6.2 Rydberg Equation
6.2 Rydberg Equation
Planck’s and Rydberg’s Principles
Freshman students are taking a general chemistry class at Georgia State University. One day during the semester, their professor decided to take them to Las Vegas for field trip and to visit the University of Nevada, Las Vegas department of chemistry and look at their labs and undergraduate research.
They took the plane and arrived Las Vegas in the morning. They took the whole morning a calm rest at one of the Las Vegas Hotels.
At night they walked through Las Vegas Down Town. Las Vegas was looking very impressive with lots of fancy lights with different colors.
Figure 6.17 Night Sky of Las Vegas
Ref: Commons.wikimedia.org/
Las Vegas at night
Reference: https://www.flickr.com/photos/whsieh78/8222313719
The students were very excited and impressed with all the colors of light they saw.
One student asked his professor about the origins of these different colors of lights they saw.
The professor replied to his class: “Let us enjoy the night tonight and tomorrow after the breakfast and before visiting the chemistry department at the University of Nevada, I will gather everyone and will go over the origin of these beautiful lights”.
Next day, the professor and his class gathered around the professor’s laptop after the breakfast.
He started explaining the emission and absorption concept to his students. Luckily, they covered some of the materials covering the aspects of emission and absorption.
The professor explained them the concept of the emission and absorption.
Absorption: When an atom is absorbing energy from outside source and the valence electrons get excited to higher levels. This excitation is not stable and the valence electrons can give away the absorbed energy and the valence electrons go back to their original levels. This process is called emission because the valence electrons emit photons (releasing the absorbed energy). Each element has specific transitions of absorption and emission.
Figure 6.18 H-Atom Emission Spectrum
Ref: Commons.wikimedia.org/
In the figure above, electron energy levels for the hydrogen atom. The arrows indicate the electron transitions from higher energy levels to lower energy levels. The energies of the emitted photons are the same as the energy difference between two energy levels.
The absorption process (arrows up) can be considered as the opposite process of the emission process (arrows down)..
Figure 6.19 Different Wavelengths of Absorption and Emission Energy
Ref: Commons.wikimedia.org/
ΔE(electron) = E(final) − E(initial) = h x ϑ = [h x c / λ ]
The above formula is called Planck’s formula or Principle
ΔE(electron) = Photon’s energy is equal to the energy difference between the two energy levels (final – initial).
h = Planck’s constant
ϑ = frequency
c = speed of light
λ = wavelength
Each specific photon frequency (or wavelength) for each transition will give very specific color. An example of this is the hydrogen spectra photon transitions in the visible range. The figure below illustrates these unique spectra. Each element in the periodic table will have its own unique line spectra.
Figure 6.20 Discrete Emission Spectrum Lines
Ref: Commons.wikimedia.org/
The professor went further and recommended his students the following YouTube videos:
.
Emission and absorption of light
Figure 6.21 Bohr’s Orbit based on Rydberg Equation
Ref: Commons.wikimedia.org/
Electron excitation, emission and absorption spectra
Modern Atomic Theory: Max Planck & Quantized Energy
The professor went on his talk and recommended his students to wait till they will hear the concept of Rydberg’s Principle.
Rydberg’s Principle
The Rydberg’s Principleconcept explains spectral lines of each element related to their frequencies which can be either the sum or the difference of the frequencies of two other lines. Lines of the spectra of elements could be predicted from existing lines.
Rydberg’s Formula
The formula for hydrogen is given in the figure below:
n2 is greater than n1
Reference: https://en.wikipedia.org/wiki/Rydberg_formula
The figure below illustrates the origin of the line spectra of hydrogen:
Figure 6.22 Spectral Lines of Bohr’s Orbit
Ref: Commons.wikimedia.org
http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html
There is another form of Rydberg Equation which is popularly used:
∆E=-2.18 *10-18 
Above equation is frequently used for energy difference and followed by Planck’s equation to determine wavelength.
For example, suppose, that an electron in a hydrogen atom relaxes from an orbital in the n=3 level to an orbital in the n=2 level. We determine ∆E, the energy difference corresponding to the transition from n=3 to n=2 as follows,
∆Eatom= E2-E3
=-2.18 *10-18(1/22)- -2.18 *10-18(1/32) = =-2.18 *10-18(1/22– 1/32)= -3.03*10-19 J
Since energy must be conserved, the exact amount of energy emitted by atom is carried by the photon.
-∆Eatom=∆Ephoton
Since wavelength of a photon is related to its energy, E= hc/λ, we calculate wavelength of the photon as:
Λ= hc/E= (6.626 *10-34 *3.00*108)/3.03 *10-19
= 6.56 *10-7 m or 656 nm
With the help of Rydberg formula, one can calculate the wavelength of each transition and the hence the colors of the line spectra will be determined as shown in the hydrogen spectrum.
At the end of his long talk (almost a lecture), the professor recommended his students the videos below which cover the Rydberg Principle and Formula and Calculations:
Bohr Model of the Hydrogen Atom, Electron Transitions, Atomic Energy Levels, Lyman & Balmer Series
Quantum Chemistry 1.3 – Rydberg Formula
Bohr Model of the Hydrogen Atom
At the of the professor talk(lecture), the students were very satisfied and happy to cover these materials with lots of details and asked the professor to go back to downtown Las Vegas and enjoy the different colors of lights and eager to know more about the elements with different emission line spectra (different colors).The professor promised his students to cover this topic as well, when they return back to Atlanta and to Georgia State University.
Figure 6.23 Rydberg Equation
Ref: Commons.wikimedia.org
It was later found that n2 and n1 were related to the principal quantum number or energy quantum number. This formula works very well for transitions between energy levels of the hydrogen atom. It is because H is with only one electron.
For atoms with more than one electron, this formula begins to break down and give incorrect results. This was due to the inaccuracy that the amount of screening for inner electrons for outer electron transitions varies.
We can use the Rydberg formula for hydrogen to obtain its spectral lines. Setting n1 to 1 and running n2 from 2 to infinity yields to get the Lyman series.
For most problems, we will deal with hydrogen so we can use the formula:
Figure 6.24 Numerical Problems on Rydberg Equation
Ref: Commons.wikimedia.org
Solution: We have the Rydberg equation:
This formula gives a wavelength in meters using this value for Rydberg’s constant.
More Practice problems:
- Determine the wavelength of the light absorbed when an electron in a hydrogen atom makes a transition from the orbital in the n=2 level to an orbital in the n=7 level.
- An electron in the n=6 level of the hydrogen atom relaxes to a lower energy level, emitting light of λ= 93.8 nm. Find the principal level to which the electron relaxed.
The Rydberg equation works only for hydrogen because it is an empirical formula that is based on the Bohr model of the hydrogen atom and can only apply to it and other hydrogenic species. The Bohr model assumes that hydrogen and all subsequent elements have quantized shells with each taking discrete principle energy levels. However, the Bohr model breaks apart past hydrogen due to the presence of two or more electrons in higher atomic numbered elements, which introduces electron-electron interactions that complicate things significantly and can only be explained by more in-depth and rigorous quantum mechanics.
The video below illustrates Rydberg equation and calculations.