Question

When an electron makes a transition between energy levels of a hydrogen atom, there are no restrictions on the initial and final values of the principal quantum number \(n\). However, there is a quantum mechanical rule that restricts the initial and final values of the orbital angular momentum \(\ell\). This is the selection rule, which states that \(\Delta \ell=\pm 1,\) that is, in a transition, the value of \(\ell\) can only increase or decrease by one. According to this rule, which of the following transitions are allowed: (a) \(1 s \longrightarrow 2 s\), (b) \(2 p \longrightarrow 1 s\) (c) \(1 s \longrightarrow 3 d\) (d) \(3 d \longrightarrow 4 f\), (e) \(4 d \longrightarrow 3 s ?\)

Short answer

The allowed transitions according to the selection rule are: (b) \(2 p \rightarrow 1 s\) and (d) \(3 d \rightarrow 4 f\). The transitions (a) \(1 s \rightarrow 2 s\), (c) \(1 s \rightarrow 3 d\), and (e) \(4 d \rightarrow 3 s\) are not allowed.

Step-by-step solution

Understanding Quantum Numbers

In Quantum Mechanics, each electron in an atom is described by four quantum numbers. The principal quantum number n, the orbital angular momentum quantum number \(\ell\), the magnetic quantum number m, and the spin quantum number s. The principal quantum number describes the size of the electron's orbit and can take any positive integer n > 0. The orbital angular momentum quantum number \(\ell\) describes the shape of the electron's orbit, and can take any integer value from 0 to n - 1. For each value of \(\ell\), the magnetic quantum number m takes integer values ranging from -\(\ell\) to +\(\ell\). The spin quantum number reflects the intrinsic spin of the electron which can take values of +1/2 or -1/2. The orbitals s, p, d, and f correspond to \(\ell\) = 0, 1, 2, 3 respectively.

Analyze Transition (a): \(1 s \rightarrow 2 s\)

In the first transition, the principal quantum number is changing from 1 to 2, but with no change in the orbital type (both s orbitals, hence, both \(\ell = 0\)). So, in this case, \(\Delta \ell = 0\). This violates the selection rule \(\Delta \ell = \pm 1\) and hence, this transition is not allowed.

Analyze Transition (b): \(2 p \rightarrow 1 s\)

In the second transition, the principal quantum number is changing from 2 to 1, with a change from a p orbital to an s orbital. This means \(\ell\) changes from 1 to 0 (p to s). So, in this case, \(\Delta \ell = - 1\). This respects the selection rule \(\Delta \ell = \pm 1\) and hence, this transition is allowed.

Analyze Transition (c): \(1 s \rightarrow 3 d\)

In the third transition, the principal quantum number is changing from 1 to 3, with a change from an s orbital to a d orbital. This means \(\ell\) changes from 0 to 2 (s to d). So, in this case, \(\Delta \ell = + 2\). This violates the selection rule \(\Delta \ell = \pm 1\) and hence, this transition is not allowed.

Analyze Transition (d): \(3 d \rightarrow 4 f\)

In the fourth transition, the principal quantum number is changing from 3 to 4, with a change from a d orbital to an f orbital. This means \(\ell\) changes from 2 to 3 (d to f). So, in this case, \(\Delta \ell = + 1\). This respects the selection rule \(\Delta \ell = \pm 1\) and hence, this transition is allowed.

Analyze Transition (e): \(4 d \rightarrow 3 s\)

In the fifth transition, the principal quantum number is changing from 4 to 3, with a change from a d orbital to an s orbital. This means \(\ell\) changes from 2 to 0 (d to s). So, in this case, \(\Delta \ell = - 2\). This violates the selection rule \(\Delta \ell = \pm 1\) and hence, this transition is not allowed.

Key concepts

Electron Transitions

Understanding electron transitions is crucial in the field of quantum chemistry. Simply put, these are the movements of an electron from one energy level to another within an atom. Such transitions are integral to the absorption or emission of light, resulting in the unique spectra of elements.

Each energy level is associated with a specific value of the principal quantum number, denoted as 'n'. When an electron transitions, it either absorbs or emits energy, which corresponds to the difference in energy levels involved. Only certain transitions are allowed, governed by specific rules, which include the quantum mechanical selection rules discussed later in this article.

In the context of our exercise example, analyzing transitions involves checking for changes in both the principal quantum number and the orbital angular momentum quantum number to ensure they comply with established selection rules.

Orbital Angular Momentum

The concept of orbital angular momentum in quantum mechanics is tied to the shape and orientation of electron orbitals. Represented by the quantum number \(\ell\), it indicates the subshell (s, p, d, f, etc.) the electron occupies, with corresponding values of 0, 1, 2, 3, and so on, respectively.

Orbital angular momentum is derived from the electron's motion and gives us more information than just the energy level the electron occupies; it provides insight into the geometry of its probability distribution around the nucleus. Understanding the allowed changes in \(\ell\), as featured in the exercise, is essential for predicting electron behavior during transitions.

Quantum Mechanical Selection Rules

Quantum mechanical selection rules are parameters that determine the probability of a transition between quantum states. These rules are critical in predicting the spectral lines resulting from electron transitions. The most recognized among these is the \(\Delta \ell = \pm 1\) rule, also highlighted in our example.

This rule dictates that for a transition to be allowed, the orbital angular momentum quantum number must change by exactly one. These rules ensure that only transitions with certain changes in angular momentum are permitted; hence, they shape the understanding of transition likelihood and enforce the intrinsic quantum nature of atomic particles.

Principal Quantum Number

The principal quantum number \(n\) is a fundamental term in quantum chemistry that symbolizes the size and energy level of the electron orbit. It takes positive integer values (1, 2, 3,...), with each increase in \(n\) signifying a leap to a higher energy level and a larger orbit.

In any electron transition, the change in the principal number signifies a leap or drop across energy levels, which essentially means absorbing or emitting energy. This quantum number is the first in the sequence of quantum numbers that together describe the state of an electron within an atom. It's crucial to note that while there are no restrictions on the initial and final values of \(n\), as seen in our textbook exercise, they must still comply with the orbital angular momentum selection rule to validate an electron transition.