Problem 131 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The orbital angular momentum of a 4p electron will be (a) 4. \(\frac{h}{2 \pi}\) (b) \(\sqrt{2} \cdot \frac{h}{2 \pi}\) (c) \(\sqrt{6} \cdot \frac{h}{4 \pi}\) (d) \(\sqrt{2} \cdot \frac{h}{4 \pi}\)

Step-by-Step Solution

Answer

(b) \(\sqrt{2} \cdot \frac{h}{2\pi}\)

Step 1: Identify Quantum Numbers

For a 4p electron, the principal quantum number is n=4 and the azimuthal (angular momentum) quantum number is l=1, since it is a p-orbital.

Step 2: Use Angular Momentum Formula

The formula to calculate the orbital angular momentum is \(L = \sqrt{l(l+1)} \cdot \frac{h}{2\pi}\), where \(l\) is the azimuthal quantum number and \(h\) is Planck's constant.

Step 3: Calculate Angular Momentum

Substitute \(l=1\) into the formula: \(L = \sqrt{1(1+1)} \cdot \frac{h}{2\pi} = \sqrt{2} \cdot \frac{h}{2\pi}\). The correct answer is therefore (b) \(\sqrt{2} \cdot \frac{h}{2\pi}\).

Key Concepts

Quantum Numbers

Quantum numbers play an essential role in the quantum mechanical model of atoms. They are like a detailed address system that tells us the location and characteristics of an electron within an atom. There are four types of quantum numbers: principal (n), azimuthal (l), magnetic (m_l), and spin (s).

The principal quantum number, n, indicates the main energy level and distance of an electron's orbit from the nucleus; values can be any positive integer. The azimuthal quantum number, often referred to as the angular momentum quantum number l, defines the shape of the electron's orbital and corresponds to the subshells (s, p, d, f). The magnetic quantum number, m_l, describes the orientation of the orbital in space, while the spin quantum number determines the direction of the electron's spin. Understanding these numbers helps us predict electron configuration and the chemical properties of elements.

P-Orbital

The p-orbital is one of the shapes that electron clouds can take and is denoted by the azimuthal quantum number l = 1. Unlike the spherically symmetric s-orbital, p-orbitals have a distinct dumbbell shape, with a nodal plane where the probability of finding an electron is zero. Each p-orbital consists of two lobes located on either side of the nucleus, oriented along an axis.

There are three p-orbitals for each energy level, once you get past the first, which correspond to the magnetic quantum numbers -1, 0, and 1. These are oriented along the x, y, and z axes respectively and are named as such (p_x, p_y, p_z). Electrons in p-orbitals have higher energy than those in s-orbitals of the same principal quantum number and can considerably influence the atom's reactivity and bonding behavior.

Planck's Constant

Planck's constant, denoted by h, is a fundamental constant in quantum mechanics. It represents the quantization of energy, which were pivotal in the development of quantum theory. The constant has a value of approximately 6.626 x 10^-34 J⋅s (joule-seconds) and is used to describe the smallest amount of energy that can be emitted or absorbed as electromagnetic radiation.

Understanding Planck's constant is crucial when dealing with phenomena on the atomic or subatomic scale, such as computing the energy of photons or the angular momentum of electrons in an atom. The expression h/2π often appears in quantum mechanics formulas because it simplifies calculations when dealing with wavefunctions and their corresponding angular momenta. This reduced constant is commonly represented by the symbol ħ (h-bar) and reflects the wave-particle duality of matter.