Rydberg Constant - Definition, Value, Formula, FAQs

Q: What is the rydberg constant for hydrogen and give the value of the rydberg constant in cm and rydberg constant in joules?

In nm, the Rydberg constant is 10 973 731.568 548(83) m -1 . In Joules, the Rydberg constant is 2.179 871 10 -18 J

Q: What is the value of Rydberg Constant value in nm?

The value of Rydberg constant in nm is $10973731.568548(83) \mathrm{m}^{-1}$.

Q: What is the Rydberg formula?

The Rydberg formula is $\frac{1}{\lambda}=R_{\infty}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$, where $\lambda$ is the wavelength of emitted or absorbed light, and $n_1$ and $n_2$ are integers representing energy levels.

Rydberg Constant - Definition, Value, Formula, FAQs

Edited By Vishal kumar | Updated on Dec 24, 2024 05:57 PM IST

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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What is the Rydberg Constant?
Rydberg Constant Formula
Units of Rydberg Constant and Their Values

Rydberg Constant - Definition, Value, Formula, FAQs

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 $R_{H}$ , is a wavenumber associated with each Electromagnetic spectrum. The Rydberg constant is measured in cm and ranges from $109, 678 cm$ - 1 to $109, 737 {cm}^{- 1}$ .

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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

$R_{\infty} = \frac{m_{e} e^{4}}{8 ϵ_{0}^{2} h^{3} c}$

Where,

$m_{e}$ is the rest mass of an electron
$h$ is the Planck Constant
$c$ is the velocity of light in a vacuum
$ε_{0}$ is the permittivity of free space
$e$ is the elementary charge

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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 $E =$ 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 $(E)$ ,
$E = \frac{1}{2} m v^{2} + \frac{k q_{1} q_{2}}{r^{2}}$
$q_{1} = + e, q_{2} = - e$
+e is the charge of a proton and -e is the charge of an electron

$E = \frac{1}{2} m v^{2} - \frac{k e^{2}}{r^{2}} - equation (1)$
The centripetal force is equal to the electrostatic force,
$\begin{aligned} F_{c} & = F_{e} \\ F_{c} & = m a_{c} \end{aligned}$
And $F_{e} = \frac{k | q_{1} | q_{2} ∣}{r^{2}}$

$m a_{c} = \frac{m v^{2}}{r}$
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)

$\begin{aligned} E = \frac{1}{2} \frac{k e^{2}}{r} - \frac{k e^{2}}{r^{2}} \\ E = \frac{- 1}{2} \frac{k e^{2}}{r} - equation (3) \end{aligned}$
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,

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

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

$r = \frac{n^{2} h^{2}}{m k e^{2}}$ 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,

$\begin{aligned} E_{i} - E_{f} = h f \\ h f = 2 π h \frac{c}{λ} \\ \frac{1}{λ} = \frac{E_{i} - E_{f}}{2 π h c} \\ \frac{1}{λ} = [- \frac{1}{2} \frac{m_{e} e^{2} e^{4}}{h^{2}} \times \frac{1}{n_{i}^{2}} - (\frac{- 1}{2} \frac{m_{e} k^{2} e^{4}}{h^{2}} \times \frac{1}{n_{f}^{2}})] \times \frac{1}{2 π h c} \\ \frac{1}{λ} = (\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}}) \times \frac{1}{4} \frac{m_{e} k^{2} e^{4}}{π h^{3} c} \end{aligned}$

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

$\frac{1}{λ} = R_{H} (\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}})$

Units of Rydberg Constant and Their Values

Unit	Value
Meters	$R_{H} = 1.097 \times 10^{7} m^{- 1}$
Electron volt	13.605 eV
Joules	$R_{H} = 2.178 \times 10^{- 18}$
Angstrom	$R_{H} = 1.0973 \times 10^{7} m^{- 1}$

Rydberg Constant Dimensional Formula

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

Frequently Asked Questions (FAQs)

1. What is the purpose of the Rydberg constant?

The Rydberg constant is used to compute the wavelengths in the hydrogen spectrum - the energy absorbed or emitted as photons when electrons migrate between shells in the hydrogen atom.

2. How does Rydberg's law work?

This formula, f = c/λ = (Lyman-alpha frequency)(Z 1)2, is known as Moseley's law (having added a component c to convert wavelength to frequency), and it can be used to forecast wavelengths of the K (K-alpha) X-ray spectral emission lines of chemical elements ranging from aluminium to gold.

3. Is it possible for the Rydberg constant to change?

Because every element in the periodic table has a different atomic number, the Rydberg constant value varies. As a result, the Rydberg constant varies depending on the element. The Rydberg constant is not a universal constant. Its value is determined by the atomic number.

4. How many Rydberg are there in a single Rydberg?

The energy of a photon with the Rydberg constant as its wavenumber, i.e. the ionisation energy of the hydrogen atom in a simplified Bohr model, is represented by the unit of rydberg constant energy, symbol Ry, in atomic physics.

5. How is the Rydberg constant calculated?

You will observe the visible wavelengths of light created by electric discharge in helium gas using a diffraction grating. The Rydberg constant, an important physical constant, can be determined by fitting these measured wavelengths to a curve.

6. What is the rydberg constant for hydrogen and give the value of the rydberg constant in cm and rydberg constant in joules?

In nm, the Rydberg constant is 10 973 731.568 548(83) m^-1.
In Joules, the Rydberg constant is 2.179 871 10^-18J

7. What is the Rydberg Constant?

The Rydberg Constant is a physical constant related to the wavelengths of spectral lines in hydrogen and other elements, denoted by the symbol $R_{\infty}$ .

8. What is the value of Rydberg Constant value in nm?

The value of Rydberg constant in nm is $10973731.568548 (83) m^{- 1}$ .

9. What is the Rydberg formula?

The Rydberg formula is $\frac{1}{λ} = R_{\infty} (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}})$ , where $λ$ is the wavelength of emitted or absorbed light, and $n_{1}$ and $n_{2}$ are integers representing energy levels.