Do the Balmer series and the Lyman series overlap? Why? Why not? (Hint: calculate the shortest Balmer line and the longest Lyman line.)

The Balmer series is a set of spectral emission lines of the hydrogen atom that are due to electronic transitions from higher energy levels to the second energy level (denoted as n=2). This sequence of lines in the hydrogen spectrum is observable in the visible spectrum, which is why it can typically be seen with a simple optical spectroscope. It's named after Johann Balmer, who discovered the Balmer formula in 1885.

The specific wavelengths of the Balmer series are given by the Rydberg formula, which involves the principal quantum numbers of the initial and final energy levels. The series begins technically with the n=3 to n=2 transition, called H-alpha, and extends indefinitely with the lines growing closer together as the frequency increases. Because the human eye perceives different wavelengths of light as different colors, the Balmer series can be seen as a range of colors ranging from red (at the longest wavelength, H-alpha) through to violet. Notably, the shortest wavelength in the Balmer series corresponds to the transition from infinity to n=2 and lies just outside the visible spectrum.

On the other hand, the Lyman series consists of lines in the hydrogen spectrum that result from electron transitions from higher energy levels to the ground state (n=1). Unlike the Balmer series, the spectral lines in the Lyman series are located in the ultraviolet part of the electromagnetic spectrum and are therefore not visible to the naked eye. The series was named after Theodore Lyman, an American physicist who discovered it in 1906.

The transitions to the ground state from higher levels produce higher energies, and thus the series is characterized by its high frequencies. It starts with the n=2 to n=1 transition, known as the Lyman-alpha line, and continues with transitions from increasingly higher levels, creating more spectral lines. Since these transitions are between high energy levels and the ground state, the radiated energy is significantly larger than that of the Balmer series, hence their ultraviolet placement.

The Rydberg formula is a mathematical equation used to predict the wavelength of light resulting from an electron moving between energy levels of an atom. The formula was proposed by Johannes Rydberg in 1888 and is given by:
\[ \frac{1}{\lambda} = R \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]
where \(\lambda\) is the wavelength, \(R\) is the Rydberg constant, and \(n_1\) and \(n_2\) are the principal quantum numbers of the initial and final energy levels respectively, with \(n_2 > n_1\). The Rydberg constant is a fundamental physical constant and has the value of approximately 1.097 x 10^7 m^-1 in air.

This formula is instrumental in quantum mechanics and has been fundamental in the development of atomic physics. It's particularly important because it was derived before the development of quantum theory, yet it accurately describes the behavior of electrons in an atom as quantum mechanics later explained. By using this formula, one can calculate the wavelengths of different series of hydrogen, such as Balmer and Lyman, as depicted in the step-by-step solution.