electron probability for s orbitals is influenced by the distance \(r\) from the nucleus and does not relate to the azimuthal configuration, the correct answer is that it does depend on the distance from the nucleus. Thus, \(\psi^2\) does not depend on (a) the azimuthal quantum number.

Key Concepts

s-orbital

In quantum chemistry, the concept of orbitals is fundamental for understanding the behavior and arrangement of electrons in atoms. An s-orbital is one of these basic building blocks. It is the simplest type of atomic orbital and is characterized by its unique shape and properties. In its most common form, it appears as a perfect sphere centered around the nucleus.

Unlike other types of orbitals, such as p, d, or f orbitals, the s-orbital has no directional preference. This means it is spherically symmetrical. The term "s-orbital" comes from the word 'spherical'. It doesn't depend on any angular quantum numbers, making it unique in its perfect symmetry.

• Shape: Spherical
• Symmetry: Perfectly spherical
• Angular Dependence: None
• Position: Centered around the nucleus

The wave function of an s-orbital, denoted as ps, forms the basis for calculating the probability density, a concept we'll explore further. This symmetric nature of the s-orbital means calculations for probabilities are straightforward, focusing mainly on the distance from the nucleus, unlike other orbitals with complex angular dependencies.

probability density

Probability density is a concept used in quantum chemistry to describe the likelihood of finding an electron in a specific region in space around the nucleus. For s-orbitals, this is particularly straightforward due to their spherical shape.

The probability density is given by ps^2, which is essentially the square of the wave function ps. This value signifies how likely it is to find an electron at a certain distance from the nucleus. In s-orbitals, unlike others, probability density is not influenced by angles or directions but primarily by the radial distance \(r\) from the nucleus.

• Radial dependence: The probability decreases as you move further from the nucleus.
• Angular dependence: None for s-orbitals, leading to spherical symmetry.

Understanding probability density helps in predicting electron distribution and influences how atoms and molecules interact. There are peaks and nodes in the probability density of an electron around the nucleus, shaping the chemical behavior of elements.

quantum numbers

Quantum numbers are pivotal in quantum chemistry, describing properties of electrons in atoms. There are four quantum numbers, each describing a unique characteristic of the electron, but not all affect the s-orbital properties like electron probability density.

• Principal Quantum Number (\(n\)): Determines the energy level and size of the orbital. Higher \(n\) values indicate larger orbitals.
• Azimuthal Quantum Number (\(l\)): Determines the shape of the orbital. For s-orbitals, \(l = 0\), contributing to their spherical shape.
• Magnetic Quantum Number (\(m_l\)): Affects the orientation of orbitals in space. Not relevant for s-orbitals since they have no preferred direction.
• Spin Quantum Number (\(m_s\)): Describes the spin of the electron, which is a fundamental property of electrons but does not influence orbital shapes.

In the context of s-orbitals, while the electron energy is influenced by the principal quantum number, the other numbers have specific roles. However, the azimuthal and magnetic numbers are less impactful due to the spherical nature of s-orbitals. The understanding of these numbers is crucial for predicting electron configurations and chemical bonding behavior.