## Table of Contents

# Rydberg Constant Definition

The Rydberg formula is a mathematical formula that is used to measure the wavelength of light resulting from an electron jumping between the energy states of an atom.

When an electron moves from one atomic orbital to another orbital, the energy of electron changes. When the electron moves from a high-energy orbital to a lower energy orbit, then a photon of light is produced. When the electron jumps from a low energy level to a higher energy level, then a photon of light is absorbed by the atom.

## What is Rydberg Constant?

Rydberg constant (or represented by the symbol R∞ or RΗ ), where R∞ is for heavier atoms and RH for hydrogen atoms, is a fundamental constant of atomic physics that was proposed by the Swedish physicist named Johannes Rydberg, who defined the wavelengths or frequencies of light in various sequences of associated spectral lines, most remarkably those emitted by hydrogen atoms in the Balmer series. The value of this constant is usually based on the fact that the light emitted by the nucleus of the atom is exceptionally enormous as compared with a single orbiting electron (hence the infinity symbol ∞ is used).

The constant can be expressed as follows;

**R _{∞} = m_{e}e^{4}/8ε^{2}_{0}h^{3}c**

Here;

R_{∞} represents Rydberg constant

M_{e} represents the rest mass of the electron

e represents the elementary charge

ℇ_{0} represents permittivity of free space

h represents Planck constant

c represents the speed of light in a vacuum

The value of the Rydberg constant R∞ is equal to 10,973,731.56816 per metre. When it is used in this form in the mathematical explanation of series of spectral lines, then the result we get will be equal to the number of waves per unit length or the wavenumbers. Thus, multiplication with the speed of light results in the frequencies of the spectral lines.

### Value of Rydberg Constant

The accepted values of the Rydberg constant, R∞, are as follows;

Rydberg Constant represented in nm is 10 973 731.568 548(83) m^{-1}.

Rydberg Constant represented in Joules is 2.179 871 90(17).10^{-18} J.

Rydberg Constant represented in Electron Volt is 13.605662285137 eV.

Rydberg Constant represented in Tons of TNT is 5.2100191204589E-28 tTNT.

Rydberg Constant represented in ergs is 2.179872E-11 ergs.

#### Rydberg Constant and Hydrogen Spectrum

The hydrogen atoms of the molecule dissociates when an electric discharge is passed through a gaseous molecule of hydrogen.Thus, it results in the release or emission of electromagnetic radiation produced by the energetically excited hydrogen atoms. The hydrogen emission spectrum generally contains radiation of discrete frequencies.

This series of the hydrogen emission spectrum is termed as the Balmer series. This is the only series of electromagnetic spectrum that lies in the visible region. The numeric value, 109,677 cm^{-1}, is termed as the Rydberg constant for hydrogen. The Balmer series is fundamentally the part of hydrogen emission spectrum accountable for the excitation electron that moves from higher energy shell to the second shell. Likewise, other transitions also have their own series names.

Some of them are listed below,

• The transition of electron from higher shell to first shell is termed as Lyman series

• The transition of electron from other higher shell to second shell is termed Balmer series

• The transition of electron from other higher shell to third shell is termed Paschen series

• The transition of electron from other higher shell to 4th shell is termed Bracket series

• The transition of electron from other higher shell to 5th shell is termed Pfund serie

##### Rydberg Constant Equation

Atomic hydrogen demonstrates emission spectrum. This spectrum enfolds several spectral series. Once the electrons in the gas are excited, then they make transitions between different energy levels. These spectral lines are thus the consequence of such electron transitions between different energy levels exhibited by Neils Bohr. The wavelengths of these spectral series is calculated by Rydberg formula.

Johannes Rydberg tried to find a mathematical relationship between spectral lines. He ultimately discovered that there is an integer relationship between the wavenumbers of consecutive lines.

His conclusions were linked with Bohr’s model of the atom to create this formula which is;

1/λ = RZ^{2}(1/n_{1}^{2} – 1/n_{2}^{2})

Where,

λ represents the wavelength of the photon (wavenumber = 1/wavelength)

R represents Rydberg’s constant (1.0973731568539(55) x 10^{7} m^{-1})

Z represents atomic number of the given atom

n_{1} and n_{2} are the integers for orbits or shell where n_{2} > n_{1}.

It was later found that n_{2} and n_{1} were associated with the principal quantum number or also termed as energy quantum number. This formula hence, works quite well for transitions between energy levels of a hydrogen atom where only one electron is involved. For atoms with multiple electrons or heavier atoms, this method starts to break down and often give inappropriate results. The reason for the inaccuracy is because the amount of screening for inner electrons or outer electron transitions often varies. The equation is too basic to compensate for these differences.

For most questions, we will deal with hydrogen so the formula which can be used is as follows:

1/λ = R_{H}(1/n_{1}^{2} – 1/n_{2}^{2})

where;

RH represents Rydberg’s constant

The value of Z for hydrogen is 1

##### Rydberg Constant

The Rydberg constant holds high position in atomic physics as it is linked with basic fundamental atomic constants, that are e, h, c, and me.

** Examples:**

1. Determine the wave length of light emitted by the electron in Hydrogen atom when electron jumps from n=4 energy level to n=2 energy level.

Using Rydberg equation;

1/ λ = R (1/2^{2}-1/n^{2})

1/ λ =1.0974×10^{7}(1/4-1/16)

= 2057625m^{-1 }

Thus,

λ = 1/2057625 = 4.86×10^{-7}m =486 nm.

2. Determine the wavelength of the electromagnetic radiation that is emitted from an electron when it moves from n = 3 to n = 1.

Using the Rydberg equation;

1/λ = R (1/n1^{2} – 1/n2^{2})

Now puting the values of n_{1} and n_{2}

n_{1} =1, n_{2} is equal to 3

R for Rydberg constant = 1.0974 x10^{7}m-1

Thus,

1/λ = (1.0974 x 10^{7}) (1/1^{2} – 1/3^{2})

The wavelength (λ) = 1.025 x 10^{-7} m

##### Rydberg Constant Citations

- New value for the Rydberg constant from the hydrogen Balmer- beta transition. Phys Rev Lett . 1987 Mar 30;58(13):1293-1295.
- Rydberg constant and fundamental atomic physics. Phys Rev A Gen Phys . 1989 Mar 15;39(6):2888-2898.
- The Rydberg constant and proton size from atomic hydrogen. Science . 2017 Oct 6;358(6359):79-85.

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