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The 'Rydberg constant', named after physicist Johannes Rydberg, is a physical constant that appears in the Rydberg formula. It was discovered when measuring the spectrum of hydrogen, and building upon results from Anders Jonas Ångström and Johann Balmer.
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, and can thus be used to stringently test physical theories such as quantum electrodynamics.
Each chemical element has its own Rydberg constant. For all Hydrogen-like atoms (atoms with a single electron in the outer most orbit) the Rydberg constant $R\_M$ can be derived from the "infinity" Rydberg constant, as follows:
:
::where,
:::$R\_M$ is the Rydberg constant for a certain atom with one electron with the rest mass $m\_e$
:::$M$ is the mass of its atomic nucleus.
The "infinity" Rydberg constant is (according to 2002 CODATA results):
:
::where,
::: $hbar$ is the reduced Planck's constant,
::: $m\_e$ is the rest mass of the electron,
::: $e$ is the elementary charge,
::: $c$ is the speed of light in vacuum, and
::: $epsilon\_0$ is the permittivity of free space.
This constant is often used in atomic physics in the form of an energy:
:$h\; c\; R\_infty\; =\; 13.6056923(12)\; ,mathrm\{eV\}\; equiv\; 1\; ,mathrm\{Ry\}$

Contents

Alternate expressions

Rydberg constant for hydrogen

Derivation of Rydberg constant

See also

References

Alternate expressions

The Rydberg constant can also be expressed as the following equations.
:
and
:
where
:: $h$ is Planck's constant,
:: $c$ is the speed of light in a vacuum,
:: is the fine-structure constant,
:: $lambda\_e$ is the Compton wavelength of the electron,
:: $f\_C$ is the Compton frequency of the electron,
:: $hbar$ is the reduced Planck's constant, and
:: $omega\_C$ is the Compton angular frequency of the electron.

Rydberg constant for hydrogen

Plugging the 2002 CODATA value for the electron-proton mass ratio of $m\_e\; /\; m\_p\; =\; 5.446\; 170\; 2173(25)\; cdot\; 10^\{-4\}$, into the general formula for the Rydberg constant for any Hydrogen-like element $R\_M$, we find the Rydberg constant for hydrogen, $R\_H$.
:$R\_H\; =\; 10\; 967\; 758.341\; pm\; 0.001,mathrm\{m\}^\{-1\}$
Plugging $R\; =\; R\_H$ into the Rydberg formula for the Hydrogen-like atoms, we can obtain the emission spectrum of hydrogen,
:
ight)
Where
:$lambda\_\{mathrm\{vac\}\}$ is the wavelength of the light emitted in vacuum,
:$R\_\{mathrm\{H\}\}$ is the Rydberg constant for hydrogen,
:$n\_1$ and $n\_2$ are integers such that ,
: ''Z'' is the atomic number, which is 1 for hydrogen.

Derivation of Rydberg constant

The Rydberg constant for hydrogen can be derived using Bohr's condition, centripetal force, electric force, and electric potential energy of an electron in orbit around a proton (corresponding to the case for the hydrogen atom).

• Bohr's condition,
  : The angular momentum of the electron can only have certain discrete values:
  :::
  ::where ''n'' = 1,2,3,… (some integer) and is called the principal quantum number, ''h'' is Planck's constant, and $hbar=h/(2pi)$.
  ::$r$ is the radius of the electron's orbit
  


• Force necessary to maintain circular motion (a.k.a. centripetal force),
  :
  where
  :: $m\_e$ is the rest mass of the electron, and $v$ is the electron's velocity
  


• Electric Force of Attraction between an electron and a proton
  :
  where
  :: $e$ is the elementary charge,
  :: $epsilon\_0$ is the permittivity of free space.
  


• The expression for the total electric potential energy of an electron some distance $r$ from a proton is
  
  

To begin, we take Bohr's primary condition and solve it in terms of the electron's permitted orbital velocity $v$:

Since the electric force attracting the electron to the nucleus is the (centripetal) force driving the electron into a circular orbit around the proton, we can set $F\_mathrm\{centripetal\}\; =\; F\_mathrm\{electric\}$ to obtain

Substitute our previous expression for the electron orbital velocity $v$ in and solve for $r$ to obtain

This value of $r$ supposedly represents the only allowed values for the orbital radius of an electron in orbit around a proton assuming the Bohr condition holds for the wave nature of the electron. If we now substitute $r$ into the expression for the electric potential energy of an electron some distance from a proton and we get

Therefore a change in energy in an electron changing from one value of $n$ to another is

ight)
We simply change the units to wavelength
ightarrow Delta{E} = hc Delta left( rac{1}{lambda}
ight)
ight) and we get

ight) = rac{ m_e e^4}{8 epsilon_0^2 h^3 c} left( rac{1}{n_mathrm{initial}^2} - rac{1}{n_mathrm{final}^2}
ight)
where
:: $h$ is Planck's constant,
:: $m\_e$ is the rest mass of the electron,
:: $e$ is the elementary charge,
:: $c$ is the speed of light in vacuum, and
:: $epsilon\_0$ is the permittivity of free space.
:: $n\_mathrm\{initial\}$ and $n\_mathrm\{final\}$ being the electron shell number of the hydrogen atom
We have therefore found the Rydberg constant for Hydrogen to be

See also

★ Rydberg formula

References

Mathworld