Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Rydberg constant

The Rydberg constant, named after physicist Johannes Rydberg, is a physical constant that apperas in the Rydberg formula. It was discovered when measuring the spectrum of hydrogen, and building upon results from Anders Jonas Ångström and Johann Balmer. Each chemical element has its own Rydberg constant, which can be derived from the "infinity" Rydberg constant.

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 confirms the proportions of the values of the other physical constants that define it.

For a series of discrete spectral lines emitted by atomic hydrogen,

.

The "infinity" Rydberg constant is (according to 2002 CODATA results):

where
is the reduced Planck's constant,
is the rest mass of the electron,
is the elementary charge,
is the speed of light in vacuum, and
is the permittivity of free space.

This constant is often used in atomic physics in the form of an energy:

The "infinity" constant appears in the formula:

where
is the Rydberg constant for a certain atom with one electron with the rest mass
is the mass of its atomic nucleus.

Alternate expressionsEdit

The Rydberg constant can also be expressed as the following equations.

 

and

 

where

  is Planck's constant,
  is the speed of light in a vacuum,
  is the fine-structure constant,
  is the Compton wavelength of the electron,
  is the Compton frequency of the electron,
  is the reduced Planck's constant, and
  is the Compton angular frequency of the electron.

Rydberg Constant for hydrogenEdit

Plugging in the rest mass of an electron and an atomic mass   of 1 for hydrogen, we find the Rydberg constant for hydrogen,  .

 

Plugging this constant into the Rydberg formula, we can obtain the emission spectrum of hydrogen.

Derivation of Rydberg ConstantEdit

The Rydberg Constant can be derived using Bohr's condition, centripetal acceleration, and Potential Energy of the electron to the nucleus.

Bohr's condition,

 

where

  is some integer
  is the radius of some atom

Centripetal Acceleration,

 

where

  is the rest mass of the electron,
  is the electron's velocity

PE of Attraction between Electron and Nucleus

 

where

  is the elementary charge,
  is the permittivity of free space.

Firstly we substitute   into Bohr's condition, then solve for   We obtain  

We equate centripetal acceleration and attraction between nucleus to obtain  

Substitute   in and solve for   we obtain  

We know that  

Substitute   into this energy equation and we get  

Therefore a chance in energy would be  

We simply change the units to wave number ( ) and we get   where

  is Planck's constant,
  is the rest mass of the electron,
  is the elementary charge,
  is the speed of light in vacuum, and
  is the permittivity of free space.
  and   being the excited states of the atom

We have therefore find the Rydberg constant for Hydrogen to be

 

ReferencesEdit

Mathworld