Question

The Lyman series of the hydrogen spectrum can be represented by the equation $$\nu=3.2881 \times 10^{15} \mathrm{s}^{-1}\left(\frac{1}{1^{2}}-\frac{1}{n^{2}}\right)(\text { where } n=2,3, \ldots)$$ (a) Calculate the maximum and minimum wavelength lines, in nanometers, in this series. (b) What value of \(n\) corresponds to a spectral line at 95.0 nm? (c) Is there a line at \(108.5 \mathrm{nm} ?\) Explain.

Short answer

For part (a), the maximum wavelength is approximately 121.5 nm, and minimum value is 0. The value of \(n\) in part (b) that corresponds to a spectral line at 95.0 nm is 3. For part (c), the calculated value of \(n\) is an integer, so there is a line at 108.5 nm.

Step-by-step solution

Calculate Maximum and Minimum Wavelength

To achieve this, remember that frequency and wavelength are inversely related (ν = c/λ), where c is the speed of light with a known value of \(3.0 × 10^8 \, m/s\). Also consider that nanometer is \(10^{-9}\,m\). Let's first calculate the maximum wavelength which corresponds to the smallest frequency, that is \(n = 2.\) Plug in \(n = 2\) into the frequency equation and obtain frequency. Now find the wavelength by plugging the obtained frequency into the wavelength equation and solving it for max wavelength. Now, to calculate the minimum wavelength which corresponds to the maximum frequency when \(n\) approaches infinity. Calculate the wavelength using the frequency obtained when \(n\) is quite large.

Find the Value of n that Corresponds to Wavelength of 95.0 nm

Given the wavelength, find frequency by use frequency-wavelength relationship. Then substitute the obtained frequency in given frequency-equation and obscure \(n\). Solve the equation to get the value of \(n\).

Is There a Line at 108.5 nm?

To check if 108.5 nm is a valid wavelength in this series, firstly convert the wavelength to frequency using frequency-wavelength relation. Then, put the frequency in given frequency-equation and solve for \(n\). You will find real number and if it is a natural number then there is a spectral line at 108.5 nm.

Key concepts

Lyman Series

The Lyman Series is a collection of spectral lines in the hydrogen spectrum associated with transitions of electrons from higher energy levels down to the innermost energy level, specifically the first energy level, also known as the ground state or 1n level. These lines are in the ultraviolet region of the electromagnetic spectrum.
When an electron in the hydrogen atom descends from a higher energy level (denoted by n) to the n=1 level, a photon is emitted.
The energy difference between the initial and final states of the electron determines the frequency and wavelength of the emitted light.
In the case of the Lyman Series:

• The electron transitions from n=2, 3, 4, ... to n=1.
• The emitted photons are in the ultraviolet spectrum, less visible to the naked eye but crucial for understanding hydrogen's electronic transitions.

This series provides a great insight into quantum mechanics and the energy levels of atoms by demonstrating how electrons move between different states.

Spectral Lines

Spectral lines are specific wavelengths at which light is emitted or absorbed by atoms and molecules. These lines appear in different regions of the electromagnetic spectrum—ultraviolet, visible, and infrared—depending on the energy changes of the electron within the atom.
For the hydrogen atom, its spectral lines play a significant role in astronomy and chemistry. In the case of the Lyman Series, the spectral lines result from electronic transitions to the lowest energy level:

• Spectral lines act like a fingerprint for elements, helping scientists identify the presence of hydrogen in stars and distant galaxies.
• Each line corresponds to a particular electron transition and has a distinct frequency and wavelength.

When we talk about spectral lines, we essentially refer to the unique pattern generated by the electron transitions that produce these lines, providing rich information about atomic structure.

Frequency and Wavelength Relationship

The relationship between frequency (\( u \)) and wavelength (\( \lambda \)) is one of the most fundamental in physics and is given by the equation \( u = \frac{c}{\lambda} \), where \(c\) is the speed of light, approximately \(3.0 \times 10^8\, m/s\).

• This equation illustrates that frequency and wavelength are inversely proportional to each other: as one increases, the other decreases.
• For the Lyman Series, when an electron transitions to the ground state, the energy released manifests as specific spectral lines characterized by particular frequencies and wavelengths.

To find the wavelength of emission or absorption in the Lyman series, we need to calculate the frequency using the main spectral formula, and then use the frequency-wavelength relationship to find the wavelength.
This relationship is crucial because it allows scientists to determine the specific transitions happening within electrons in an atom when only a frequency or wavelength is measured, like in the original exercise with hydrogen's spectral lines.