Which spectral line of the hydrogen Paschen series \(\left(n_{2}=3\right)\) has wavelength \(1282 \mathrm{nm} ?\)

The Rydberg formula is a valuable tool in the field of spectroscopy, which deals with the study of the interaction between matter and electromagnetic radiation. It is especially critical for understanding the spectral lines of hydrogen, as seen in the hydrogen Paschen series problem.

The formula is expressed as \[\begin{equation}\frac{1}{\lambda} = R \left(\frac{1}{{n_1}^2} - \frac{1}{{n_2}^2}\right)\end{equation}\]where \(\lambda\) is the wavelength of the light emitted or absorbed, \(R\) is the Rydberg constant, approximately equal to 1.097 x \(10^7\) per meter, and \(n_1\) and \(n_2\) are the principal quantum numbers of the energy levels involved.

For the hydrogen Paschen series, the energy level \(n_2\) is fixed at 3 while \(n_1\) is a variable representing energy levels higher than 3. By entering the known values into the Rydberg formula and rearranging the equation, we can calculate the missing principal quantum number, \(n_1\), which represents the initial energy level before the transition that emits the spectral line with the known wavelength.

When it comes to identifying a spectral line, its wavelength is the signature that helps us pinpoint which transition it corresponds to.

In the context of our problem, the specific wavelength in question is 1282 nm (nanometers), which represents a particular spectral line of the hydrogen Paschen series. Spectral line wavelengths are highly characteristic and serve as fingerprints of the elements that emit or absorb them.

To solve our problem, we first convert the wavelength to meters since the units must match those used in the Rydberg constant. Conversion is straightforward: \(\lambda_{meters} = \lambda_{nanometers} \times 10^{-9}\).With \(\lambda_{meters}\) calculated, it serves as a crucial piece of information to work backwards from the observed spectral line to the energy level transition that produced it, using the Rydberg formula.

An energy level transition in an atom occurs when an electron moves between defined energy states or levels. In the case of the Paschen series, these transitions involve an electron falling from higher energy levels (\(n_1 > 3\)) to the third energy level (\(n_2 = 3\)).

Every transition corresponds to a photon of light with a specific energy, and consequently, a specific wavelength, which is emitted. This is a core concept of quantum mechanics, where the energy difference between levels is quantized.

In practice, to identify the energy level the electron transitioned from, we use the observed wavelength of the emitted photon in the Rydberg formula. By rearranging the formula to solve for \(n_1\), we can determine the precise energy level transition that led to the observed wavelength of 1282 nm. It's an elegant interplay between the quantized nature of atomic energy states and the classical wave property of light.