Rydberg Constant - Definition, Value, Formula, FAQs

The Rydberg constant is a physical constant that is related to atomic spectra in the field of spectroscopy. R stands for heavy atoms, and RH stands for the hydrogen. The Rydberg constant first appeared as a fitting parameter in the Rydberg formula. It was later determined by Neils Bohr using fundamental constants. Let's discuss Rydberg's Constant in detail.

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This Story also Contains

1. What is the Rydberg Constant?
2. Rydberg Constant Formula
3. Units of Rydberg Constant and Their Values

What is the Rydberg Constant?

The energy of an electron changes when it moves from one atomic orbit to another. A photon of light is formed when an electron changes from a higher energy state to a lower energy level. When an electron moves from a low to a higher energy state, the atom absorbs a photon of light. Every element has its unique spectral fingerprint.

The Rydberg constant, abbreviated R or $\mathrm{R}_H$, is a wavenumber associated with each Electromagnetic spectrum. The Rydberg constant is measured in cm and ranges from $109,678 \mathrm{~cm}$ - 1 to $109,737 \mathrm{~cm}^{-1}$.

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Rydberg Constant Formula

The Rydberg constant formula is a mathematical expression describing the wavelength of light emitted by an electron moving between energy levels within an atom. Rydberg's findings, combined with Bohr's atomic model, resulted in the following formula

$R_{\infty}=\frac{m_e e^4}{8 \epsilon_0^2 h^3 c}$

Where,

• $m_e$ is the rest mass of an electron
• $h$ is the Planck Constant
• $c$ is the velocity of light in a vacuum
• $\varepsilon_0$ is the permittivity of free space
• $e$ is the elementary charge

Rydberg Constant Derivation

The energy of an atom of hydrogen is derived using the Rydberg constant in the following context.

Think about an electron in an orbit where the coulomb's force and the centripetal force are balanced.

Total energy $\mathrm{E}=$ kinetic energy ( KE$)+$ potential energy (PE)
$K \cdot E=\frac{1}{2} m v^2$
The electric potential energy
$P. E=k \frac{q_1 q_2}{r^2}$
Total energy $(\mathrm{E})$,
$E=\frac{1}{2} m v^2+\frac{k q_1 q_2}{r^2}$
$q_1=+e, q_2=-e$
+e is the charge of a proton and -e is the charge of an electron

$
E=\frac{1}{2} m v^2-\frac{k e^2}{r^2}-\text { equation (1) }
$
The centripetal force is equal to the electrostatic force,
$
\begin{aligned}
F_c & =F_e \\
F_c & =m a_c
\end{aligned}
$
And $F_e=\frac{k\left|q_1\right| q_2 \mid}{r^2}$

$
m a_c=\frac{m v^2}{r}
$
And $k \frac{q_1 q_2}{r^2}=\frac{k e^2}{r^2}$

$
m v^2=\frac{k e^2}{r}
$

equation (2)

Substituting equation (2) in equation (1)

$
\begin{aligned}
& E=\frac{1}{2} \frac{k e^2}{r}-\frac{k e^2}{r^2} \\
& E=\frac{-1}{2} \frac{k e^2}{r}-\text { equation (3) }
\end{aligned}
$
Is the equation for the total energy equation of an electron.
Since we are working with Bohr's radius, quantized angular momentum shall be given by,

$
\begin{aligned}
& \mathrm{L}=\mathrm{mvr} \\
& \mathrm{mvr}=\mathrm{nh},
\end{aligned}
$
Where $\mathrm{n}=$ energy level of an atomic spectrum and $\mathrm{h}=$ planck's constant.
$
v=\frac{n h}{m r} \text {-equation (4) }
$
Substituting equation (3) in equation (2),

$
\begin{aligned}
& m\left(\frac{n h}{m r}\right)^2=\frac{k e^2}{r} \\
& \left(\frac{m n^2 h^2}{m^2 r^2}\right)=\frac{k e^2}{r} \\
& \left(\frac{n^2 h^2}{m r}\right)=k e^2
\end{aligned}
$

$r=\frac{n^2 h^2}{m k e^2}$ This equation is known as the Bohr radius for a hydrogen atom. - equation (5)
Substituting equation (5) in equation (3)

$
E=-\frac{1}{2} \frac{k e^2}{\frac{n^2 h^2}{m k_2^2}}
$
Hence the energy level of an nth orbit,

$
E_n=-\frac{1}{2} \frac{m_c e^2 e^4}{h^2} \frac{1}{n^2}
$

$m_e$ is the mass of an electron.

The difference between the energy levels for an electron from its initial energy and final energy in atomic spectra,

$
\begin{aligned}
& E_i-E_f=h f \\
& h f=2 \pi h \frac{c}{\lambda} \\
& \frac{1}{\lambda}=\frac{E_i-E_f}{2 \pi h c} \\
& \frac{1}{\lambda}=\left[-\frac{1}{2} \frac{m_e e^2 e^4}{h^2} \times \frac{1}{n_i^2}-\left(\frac{-1}{2} \frac{m_e k^2 e^4}{h^2} \times \frac{1}{n_f^2}\right)\right] \times \frac{1}{2 \pi h c} \\
& \frac{1}{\lambda}=\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right) \times \frac{1}{4} \frac{m_e k^2 e^4}{\pi h^3 c}
\end{aligned}
$

where,
$R_H=\frac{1}{4} \frac{m_e k^2 e^4}{\pi h^3 c}$ is the value of Rydberg's constant.
The equation for wavelength becomes,

$
\frac{1}{\lambda}=R_H\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)
$

Units of Rydberg Constant and Their Values

Rydberg Constant Dimensional Formula

The dimensional formula for the Rydberg constant is $\left[M^0 L^{-1} T^0\right]$.

Related Topic,

The Rydberg constant, represented with R or R H $Ryd$ as shown in this Rydberg Relation is a basic constant belonging to spectroscopy particularly concerning line spectra of element provisions. It assists in finding out the wavenumber for the photon of an atom which is gained or lost by an electron making a step from one energy level to another inside the atom. Rydberg constant can also be formed by the use of the mass of the electron, Planck’s constant, speed of light, vacuum permittivity, and elementary charge.