What is the wavelength of light emitted when the electron in a hydrogen atom undergoes transition from an energy level with \(n=4\) to an energy level with \(n=2 ?\)

To understand how light is emitted from a hydrogen atom, we need to dive into the mechanics of electron transitions. An electron transition occurs when an electron in an atom absorbs or emits energy and moves between quantized energy levels. These levels are like steps on a ladder that the electron can 'climb' up or down.

In a hydrogen atom, which is the simplest atom with only one electron, these energy levels are well-defined and labelled by an integer, 'n', known as the principal quantum number. The different levels correspond to different distances from the nucleus, with 'n=1' being the closest. When an electron moves from a higher level (which means it is more 'excited') to a lower level, it emits energy. This energy is released in the form of a photon, a particle of light. The wavelength of the emitted light is directly related to the difference in energy between the two levels.

Each possible electron transition has a specific energy difference, and therefore emits a photon with a specific wavelength, which can be calculated using the Rydberg formula. This is essential in understanding atomic structure and spectral lines seen in spectrometry.

Calculating the wavelength of emitted light is critical in the field of spectroscopy, where we study the spectrum of light to gain information about the atomic structure. The Rydberg formula provides a mathematical way to predict this wavelength based on the initial and final energy levels of an electron in an atom.

To calculate the wavelength of light emitted during an electron transition, we employ the inverse relationship given by the Rydberg formula: \[\begin{equation}\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\end{equation}\]Here, \(\lambda\) is the wavelength of the emitted light, \(R_H\) is the Rydberg constant for hydrogen, and \(n_1\) and \(n_2\) are the lower and higher energy levels, respectively. After simplifying the terms and calculating, the inverse of the result gives us the wavelength. The solution process shows that even small changes in the energy levels can result in different wavelengths of light, which is why hydrogen's emission spectrum is composed of distinct lines, each corresponding to specific transitions.

The energy levels of the hydrogen atom are a fundamental concept in quantum mechanics and chemistry. The difference between these levels determines the amount of energy absorbed or emitted when an electron transitions. As per the Bohr model of the hydrogen atom, energy levels are quantized and inversely proportional to the square of the principal quantum number (n), known as an integer denoting the level.

These energy levels are often represented by diagrams where each level is a horizontal line, with the ground state at the bottom. When an electron jumps between these levels, it must either absorb energy (to move up) or release energy (to move down), which is strictly quantified by Planck's relation (E=hf), where 'E' is the energy difference, 'h' is Planck's constant, and 'f' is the frequency of light.

In the simplified case of the hydrogen atom, it’s possible to precisely determine these energy differences since it has only one electron, and as a result, its emission spectrum, a physical manifestation of these levels, forms clear, distinct lines. Understanding these energy levels is essential for explaining a wide range of physical phenomena, from the inner workings of lasers to the process of nuclear fusion in stars.