Problem 105 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

An electron jumps from the fourth orbit to the first orbit in a H-atom. The number of photons liberated out will be (a) 1 (b) 2 (c) 3 (d) 6

Step-by-Step Solution

Answer

The number of photons liberated will be 1.

Step 1: Identify The Transition

Determine the initial and final orbit numbers of the electron. Here, the electron is initially in the fourth orbit (n=4) and ends in the first orbit (n=1).

Step 2: Understand the Emission Process

When an electron transitions from a higher energy level to a lower energy level, it emits a photon. Each transition results in the emission of one photon.

Step 3: Calculate the Number of Photons

Since the electron transitions directly from the fourth orbit to the first orbit, there is only one such transition, and consequently, only one photon will be emitted.

Key Concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a comprehensive framework for understanding the very small scales of energy levels within atoms. At the heart of quantum mechanics is the notion that energy is not continuous, but rather quantized, meaning it comes in discrete amounts called 'quanta'.

In the context of the hydrogen atom, an electron is not free to possess any amount of energy but is restricted to certain energy levels. These levels are analogous to the rungs of a ladder; an electron can only inhabit a rung, not the spaces between the rungs. When an electron in a hydrogen atom changes its orbit from one energy level to another, this transition involves discrete changes in energy.

Photon Emission

Photon emission occurs when an electron moves from a higher to a lower energy level within an atom. According to quantum mechanics, the atom loses energy through the emission of a photon, which is a particle of light. The energy of the emitted photon is equal to the difference between the two energy levels.

In our example of the hydrogen atom, the single transition of an electron from the fourth to the first orbit results in the release of energy in the form of one photon. The photon's energy will correspond to the specific energy interval between these two orbits. Importantly, the emission of a photon is a single event per transition; hence, for a direct transition, only one photon is emitted.

Energy Levels

The concept of energy levels is integral to understanding the electronic structure of an atom. In a hydrogen atom, the energy levels are indexed by the principal quantum number 'n'. These levels dictate the potential energy of an electron within the atom and are inversely related to the energy: the lower the value of 'n', the lower the energy level.

For a hydrogen atom, the energy levels can be represented by the formula \[\begin{equation} E_n = -\frac{13.6 \text{ eV}}{n^2} \end{equation}\]where 'n' is the principal quantum number and 'E_n' is the energy of the nth level. As 'n' increases, the energy levels get closer together, which explains why the energy released in transitions between high-level orbits is less than that released in transitions involving low-level orbits.