According to the quantum-mechanical model for the hydrogen atom, which electron transition produces light with the longer wavelength: $$3p \longrightarrow 2 s \quad {\text{or}} \quad 4p \longrightarrow 3 p ?$$

The quantum-mechanical model is a cornerstone of modern physics, providing a detailed explanation of how particles such as electrons move and interact. Unlike classical physics, which treats energy as a continuous flow, the quantum-mechanical model postulates that energy exists in discrete packets called quanta. Electrons orbit the nucleus of an atom in specific energy levels or shells. Each shell corresponds to a principal quantum number (). These shells are not to be imagined as simple orbits but rather as complex, three-dimensional standing waves — think of them as ‘clouds’ of probability where an electron is likely to be found.

When an electron transitions between these shells, it must absorb or emit energy equivalent to the difference between the energy levels. This is fundamentally different from classical transitions, which could theoretically have any value. Because of this quantized nature of energy levels, electrons transitioning between them create or absorb photons with very specific energies, which corresponds to specific wavelengths of light. Therefore, these electron transitions are responsible for the line spectra we see in elements, such as the characteristic colors of neon signs or the spectral lines used to determine the chemical composition of stars.

The Rydberg formula is a mathematical expression that allows us to calculate the wavelengths of electromagnetic spectrum lines emitted by transitions in the energy level of an electron in a hydrogen atom. It is given by the equation: \[ \frac{1}{\lambda} = R_{\infty}\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \] where \( \lambda \) is the wavelength of the emitted or absorbed light, \( R_{\infty} \) is the Rydberg constant (approximately \( 1.097 \times 10^7 \) per meter), and \( n_1 \) and \( n_2 \) are the principal quantum numbers of the initial and final energy levels, respectively. As suggested by the formula, the wavelength is inversely related to the difference in the reciprocals of the squares of the principal quantum numbers. Hence, smaller energy transitions will result in light with longer wavelengths because the difference between the squared reciprocals of \( n_1 \) and \( n_2 \) will be smaller.

Photon energy refers to the amount of energy carried by a single photon, which is the quantum of the electromagnetic field including light. The energy of a photon is directly related to its frequency and inversely proportional to its wavelength, given by the equation: \[ E = hu = \frac{hc}{\lambda} \] where \( E \) is the energy of the photon, \( h \) is Planck's constant (approximately \( 6.626 \times 10^{-34} \) joule seconds), \( u \) is the frequency, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the photon. If we know the energy levels from which an electron descends, we can determine the energy of the photon emitted and, subsequently, the wavelength of the light. This connection between electron transitions and photon energy is why understanding electron transitions is crucial for fields such as spectroscopy, which studies the interaction between matter and electromagnetic radiation.

The principal quantum number, denoted as \( n \), is one of four quantum numbers used to describe the quantum state of an electron in an atom. It delineates the energy level or shell of an electron within an atom and is always a positive integer (1, 2, 3, ...). Each increase in the principal quantum number designates a new energy level with further distance from the nucleus and therefore, greater energy. For example, an electron in the shell corresponding to \( n = 2 \) will have less energy and be closer to the nucleus than an electron in the shell with \( n = 3 \).

The significance of the principal quantum number lies in its direct impact on the properties of the atom. As \( n \) changes, so do the energy and the average distance of the electron from the nucleus. These changes can significantly affect atomic behavior, reactivity, and the emission or absorption spectra produced during electron transitions. For students studying chemistry and physics, understanding the principal quantum number is essential for comprehending the complexities of atomic structure and the patterns within the periodic table.