When an electron makes a transition from the \(n=4\) to the \(n=2\) hydrogen atom Bohr orbit, the energy difference between these two orbits \(\left(4.1 \times 10^{-19} \mathrm{~J}\right)\) is emitted as a photon of light. The relationship between the energy of a photon and its wavelength is given by \(E=h c / \lambda\), where \(E\) is the energy of the photon in \(\mathrm{J}, h\) is Planck's constant \(\left(6.626 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\right)\), and \(c\) is the speed of light \(\left(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\). Find the wavelength of light emitted by hydrogen atoms when an electron makes this transition.

The Bohr atomic model was developed by Niels Bohr in 1913. This model revolutionized the understanding of atomic structure by introducing the idea that electrons orbit the nucleus at certain discrete distances, or energy levels, rather than randomly. Each orbit corresponds to a specific energy level, and electrons can move between these orbits by absorbing or emitting energy in the form of photons.

According to the Bohr model, when an electron jumps from a higher to a lower energy level, it emits a photon of light. The energy of this photon is precisely the difference in energy between the two orbits. This quantization of energy levels explains the discrete lines observed in the atomic spectra of elements and is fundamental to the study of quantum mechanics.

For the hydrogen atom, the simplest atom with one electron, the energy levels are calculated using the formula derived by Bohr, taking into account the electron's quantized angular momentum. The model is particularly successful in explaining the hydrogen spectral lines. Nevertheless, it does not fully account for the complexities of multi-electron atoms or the wave nature of electrons, for which a more advanced quantum mechanical approach is required.

The relationship between photon energy and wavelength emerges from the wave-particle duality of light. Light can be described both as a wave and a particle known as a photon. The energy (\(E\)) of a photon is inversely proportional to its wavelength (\(\lambda\)), which represents the distance between two peaks of the light wave.

The equation that connects these two properties is \(E = \frac{hc}{\lambda}\), where \(E\) is the energy in joules, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength in meters. From this equation, it can be seen that as the wavelength increases, the energy of the photon decreases, and vice versa.

This relationship is crucial in many areas of physics and chemistry, including the study of atomic spectra, chemical reactions initiated by light, and even in technologies such as lasers and LEDs. Understanding this correlation helps us predict the type of light (visible, ultraviolet, etc.) emitted or absorbed by atoms and molecules based on energy transitions.

Planck's constant (symbolized as \(h\)) is a fundamental constant in quantum mechanics, which was first introduced by Max Planck. It has a value of approximately \(6.626 \times 10^{-34} \text{J} \cdot \text{s}\) and represents the quantization of energy. This constant is a central part of the energy-wavelength relationship for photons and dictates the size of the energy packets or 'quanta' that can be absorbed or emitted by an atom.

\(h\) is used to calculate the energy of individual photons as well as the energy levels of electrons within atoms. It appears in numerous formulas throughout quantum physics and is essential for calculations like those needed for the Bohr atomic model. Planck's constant also plays a pivotal role in the photoelectric effect, which further demonstrates the particle-like behavior of light and was a crucial piece of evidence for quantum theory.

The speed of light (denoted \(c\)) is a constant that represents the maximum speed at which all energy, matter, and information in the universe can travel. It has a value of approximately \(3.00 \times 10^{8} \text{m/s}\) in a vacuum. This constant is not only essential for the theories of relativity but also for quantum mechanics and the study of electromagnetic waves.

In the context of the photon energy-wavelength relationship, the speed of light links the wave and particle natures of light. It allows us to calculate the energy of photons given their frequency or wavelength and underpins many of the calculations involving electromagnetic radiation. From the design of telescopes to the synchronization of GPS satellites, the unvarying speed of light is an essential factor in a wide array of scientific and technological applications.