Problem 49 - A Complete Resource Book in Chemistry for JEE Main

Question

When the \(\mathrm{e}^{-}\)of a \(\mathrm{H}\)-atom moves from \(\mathrm{n}=4\) to \(\mathrm{n}=1\) the number of spectral lines emitted is (a) 3 (b) 6 (c) 9 (d) 15

Step-by-Step Solution

Answer

6 spectral lines are emitted.

Step 1: Understand the concept

When an electron in a hydrogen atom transitions from a higher energy level (n=4) to a lower energy level, it emits photons as it moves. The number of distinct spectral lines is determined by the possible transitions the electron can make as it descends from the initial level to the ground state.

Step 2: Determine possible transitions

From level n=4, the electron can drop to n=3, n=2, or n=1. From n=3, it can move to n=2 or n=1. Finally, from n=2, it can move to n=1.

Step 3: Calculate total transitions

The number of possible spectral lines can be calculated using the formula \( \frac{n(n-1)}{2} \), where \( n \) is the initial energy level. For n=4, the calculation is \( \frac{4(4-1)}{2} = 6 \).

Step 4: Conclude the number of spectral lines

The number of spectral lines produced by an electron moving from n=4 to n=1 is 6, corresponding to the different ways the electron can transition down to the ground state.

Key Concepts

Spectral Lines

When an electron in a hydrogen atom transitions between energy levels, it emits or absorbs energy in the form of light. These transitions are displayed as spectral lines, which are essentially the fingerprints of each element. Each spectral line corresponds to a specific transition, and can be observed in a spectrum when this energy is emitted as light.

It occurs because electrons do not move continuously between energy levels, they rather jump from one level to another. Each of these jumps results in either the emission or absorption of a specific amount of energy corresponding to the difference between the two levels. Thus, the resulting spectrum consists of lines, each representing a unique energy difference.

• The greater the energy difference, the shorter the wavelength of the emitted light.
• Different series of spectral lines exist in hydrogen, such as Balmer and Lyman, which represent transitions to specific energy levels.
• Each line can be calculated with precise wavelengths due to these fixed energy differences.

Understanding these spectral lines allows scientists to determine the presence of hydrogen in distant stars due to their distinct patterns.

Energy Levels

In an atom, electrons are organized into various energy levels or shells that are denoted by the principal quantum number, typically represented by the letter "n". This number indicates the level or shell an electron occupies, where each level has a specific energy value. The lowest energy level, known as the ground state, is the most stable.

The energy levels exist because electrons are confined to specific energy states as dictated by quantum mechanics. When an electron absorbs energy, it moves to higher levels and when it emits energy, it drops to lower ones.

• The spacing between energy levels decreases as "n" increases, which means higher levels get closer together.
• Energy is absorbed when electrons move to a higher energy level, while they emit energy moving to a lower level.
• In hydrogen, transitions returning to the ground state create visible light emissions known as the Lyman series.

These transitioning electrons between levels form the basis for the observation of spectral lines in atomic spectra.

Photon Emission

Photon emission occurs when an electron transitions from a higher energy level to a lower one within an atom. This process results in the release of energy in the form of a photon, which is a particle of light. The energy of the emitted photon corresponds to the difference in energy between the initial and final energy levels.

In the context of a hydrogen atom, an electron starting at an energy level of 4 and moving to 1 will emit photons that can be detected as distinct spectral lines.

• The energy of a photon is given by the equation \( E = h u \), where \( E \) is the energy, \( h \) is Planck's constant, and \( u \) is the frequency of the photon.
• Each transition can result in a photon of different energy, leading to different spectral lines.
• Photon emission is crucial for understanding phenomena such as fluorescence and laser operation, as both rely on electron transitions and energy release.

These emissions provide critical insights into the energy arrangement within atoms, aiding in the development of quantum theory and spectroscopy.