Rydberg constant

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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. 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,

$\frac{1}{\lambda}=R\left(\frac{1}{m^2}-\frac{1}{n^2}\right) \$ .

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

$R_\infty = \frac{m_e e^4}{(4 \pi \epsilon_0)^2 \hbar^3 4 \pi c} = \frac{m_e e^4}{8 \epsilon_0^2 h^3 c} = 1.0973731568525(73) \cdot 10^7 \,\mathrm{m}^{-1}$

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} \$

The "infinity" constant appears in the formula:

$R_M = \frac{R_\infty}{1+m_e/M} \$

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.

1 Alternate expressions
2 Rydberg Constant for hydrogen
3 Derivation of Rydberg Constant
4 See also
5 References

[edit] Alternate expressions

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

$R_\infty = \frac{\alpha^2 m_e c}{4 \pi \hbar} = \frac{\alpha^2}{2 \lambda_e} \$

and

$h c R_\infty = \frac{h c \alpha^2}{2 \lambda_e} = \frac{h f_C \alpha^2}{2} = \frac{\hbar \omega_C}{2} \alpha^2 \$

where

$h \$ is Planck's constant,

$c \$ is the speed of light in a vacuum,

$\alpha \$ 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.

[edit] Rydberg Constant for hydrogen

Plugging in the 2002 CODATA value for the electron-proton mass ratio of $m_e / m_p = 5.446 170 2173(25) \cdot 10^{-4} \$ , we find the Rydberg constant for hydrogen, $R_H \$ .

$R_H = 10 967 758.341 \pm 0.001\,\mathrm{m}^{-1} \$

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

[edit] Derivation of Rydberg Constant

The Rydberg Constant 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,

$2 \pi r = n \lambda \$

where

$n \$ is some integer

$r \$ is the radius of the electron's orbit
Force necessary to maintain circular motion (a.k.a. centripetal force),

$F_{centripetal}= \frac{m_ev^2}{r} \$

where

$m_e \$ is the rest mass of the electron,

$v \$ is the electron's velocity
Electric Force of Attraction between an electron and a proton

$F_{electric}= \frac{e^2}{4 \pi \epsilon_0 r^2 } \$

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 $E_\mathrm{total} = - \frac {e^2}{ 8 \pi \epsilon_0 r} \$

Firstly we substitute $\lambda = \frac{h}{p} =\frac{h}{mv} \$ into Bohr's condition, then solve for the electron orbital velocity $v$ :

$v = \frac {n h}{2 \pi r m} \$

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 $F c e n t r i p e t a l = F e l e c t r i c$ to obtain

$\frac{m v^2}{r} = \frac{e^2}{4 \pi \epsilon_0 r^2 } \$

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

$r = \frac{n^2 h^2 \epsilon_0 }{ \pi m e^2} \$

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

$E_\mathrm{total} = \frac{- m e^4}{8 \epsilon_0^2 h^2}. \frac{1}{n^2} \$

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

$\Delta E = \frac{ m e^4}{8 \epsilon_0^2 h^2} \left( \frac{1}{n_\mathrm{initial}^2} - \frac{1}{n_\mathrm{final}^2} \right) \$

We simply change the units to wavelength $\left( \frac{1}{ \lambda} = \frac {E}{hc} \rightarrow \Delta{E} = \frac{hc}{\Delta\lambda}\right) \$ and we get