Problem 91 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The orbital angular momentum of \(2 \mathrm{p}\) and \(3 \mathrm{p}\) -orbitals (a) are same (b) are different, more for 2p-orbital (c) are different, more for 3 p-orbital (d) depends on the type of atom or ion

Step-by-Step Solution

Answer

The orbital angular momentum for 2p and 3p orbitals are the same because they depend only on the azimuthal quantum number (l), which is 1 for all p-orbitals.

Step 1: Understanding Orbital Angular Momentum

Recognize that the orbital angular momentum (L) is associated with the motion of electrons in atomic orbitals and is quantized. It depends on the principal quantum number (n) and the azimuthal (or angular momentum) quantum number (l). The formula is given by L = \( \sqrt{l(l+1)} \hbar \), where \( \hbar \) is the reduced Planck constant.

Step 2: Apply Quantum Numbers for 2p and 3p Orbitals

For a 'p' orbital, the azimuthal quantum number (l) is always 1, regardless of the principal quantum number (n). Thus for both 2p and 3p orbitals, l = 1.

Step 3: Calculate Orbital Angular Momentum for 2p and 3p Orbitals

Using the formula for orbital angular momentum, calculate L for both orbitals: L(2p) = L(3p) = \( \sqrt{1(1+1)} \hbar = \sqrt{2} \hbar \). Since the values of L for 2p and 3p orbitals are determined solely by the value of l, and that is constant for p-orbitals, L is the same for both orbitals.

Key Concepts

Atomic Orbitals

Understanding atomic orbitals is pivotal when studying chemistry and physics. These are regions in an atom where there is a high probability of finding an electron. Each orbital has a unique shape and size, representing the electron's behavior within an atom.

Orbitals are categorized into several types, such as s, p, d, and f, depending on their shape. The p-orbitals, for instance, have a dumbbell shape and can accommodate a maximum of six electrons in three spatial orientations. The 2p and 3p orbitals mentioned in the exercise are from the second and third shells of an atom, respectively, with their principal quantum numbers being 2 and 3.

Quantum Numbers

The concept of quantum numbers arises from the quantum mechanical model of the atom. These numbers describe the properties of atomic orbitals and the electrons in those orbitals, much like an atomic address system. There are four key quantum numbers: principal (n), azimuthal (l), magnetic (m_l), and spin (m_s).

Each quantum number gives us unique information: the principal quantum number indicates the shell, the azimuthal gives the subshell and shape, the magnetic describes the orbital's orientation in space, and the spin quantum number tells us the electron's spin direction. Together, these quantum numbers ensure that no two electrons in an atom have the exact same set of quantum numbers, which is the essence of the Pauli Exclusion Principle.

Principal Quantum Number

Diving deeper, the principal quantum number (n) indicates the energy level of an electron within an atom. It can take integer values starting from 1, so n = 1, 2, 3, and so on. As the value of n increases, so does the energy of the electron and the size of the orbital.

A higher principal quantum number means the electron is further from the nucleus and has a higher energy, making the atom larger. For example, in the 2p and 3p orbitals referenced in the exercise, the principal quantum numbers are 2 and 3, signifying that the 3p orbitals are higher in energy and larger in size compared to the 2p orbitals.

Azimuthal Quantum Number

The azimuthal quantum number (l) is tied to the shape of the atomic orbital and is integral to understanding the orbital's angular momentum. It has integer values ranging from 0 to n-1 for each principal quantum number. The azimuthal quantum number determines the subshell: 0 for 's', 1 for 'p', 2 for 'd', and 3 for 'f' subshells.

For any 'p' orbital, like the ones in our exercise, the azimuthal quantum number is always 1. This is crucial in determining the orbital's angular momentum, which, as the solution reveals, does not differ between the 2p and 3p orbitals—both have an azimuthal quantum number of 1, resulting in the same angular momentum value.