91Ó°ÊÓ

When an electron makes a transition between energy levels of an atom, there are no restrictions on the initial and final values of the principal quantum number \(n\). According to quantum mechanics, however, there is a rule that restricts the initial and final values of the orbital quantum number \(\ell\). This rule is called a selection rule and states that \(\Delta \ell=\pm 1 .\) In other words, when an electron makes a transition between energy levels, the value of \(\ell\) can only increase or decrease by one. The value of \(\ell\) may not remain the same nor increase or decrease by more than one. According to this rule, which of the following energy level transitions are allowed: (a) \(2 \mathrm{~s} \rightarrow 1 \mathrm{~s}\), (b) \(2 \mathrm{p} \rightarrow 1 \mathrm{~s}\), (c) \(4 \mathrm{p} \rightarrow 2 \mathrm{p}\), (d) \(4 \mathrm{~s} \rightarrow 2 \mathrm{p}\), and (e) \(3 \mathrm{~d} \rightarrow 3 \mathrm{~s} ?\)

Step by step solution

01

Understanding Quantum Number Changes

For a transition to be allowed, the change in the orbital quantum number, \( \Delta \ell \), must be \(+1\) or \(-1\), according to the selection rule \( \Delta \ell = \pm 1 \). This means if the initial orbital quantum number is \( \ell_i \) and the final is \( \ell_f \), then \( \ell_f = \ell_i \pm 1 \).

02

Analyzing Transition (a)

The transition \( 2s \rightarrow 1s \) involves \( \ell_i = 0 \) (for \(s\)), and \( \ell_f = 0 \) (for \(s\)). The change here is \( \Delta \ell = 0 \). Thus, it is not allowed as \( \Delta \ell eq \pm 1 \).

03

Analyzing Transition (b)

For \( 2p \rightarrow 1s \), \( \ell_i = 1 \) (for \(p\)), \( \ell_f = 0 \) (for \(s\)). The change is \( \Delta \ell = 1 - 0 = 1 \), which satisfies \( \Delta \ell = +1 \). Therefore, this transition is allowed.

04

Analyzing Transition (c)

The transition \( 4p \rightarrow 2p \) involves \( \ell_i = 1 \) (for \(p\)), and \( \ell_f = 1 \) (for \(p\)). The change is \( \Delta \ell = 1 - 1 = 0 \). Thus, it is not allowed as \( \Delta \ell eq \pm 1 \).

05

Analyzing Transition (d)

For \( 4s \rightarrow 2p \), \( \ell_i = 0 \) (for \(s\)), and \( \ell_f = 1 \) (for \(p\)). The change is \( \Delta \ell = 1 - 0 = 1 \), which satisfies \( \Delta \ell = +1 \). Therefore, this transition is allowed.

06

Analyzing Transition (e)

The transition \( 3d \rightarrow 3s \) involves \( \ell_i = 2 \) (for \(d\)), and \( \ell_f = 0 \) (for \(s\)). The change is \( \Delta \ell = 0 - 2 = -2 \). Thus, it is not allowed as \( \Delta \ell \) is neither \(+1\) nor \(-1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantum Mechanics

Quantum mechanics is the branch of physics that seeks to explain the physical behaviors of matter and energy on the atomic and subatomic levels. It provides a fundamental framework through which we understand how particles such as electrons, protons, and photons behave. Unlike classical physics, quantum mechanics incorporates the concepts of probability and wave-particle duality. These concepts are essential for explaining phenomena that cannot be understood using just classical theories.

In quantum mechanics, electrons do not follow fixed paths around the nucleus. Instead, they exist in certain probability zones known as atomic orbitals. The possible locations and behaviors of an electron are determined by various quantum numbers, which are part of a system that quantifies the properties of a particle. Understanding these quantum numbers is crucial to unravel the mysteries of atomic behavior and energy level transitions.

Orbital Quantum Number

The orbital quantum number, represented by the symbol \( \ell \), is one of the four quantum numbers used to describe an electron's state in an atom. It determines the shape of the electron's orbital and is related to the angular momentum of the electron. The value of \( \ell \) is an integer that ranges from \( 0 \) to \( n-1 \), where \( n \) is the principal quantum number. Each integer value corresponds to a specific type of orbital (\(s, p, d, \) or \(f\)).

• \( \ell = 0 \) corresponds to an \(s\) orbital.
• \( \ell = 1 \) corresponds to a \(p\) orbital.
• \( \ell = 2 \) corresponds to a \(d\) orbital.
• \( \ell = 3 \) corresponds to an \(f\) orbital.

In energy level transitions, the selection rule \( \Delta \ell = \pm 1 \) states that the orbital quantum number can only increase or decrease by one during a transition. This rule helps us determine which energy transitions are allowed or forbidden.

Energy Level Transitions

Energy level transitions in atoms occur when electrons move between different orbitals. This movement is typically in response to absorbing or emitting a photon. A photon is a quantum of light energy, and its energy must match the difference between the energy levels to initiate a transition.

During this transition, electrons can only move under certain conditions defined by quantum mechanics. The selection rule, \( \Delta \ell = \pm 1 \), is a crucial condition governing the transition between orbitals. Not all transitions are allowed; only those that comply with this rule can occur. For example:

• An \(2p\) to \(1s\) transition is allowed because \( \Delta \ell = +1 \).
• A \(3d\) to \(3s\) transition is forbidden because \( \Delta \ell = -2 \).

Principal Quantum Number

The principal quantum number, denoted as \( n \), indicates the main energy level occupied by an electron in an atom. It is a positive integer, and as \( n \) increases, so does the energy and average distance of the electron from the nucleus. The higher the value of \( n \), the higher the energy level and the further away the electron is likely to be from the nucleus.

Though the principal quantum number defines the general size and energy, it is the orbital quantum number \( \ell \) that governs the sub-levels within each energy level. Unlike the orbital quantum number, there are no selection rules on \( n \). This means electrons can transition between different principal quantum levels without restrictions on the \( n \) values, as long as other quantum conditions, such as the orbital selection rule, are satisfied.