A student in a physics laboratory observes a hydrogen spectrum with a diffraction grating for the purpose of measuring the wavelengths of the emitted radiation. In the spectrum, she observes a yellow line and finds its wavelength to be \(589 \mathrm{nm}\). (a) Assuming that this is part of the Balmer series, determine \(n_{\mathrm{i}},\) the principal quantum number of the initial state. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

The hydrogen spectrum is a sequence of visible, infrared, and ultraviolet light that hydrogen atoms emit when their electrons transition between energy levels. This spectrum is not continuous; rather, it consists of discrete lines at specific frequencies corresponding to the energy differences between levels. In a classroom setting, the Balmer series is often observed because it lies within the visible range.

When an electron in a hydrogen atom drops from a higher energy level to the second level, it emits light that forms part of the Balmer series. The colors observed depend on the initial energy level the electron falls from, producing a unique pattern that is like a fingerprint for hydrogen. This emission spectrum is pivotal in understanding electronic transitions and energy quantization in atoms.

A diffraction grating is an optical component marked with a pattern of parallel lines or grooves, designed to disperse light into its individual colors or wavelengths. When a monochromatic light source, such as light from a hydrogen discharge tube, passes through a diffraction grating, it creates an interference pattern that allows for precise determination of the light's wavelength.

To study the hydrogen spectrum, a student can use a diffraction grating to split light into its component wavelengths, which appear as distinct lines. Each line corresponds to a specific transition between energy levels in the hydrogen atom. By measuring the angle at which the light is diffracted, the student can accurately calculate the wavelength using the grating equation.

The principal quantum number, denoted as 'n', is an integer value that defines the energy level of an electron in an atom. The energy levels can be thought of as the 'rungs' of a ladder that electrons can 'step' on: the higher the energy level, the further it is from the nucleus.

For the Balmer series, the final state of electron transition is always to the second energy level ( = 2). However, the initial energy level, or the 'start' of the electron's jump, has a principal quantum number (i) greater than 2. When problem-solving, it's essential to remember that must always be an integer, as fractional energy levels do not exist within the atomic model.

The Rydberg constant (RH) is a fundamental physical constant related to the energy of photon emissions in atomic spectra. Specifically, for hydrogen, it denotes the highest possible wavenumber (inverse of wavelength) of any photon that can be emitted from the hydrogen atom, corresponding to an electron's transition from infinity to the first energy level.

The Rydberg constant is a crucial element in the Rydberg formula, which predicts the wavelengths of the hydrogen spectrum. By using the Rydberg constant and knowing the principal quantum numbers involved in the transition ( and i), the exact wavelength of light emitted can be calculated. This constant is instrumental in understanding atomic structure and spectral lines.