Electron Transitions In Hydrogen Atoms: Calculating Photon Frequency

i. What is the Frequency of this Transition? (1 Point)

Understanding Electron Transitions and Energy Emission

In the fascinating world of quantum mechanics, the hydrogen atom serves as a fundamental model for understanding the behavior of electrons within atoms. This seemingly simple system, consisting of just one proton and one electron, exhibits a rich array of phenomena, including the ability of its electron to transition between energy levels. When an electron in a hydrogen atom jumps from a higher energy level to a lower one, it releases the excess energy in the form of a photon, a particle of light. The energy of this photon is directly related to the difference in energy between the two levels, a concept that is pivotal in understanding atomic spectra and the interaction of light with matter. Understanding these electron transitions requires a firm grasp of the Bohr model and the quantum mechanical framework that governs atomic behavior.

The transition described in this problem, where an electron moves from the n = 6 energy level to the n = 1 level, is a significant one. The principal quantum number, n, dictates the energy level of the electron, with higher values of n corresponding to higher energy levels. The n = 1 level, also known as the ground state, represents the lowest energy state the electron can occupy. When the electron drops from n = 6 to n = 1, it releases a substantial amount of energy, as evidenced by the emitted photon's energy of 2.11 x 10^-18 J. This energy release is not arbitrary; it's dictated by the fundamental laws of quantum mechanics, which state that electrons can only exist in specific, quantized energy levels. The energy difference between these levels determines the energy of the emitted photon. To fully appreciate this phenomenon, one must consider the relationship between energy, frequency, and wavelength of electromagnetic radiation, as described by Planck's equation and the speed of light equation. The emitted photon's frequency, which we aim to calculate, is a key characteristic that defines its position within the electromagnetic spectrum, potentially ranging from ultraviolet to visible light, depending on the magnitude of the energy released.

Calculating the Frequency of the Emitted Photon

To determine the frequency of the emitted photon, we turn to one of the cornerstone equations of quantum mechanics: Planck's equation. This equation, expressed as E = hν, elegantly connects the energy (E) of a photon to its frequency (ν) through Planck's constant (h). Planck's constant, approximately 6.626 x 10^-34 Joule-seconds (J·s), is a fundamental constant of nature that quantifies the relationship between energy and frequency at the atomic level. In our scenario, we know the energy of the emitted photon (2.11 x 10^-18 J) and Planck's constant, allowing us to solve for the frequency (ν) using a simple algebraic manipulation. Rearranging Planck's equation, we get ν = E / h. Substituting the known values, we have ν = (2.11 x 10^-18 J) / (6.626 x 10^-34 J·s). Performing this calculation yields the frequency of the emitted photon. This frequency is a crucial piece of information, as it not only tells us about the nature of the emitted light but also provides insights into the energy level transitions within the hydrogen atom. The magnitude of the frequency will dictate where the photon lies on the electromagnetic spectrum, a critical aspect for spectroscopic analysis and understanding the interaction of light with matter.

The resulting frequency, calculated from the given energy, falls within a specific range of the electromagnetic spectrum. The higher the frequency, the more energetic the photon, and the closer it is to the ultraviolet or even X-ray region of the spectrum. Conversely, lower frequencies correspond to less energetic photons, which fall into the infrared or radio wave regions. In the case of an electron transitioning from n = 6 to n = 1 in a hydrogen atom, the substantial energy release suggests a relatively high-frequency photon. Determining the exact numerical value of the frequency allows us to pinpoint the photon's position on the spectrum and understand its potential interactions with other matter. Furthermore, this calculation highlights the quantized nature of energy transitions in atoms. The fact that the emitted photon has a specific frequency, and thus a specific energy, is a direct consequence of the electron only being able to occupy discrete energy levels within the atom. This principle is a cornerstone of quantum mechanics and differentiates it significantly from classical physics, where energy is often treated as a continuous variable.

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Electron Transitions in Hydrogen Atoms Calculating Photon Frequency

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What is the frequency of the photon emitted when an electron transitions from the n=6 to n=1 energy level in a hydrogen atom, given that the photon energy is 2.11 x 10^-18 J?