Electromagnetic radiation (photon) with highest wavelength results when an electron in the hydrogen atom falls from \(n=6\) to : (a) \(n=1\) (b) \(n=2\) (c) \(n=3\) (d) \(n=5\)

In the atomic world, quantum numbers play a crucial role in determining the properties of electrons. Think of quantum numbers as addresses that provide a way to locate and describe the electron's position within an atom. There are four types of quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (m_l), and the spin quantum number (m_s).

The principal quantum number, represented by 'n', defines the energy level of an electron in an atom and broadly describes its distance from the nucleus. Larger values of 'n' indicate that an electron is located in a higher energy level, which generally means it's further away from the nucleus. This concept is a cornerstone of the Bohr model and is essential to understand other quantum numbers and the organization of electrons in atoms.

The Bohr model, named after Niels Bohr, revolutionized our understanding of atomic structure in the early 20th century. This model depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus, similar to how planets orbit the sun. Each orbit, or shell, corresponds to a particular energy level.

Although later models have provided a more intricate picture of atomic structure, the Bohr model remains an easy-to-grasp concept explaining atomic energy levels and the emission of light. When an electron jumps between these orbits, it must absorb or emit energy, which leads to the production of electromagnetic radiation. The energy of the emitted radiation is related to the difference in energy between the initial and final orbits of the electron.

Atoms are not chaotic structures; they are well-ordered, and their electrons are organized into distinct energy levels. According to quantum theory, these levels are quantized, meaning electrons cannot exist at just any energy but must occupy specific energies or 'rungs' on the ladder of an atom’s energy structure.

Electrons can move between these levels by absorbing or emitting photons with energy equal to the difference between these levels. In the hydrogen atom, for example, the energy associated with a particular level can be calculated using the formula \(E_n = - \frac{13.6 \text{ eV}}{n^2}\), where 'n' is the principal quantum number. This quantization explains why atoms emit or absorb light at specific wavelengths, leading to the characteristic spectra by which we can identify them.

The phenomena of wavelength and frequency are intricately connected in the world of electromagnetic radiation. By definition, the wavelength (\(\lambda\)) is the distance between two consecutive peaks of a wave, while the frequency (\(u\)) is the number of waves that pass a point in space per second.

The relationship between wavelength and frequency is inversely proportional and bound by the speed of light (\(c\)), as described by the equation \(c = \lambda u\). This means if the wavelength increases, the frequency decreases, and vice versa. This relationship is vital when examining the light emitted or absorbed by atoms. For instance, when an electron falls to a lower energy level and emits a photon, the energy of that photon is related to its frequency. Therefore, when solving for transitions in a hydrogen atom, seeking the highest wavelength equates to looking for the lowest energy transition between energy levels.