Question

In a hydrogen atom, what color is the emission line observed when electrons drop from the fifth to the second energy level?

Short answer

The emission line is blue-violet.

Step-by-step solution

Understand the Balmer Series

The Balmer series of spectral lines occurs when an electron transitions to the second energy level (n=2) of the hydrogen atom from higher energy levels (n > 2). These transitions emit light in the visible spectrum.

Determine the Initial and Final Levels

In this scenario, the electron falls from the fifth energy level (n=5) to the second energy level (n=2). This transition is part of the Balmer series.

Calculate the Wavelength Using Rydberg Formula

The Rydberg Formula is given by \( \frac{1}{\lambda} = R_H ( \frac{1}{n_1^2} - \frac{1}{n_2^2} ) \), where \( \lambda \) is the wavelength, \( R_H \) is the Rydberg constant \( 1.097 \times 10^7 \text{m}^{-1} \), \( n_1 = 2 \), and \( n_2 = 5 \). Using these values, we calculate \( \lambda \) as follows: \[ \frac{1}{\lambda} = 1.097 \times 10^7 ( \frac{1}{2^2} - \frac{1}{5^2} ) = 1.097 \times 10^7 ( \frac{1}{4} - \frac{1}{25} ) \]\[ = 1.097 \times 10^7 \left( \frac{25 - 4}{100} \right) = 1.097 \times 10^7 \times \frac{21}{100} \]Calculating the exact values, we found \( \lambda \approx 434 \) nm.

Identify the Color of the Emission Line

The wavelength \( \lambda = 434 \) nm is in the visible light range and corresponds to the blue-violet color in the spectrum. Therefore, the emission line observed is blue-violet.

Key concepts

Hydrogen Atom

The hydrogen atom is the simplest and most fundamental atom in the universe. It consists of only one proton and one electron.
In hydrogen, the electron revolves around the proton much like planets orbit a sun. However, unlike planets, electrons can inhabit specific energy levels (or orbits) that correspond to quantum states.
When the electron changes its energy level by moving to a higher or lower orbit, the hydrogen atom either absorbs or emits energy in the form of light.
This absorption or emission of energy is what creates spectral lines that can be observed and analyzed.

• The hydrogen atom is essential in the study of atomic structure because of its simple composition.
• Understanding hydrogen can help us learn more about the more complicated atoms found in the universe.

A good part of modern quantum mechanics and atomic physics has been developed by studying the hydrogen atom's properties.

Energy Levels

Energy levels or electron orbits in an atom are defined by the principal quantum number, denoted as \( n \).
In the hydrogen atom, these energy levels are discrete, which means electrons can only inhabit specific levels, not the spaces between them.
When an electron transitions from one level to another, it must gain or lose an exact amount of energy, known as a quantum.

• The lowest energy level, \( n=1 \), is called the ground state and holds the electron closest to the nucleus.
• Energies increase with the increasing value of \( n \), which are higher levels (\( n=2, 3, 4, 5, \ldots \)).

The transition of an electron from a higher level to a lower one results in the emission of light.
The specific differences in energy levels determine the color or wavelength of light emitted.

Rydberg Formula

The Rydberg Formula is a fundamental mathematical equation used to predict the wavelength of light produced during transitions between energy levels of a hydrogen atom.
It is expressed as:
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Where:

• \(\lambda\) is the wavelength of the emitted light.
• \(R_H\) is the Rydberg constant, approximately \(1.097 \times 10^7 \text{m}^{-1}\).
• \(n_1\) is the lower energy level (final level).
• \(n_2\) is the higher energy level (initial level).

The formula helps calculate the exact wavelength of light emitted when an electron falls from a higher energy state to a lower one.
It is especially useful in understanding the spectral lines of the hydrogen atom, aiding in identifying elements and isotopes.