Calculate the wave number for the longest wavelength transition in the Balmer series of atomic hydrogen. \(\left(R_{11}=109677 \mathrm{~cm}^{-1}\right)\)

The Rydberg formula is central to the study of atomic spectra. It is an empirical equation discovered by the Swedish physicist Johannes Rydberg, used to predict the wavelengths of photons emitted during electron transitions between energy levels in a hydrogen atom. The formula simplifies the calculation of these wavelengths into a single, easily applied expression. Given as:

\[ u = R\bigg(\frac{1}{n_2^2} - \frac{1}{n_1^2}\bigg) \]

where \( u \) is the wave number of the emitted photon, \( R \) is the Rydberg constant approximately equal to \( 109,677 \text{cm}^{-1} \) for hydrogen, \( n_1 \) and \( n_2 \) are the principal quantum numbers of the initial and final energy levels, respectively.

The beauty and utility of the Rydberg formula lie in its connection to the energy of photons and the natural simplicity of integer use in calculating wavelength transitions for a hydrogen atom. By manipulating the formula, it's possible to find not just wavelengths but also other related values such as frequencies and energies of atomic emissions.

When hydrogen gas is excited, either by heat or electrical discharge, it emits light that, when passed through a prism, spreads into a spectrum of distinct lines, known as the hydrogen emission spectrum. This spectrum comprises several series, with the Balmer series being visible to the naked eye. The significance of the hydrogen spectrum lies in its discrete nature, which gave rise to quantum theory and changed our understanding of atomic structure.

The spectral lines correspond to specific transitions of electrons between quantum energy levels within an atom. These are often visualized as orbits or shells around the nucleus, where each shell has a defined energy level. When an electron transits from a higher to a lower energy level, it emits a photon of light. This emitted light's wavelength is distinctive to the energy difference between the two levels. For instance, in the Balmer series, all transitions end at the second energy level (n=2), but they start from higher levels – the higher the level, the more energetic the photon and the shorter its wavelength in the emitted spectrum.

Energy level transitions are changes of an electron's state within an atom from one energy level to another. These levels are quantized; that is, they can only possess certain discrete values, and the transitions between them result in the absorption or emission of photons. In the simplest terms, an electron orbiting a nucleus can be thought of as residing on steps of an energy ladder; it can leap between these steps but cannot exist in-between them.

An upward transition (absorption) occurs when an electron gains energy, typically from a photon, and moves to a higher energy level. Conversely, a downward transition (emission) results in the electron losing energy and falling back to a lower level, emitting a photon in the process. Importantly, the energy – or equivalently, the frequency – of the emitted or absorbed photons directly corresponds to the difference in energy between the initial and final levels, confirming the quantum nature of the atom. The longest wavelength within a particular series corresponds to the transition with the smallest energy difference, as seen in the Balmer series with the transition from n=3 to n=2.