Which electronic transition in a hydrogen atom, starting from the orbit \(n=7\), will produce infrared light of wavelength \(2170 \mathrm{~nm}\) ? (Given : \(R_{H}=1.09677 \times 10^{7} \mathrm{~m}^{-1}\) ) (a) \(n=7\) to \(n=6\) (b) \(n=7\) to \(n=5\) (c) \(n=7\) to \(n=4\) (d) \(n=7\) to \(n=3\)

Understanding electron transitions in hydrogen atoms is crucial for unraveling the mysteries of emission spectra. An electron transition occurs when an electron in a hydrogen atom absorbs or emits energy and moves between energy levels or orbits. These transitions are not random; they follow specific quantum mechanical rules.

For instance, when an electron falls from a higher energy level (with a higher principal quantum number, denoted as n) to a lower one, it emits a photon of light. The energy of this photon corresponds to the difference in energy between the two levels. This is what the Rydberg formula, introduced in the step-by-step solution, precisely describes.

For the exercise at hand, starting from n=7, the emitted light's wavelength can help us deduce the final energy level the electron transitioned to, which we will further investigate by employing the Rydberg formula.

Quantum numbers are the address system of electrons in an atom. Each electron in an atom is described by a set of quantum numbers which provide information about its energy, angular momentum, magnetic orientation, and spin. The most relevant in terms of the Rydberg formula and electron transitions is the principal quantum number, designated as n.

It denotes the energy level or shell in which an electron resides, with n=1 being the ground state, or the closest to the nucleus, and higher values corresponding to higher energy levels. Quantum numbers are of integral importance in predicting the electronic structure of atoms and the chemical behavior of elements.

In the case of the hydrogen atom, the electron transition from one principal quantum level to another leads to the emission or absorption of light. Analyzing these transitions helps explain how the specific lines in the hydrogen spectrum arise.

The wavelength of light emitted during an electron transition in a hydrogen atom is inversely proportional to the energy difference between the two involved energy levels. This is represented by the Rydberg formula and determines the color of the light seen in a spectrum.

To precisely calculate the wavelength of emitted light, we start by determining the wavenumber, as it's the inverse of wavelength (λ). Using the step-by-step solution provided, we have the means to calculate this for a given transition in the hydrogen atom. With R_H representing the Rydberg constant specific to hydrogen and the principal quantum numbers dictating the initial and final energy levels, we can mathematically deduce the wavelength of the emitted photon.

For educational clarity, converting units to a common standard (e.g., converting nanometers to meters) is essential prior to applying the Rydberg equation. Subsequently, you can identify the electron transition by matching the calculated wavenumber with the wavenumber derived from the wavelength given in the problem. The correct electron transition yields the known wavelength of the infrared light (2170 nm) as described in the original exercise.