- Rydberg constant
constant, named after the Swedish physicist Johannes Rydberg, is a physical constantrelating to atomic spectra in the science of spectroscopy. Rydberg initially determind its value empirically from spectroscopy, but then later, it was found that its value could be calculated from more-fundamental constants by using quantum mechanics.
The "Rydberg constant" represents the limiting value of the highest
wavenumber(the inverse wavelength) of any photon that can be emitted from the hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing the hydrogen atom from its ground state. The spectrum of hydrogen can be expressed simply in terms of the Rydberg constant, using the Rydberg formula.
Value of the Rydberg constant
Making use of the simplifying assumption that the mass of the atomic nucleus is infinite compared to the mass of the electron, the Rydberg constant is (according to 2002
:::where,::: is the
Planck's constant,::: is the rest massof the electron,::: is the elementary charge,::: is the speed of lightin vacuum, and::: is the permittivity of free space.
This constant is often used in
atomic physicsin the form of an energy::
Two complications arise. One is that one may wish to discuss a hydrogenlike ion, i.e., an atom with atomic number Z that has only one electron. In this case, the wavenumbers and photon energies are scaled up by a factor of . The other is that the mass of the atomic nucleus is not actually infinite compared to the mass of the electron. The predicted spectrum must then be corrected by substituting the
reduced massfor the mass of the electron, resulting in:
:::where,::: is the Rydberg constant for a certain
atomwith one electronwith the rest mass::: is the mass of the atomic nucleus.
The Rydberg constant is one of the most well-determined physical constants, with a relative experimental uncertainty of less than 7 parts per trillion. The ability to measure it directly to such a high precision constrains the proportions of the values of the other physical constants that define it.
The Rydberg constant can also be expressed as the following equations.
Planck's constant,:: is the speed of lightin a vacuum,:: is the fine-structure constant,:: is the Compton wavelengthof the electron,:: is the Compton frequency of the electron,:: is the reduced Planck's constant, and:: is the Compton angular frequency of the electron.
The Derivation of Rydberg constant from quantum mechanics
Historically, the Rydberg equation was found "empirically" (experimentally), and it predated the development of quantum theory. To understand its significance in terms of the quantum theory, we can start from the equation
for the energy of the electron in the nth energy state, as can be derived either from the
Bohr modelor from a fully quantum-mechanical treatment of the hydrogen atom. Therefore a change in energy in an electron changing from one value of to another is
We simply change the units to wavelength and we get
Planck's constant,:: is the rest massof the electron,:: is the elementary charge,:: is the speed of lightin vacuum, and:: is the permittivity of free space.:: and being the electron shell number of the hydrogen atomWe have therefore found the Rydberg constant for hydrogen to be
* [http://physics.nist.gov/cuu/Constants/codata.pdf CODATA recommendations 2006]
* [http://scienceworld.wolfram.com/physics/Rydberg.html Mathworld]
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