Problem 136 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The number of orbitals of \(\mathrm{g}\) -type (a) 5 (b) 7 (c) 9 (d) 11

Step-by-Step Solution

Answer

The number of orbitals of \(g\)-type is 9.

Step 1: Understanding Orbital Types

The number of orbitals for a given type can be determined using the formula \(2l + 1\), where \(l\) is the azimuthal quantum number corresponding to the orbital type. For 's' type, \(l = 0\); for 'p' type, \(l = 1\); for 'd' type, \(l = 2\); for 'f' type, \(l = 3\) and so on, increasing by one for each subsequent type. For \(g\)-type orbitals, \(l = 4\).

Step 2: Applying the Formula

Substitute \(l = 4\) for \(g\)-type orbitals into the formula to find the number of orbitals: \(2l + 1 = 2(4) + 1 = 9\).

Key Concepts

Azimuthal Quantum Number

The azimuthal quantum number, denoted by the symbol l, is one of the four quantum numbers used in quantum chemistry to describe the unique quantum state of an electron. It is also known as the angular momentum quantum number and defines the shape of the atomic orbital. The value of l can range from 0 to n-1, where n is the principal quantum number associated with the electron's energy level.

The azimuthal quantum number helps to categorize the atomic orbitals into different types such as s, p, d, f, and onward, with s having an l value of 0, p having an l value of 1, d having an l value of 2, and so forth. For the g-type orbital mentioned in the exercise, l equals 4. This quantum number influences the number of orbital types in a given energy level, which is calculated using the formula 2l + 1. This allows us to understand and predict the orbital structures found within an atom.

Quantum Chemistry

Quantum chemistry sits at the intersection of chemistry and physics, and it involves the application of quantum mechanics to chemical systems. Its goal is to understand the behavior of electrons in atoms and molecules, and it provides the theoretical foundation to describe chemical bonding and the electronic structure of atoms. The discipline relies heavily on mathematical equations and complex calculations to explain observations at the atomic and subatomic levels.

Understanding the orbital types and how electrons are configured within these orbitals is pivotal in quantum chemistry, as it determines the chemical properties of an element. This field of study progresses by solving the Schrödinger equation for molecules of interest, allowing scientists to predict how molecules will react and to design new materials and drugs. Key to this understanding is the concept of quantum numbers, including the azimuthal quantum number, which helps predict the number and types of orbitals.

Atomic Orbitals

Atomic orbitals are regions within an atom where electrons are most likely to be found. They are defined mathematically as wavefunctions and possess a range of shapes and sizes, with each shape corresponding to a particular type of orbital, as determined by the azimuthal quantum number l. The fundamental types of atomic orbitals—s, p, d, and f—each have distinctive shapes: spherical for s orbitals, dumbbell-shaped for p orbitals, and more complex shapes for d and f orbitals.

The concept of orbitals is crucial in understanding the atomic and molecular structure, and in predicting the chemical behavior of atoms. Additionally, these orbitals can combine or hybridize to form molecular orbitals, which define the shape of molecules and how atoms bond in molecules. Understanding these orbital types and their numbers, as derived using the principle of the azimuthal quantum number, is foundational to grasping the concepts of quantum chemistry.