Question

Wave number ( \(\mathrm{v}\) ) are the reciprocals of wavelengths, \(\lambda\) and are given by the expression \(\bar{v}=1 / \lambda\). For the hydrogen atom, the Bohr theory predicts that the wave number for the emission line associated with an electronic transition from the energy level having principal quantum number \(\mathrm{n}_{2}\) to that with principal quantum number \(\mathrm{n}_{1}\) is \(\bar{v}=R_{H}\left[\left(1 / n_{1}^{2}\right)-\left(1 / n_{2}^{2}\right)\right]\) where \(\mathrm{R}_{\mathrm{H}}\) is the Rydberg constant. In what region of the electromagnetic spectrum would there appear a spectral line resulting from the transition from the tenth to the fifth electronic level in hydrogen?

Short answer

The spectral line resulting from the transition from the 10th to the 5th electronic level in hydrogen has a wavelength of approximately \(6.08 \times 10^{-7}\, \text{m}\). This wavelength lies within the visible light range, so the spectral line would appear in the visible region of the electromagnetic spectrum.

Step-by-step solution

Determine the wave number

For this problem, the wave number (\(\bar{v}\)) can be calculated using the provided formula and given values for \(n_1\), \(n_2\), and \(R_H\): $\bar{v}=R_{H}\left[\left(1 / n_{1}^{2}\right)-\left(1 / n_{2}^{2}\right)\right]$ Given that \(n_1 = 5\) and \(n_2 = 10\), we can plug them into the formula: \(\bar{v}=R_{H}\left[\left(1 / 5^{2}\right)-\left(1 / 10^{2}\right)\right]\)

Calculate the value of the wave number

Now we need to compute the wave number value. Let's use the Rydberg constant value (\(R_H \approx 1.097 \times 10^7 \, \text{m}^{-1}\)): \(\bar{v}= (1.097 \times 10^7)\left[\left(1 / 5^{2}\right)-\left(1 / 10^{2}\right)\right]\) By solving the equation, we find the wave number: \(\bar{v} \approx 1.646 \times 10^6\, \text{m}^{-1}\)

Calculate the wavelength

Now, we can determine the wavelength (\(\lambda\)) of the spectral line by using the relationship between wave number and wavelength: \(\lambda = \frac{1}{\bar{v}}\) Plug the wave number value (\(\bar{v} \approx 1.646 \times 10^6\, \text{m}^{-1}\)) into the formula: \(\lambda \approx \frac{1}{1.646 \times 10^6}\, \text{m}\) After the calculation, we get: \(\lambda \approx 6.08 \times 10^{-7}\, \text{m}\)

Determine the region of the electromagnetic spectrum

The spectral line's wavelength is approximately \(6.08 \times 10^{-7}\, \text{m}\). This wavelength lies within the visible light range (approximately \(4 \times 10^{-7}\, \text{m}\) to \(7 \times 10^{-7}\, \text{m}\)). Therefore, the spectral line resulting from the transition from the 10th to the 5th electronic level in hydrogen would appear in the visible region of the electromagnetic spectrum.

Key concepts

Wave Number

The concept of a wave number is integral to understanding atomic transitions and spectra. Wave number, denoted as \( \bar{v} \), is defined as the reciprocal of the wavelength \( \lambda \), or \( \bar{v} = \frac{1}{\lambda} \). It gives the number of waves in a unit distance.
Wave numbers are particularly useful in spectroscopy as they simplify calculations that involve the relationships between energy, frequency, and wavelength. In the context of the hydrogen atom's spectral lines, the Bohr model provides a formula: \[\bar{v} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\]where \( R_H \) is the Rydberg constant, and \( n_1 \) and \( n_2 \) are principal quantum numbers of the initial and final states, respectively. This formula enables us to calculate the wave number of light emitted when electrons transition between different energy levels in a hydrogen atom.

Energy Levels

In quantum mechanics, energy levels in an atom define the specific energies that an electron can occupy. In the hydrogen atom, these levels are quantized, meaning electrons can only exist in discrete energy states. Bohr's model illustrates that the energy of an electron in hydrogen is described by the principal quantum number \( n \).
When electrons move between these levels, they emit or absorb energy in the form of electromagnetic radiation. This transition determines the spectral line's characteristics, such as its wavelength and wave number. The principal quantum number \( n \) is integral in calculating these energy transitions through the formula:\[ E_n = - \frac{R_H c h}{n^2} \]where \( E_n \) stands for energy at a particular level, and \( R_H \), \( c \), and \( h \) are constants of nature: the Rydberg constant, speed of light, and Planck's constant respectively. Lower energy levels are closer to the nucleus, and as electrons drop from a higher to a lower energy level, light is emitted, which corresponds to a certain wave number and wavelength.

Rydberg Constant

The Rydberg constant \( R_H \) is a fundamental physical constant that plays a crucial role in atomic physics, especially in calculations related to the spectral lines of hydrogen. It represents the highest wavenumber (or the limit of the series) of any photon that can be emitted from the hydrogen atom.
With a value approximately \( 1.097 \times 10^7 \, \text{m}^{-1} \), the Rydberg constant is used in the formula for calculating the wave number \( \bar{v} \) for transitions between the energy levels in hydrogen, given by:\[ \bar{v} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]Its discovery and precise calculation were significant achievements in understanding atomic spectra and verifying the accuracy of quantum mechanical models like Bohr's. This constant helps us link atomic structure with measured radiation, enabling predictions of where spectral lines will fall on the electromagnetic spectrum.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, organized by wavelength or frequency. It includes a variety of waves such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.
Spectral lines, like those emitted by hydrogen, fall into specific regions of the electromagnetic spectrum based on their wavelengths. The wavelength provides information on the type of electromagnetic radiation detected. For instance, visible light lies within the wavelength range of approximately \( 4 \times 10^{-7} \text{ m} \) to \( 7 \times 10^{-7} \text{ m} \).
When an electron in a hydrogen atom transitions from a higher to a lower energy level, it emits light whose wavelength indicates its position in the electromagnetic spectrum. In the exercise discussed, a transition from the 10th to the 5th level in hydrogen results in a spectral line in the visible region, showing the practical application of identifying regions on the electromagnetic spectrum.