Question

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

Short answer

The emission line is cyan or blue-green, around 486 nm.

Step-by-step solution

Identify the Transition

Electrons in a hydrogen atom dropping from the fourth energy level ( =4) to the second energy level ( =2) undergo a transition. This is known as the Balmer series, which is associated with visible light emissions.

Calculate Wavelength of Emission

Use the Rydberg formula for hydrogen to calculate the wavelength of the emitted light: \[\frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{4^2} \right)\]where \( R_H \approx 1.097 \times 10^7 \ \text{m}^{-1} \) is the Rydberg constant. Simplifying the expression inside the parentheses gives \( \frac{1}{4} - \frac{1}{16} = \frac{3}{16} \). Thus,\[\frac{1}{\lambda} \approx 1.097 \times 10^7 \times \frac{3}{16} \approx 2.056 \times 10^6 \ \text{m}^{-1}\],and calculate \( \lambda \approx 486 \text{ nm} \).

Determine the Color of the Wavelength

The wavelength calculated is approximately 486 nm. This wavelength falls within the visible spectrum of light and corresponds to the color cyan or blue-green.

Key concepts

Balmer Series

The Balmer Series is one of the series of lines that describe the emission spectrum of hydrogen. It is specifically related to electron transitions that result in visible light emissions. When an electron in a hydrogen atom falls from a higher energy level, such as the third, fourth, or fifth level, down to the second level, it results in the release of light that falls within the visible spectrum.

• This is why the Balmer series is so important - it allows us to see the wavelengths of light that are emitted.
• Johann Balmer discovered this series in 1885, which significantly advanced our understanding of atomic emission spectra.

When electrons transition between levels, you might notice lines of various colors. These lines are the result of different transitions that occur according to the Balmer series.
The color of the light emitted depends on the wavelength and is exact enough that it can be used to identify hydrogen gas in stars and other astronomical objects.

Electron Transition

An electron transition occurs when an electron in an atom moves between different energy levels, or shells. In the context of the hydrogen atom, these energy levels are more widely spaced at higher levels compared to those closer to the nucleus. The transition from a higher energy level to a lower one involves the release of energy in the form of light.

• The direction of the transition (i.e., from higher to lower energy levels) indicates a release of energy.
• The specific energy levels involved in the transition define the wavelength and frequency of the emitted light.

In the exercise, when an electron transitions from level four to level two, the energy difference matches specific wavelengths within the Balmer series.
This release of energy in the form of light is how we observe the distinct spectral lines in elements like hydrogen. Besides being a fascinating phenomenon, electron transitions are key to many areas of research, including quantum mechanics and astrophysics.

Rydberg Formula

The Rydberg Formula is a mathematical equation used to predict the wavelengths of photons (light) emitted or absorbed by electrons in a hydrogen atom as they transition between energy levels.
It is expressed as: \[\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\]

• Here, \( R_H \) is the Rydberg constant, approximately \( 1.097 \times 10^7 \ \text{m}^{-1} \).
• The terms \( n_1 \) and \( n_2 \) represent the lower and higher energy levels involved in the transition, respectively.

In our exercise, the transition is from \( n_2 = 4 \) to \( n_1 = 2 \), specifically part of the Balmer series.
The Rydberg formula effectively calculates the wavelength \( \lambda \) of the emitted photon, providing a clear insight into which part of the electromagnetic spectrum the light will fall into.
By examining these transitions using the Rydberg formula, scientists effectively decode the patterns of spectral lines, aiding in identifying elements in various states or locations, such as distant stars. This formula is fundamental in spectroscopy.