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

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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 atmoic 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 expressions[edit]

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 hydrogen[edit]

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 Constant[edit]

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

References[edit]

Mathworld