Calculate the wavelength of the light emitted when an electron in a hydrogen atom makes each transition and indicate the region of the electromagnetic spectrum (infrared, visible, ultraviolet, etc.) where the light is found. $$\begin{array}{ll}{\text { a. }} & {n=2 \longrightarrow n=1} \\ {\text { b. }} & {n=3 \longrightarrow n=1} \\ {\text { c. }} & {n=4 \longrightarrow n=2} \\\ {\text { d. }} & {n=5 \longrightarrow n=2}\end{array}$$

Imagine electrons as sprinters on tracks of differing lengths - these tracks being the various energy levels within an atom. An electron transition occurs when an electron leaps from one track to another. In the context of a hydrogen atom, these transitions are between energy levels characterized by quantum numbers (n=1, n=2, and so forth).

When an electron jumps to a lower energy level, it emits energy in the form of light - a process fundamental to understanding how atoms interact with light. This light has a specific wavelength, which can be found using the Rydberg formula, giving us a gateway to probing the subatomic world. For example, a transition from n=2 to n=1 releases a photon, a particle of light, with a unique wavelength correlating to energy difference between those energy levels.

The wavelength calculation is akin to finding out the exact hue of the color emitted when an electron makes its dive to a lower energy state. Using the Rydberg formula, we can calculate the wavelength by considering the energy levels the electron transitions between. Let's break it down: when we know the initial (n_i) and final (n_f) energy levels, and the Rydberg constant (R_H), we can apply the formula to get the reciprocal of the wavelength (1/λ).

After the calculation, we just take the reciprocal of that result to find the actual wavelength (λ). It's like using a mathematical magnifying glass to peer at the precise shade of light the atom emits, which can be as telling as a fingerprint in identifying the atom.

The electromagnetic spectrum is the full range of all possible light wave frequencies, including those the human eye can and cannot see. From the vast radio waves to the tiny gamma rays, this spectrum encompasses all forms of electromagnetic radiation.

The light we can see, known as the visible spectrum, is just a small part sandwiched between the infrared and ultraviolet regions. When we calculate the wavelength of light emitted from an electron transition, we can pinpoint where that light sits on this grand spectrum. It's like identifying whether the light is a deep red hue found at the end of a sunset (infrared), or the kind of invisible, skin-tanning rays at the beach (ultraviolet).

A hydrogen atom is the universe's poster child for simplicity, having only one electron. But even with its simplicity, it provides a brilliant showcase for quantum mechanics through its discrete energy levels. These levels are the defined orbits or shells where the electron can exist around the nucleus, each with increasing energy as you move further away.

Why discrete? Because the electron can only be in specified, well-defined levels, rather like steps on a ladder, no in-betweens allowed. The energy difference between these levels determines the wavelength of light emitted during an electron transition, with larger gaps corresponding to more energetic light. It's as if the electron, when jumping down from a higher step, has more potential energy to convert into light, thus emitting a higher frequency (and hence, a shorter wavelength) photon.