## Rydberg Constant in Detail

Hydrogen atoms inside a discharge lamp emit a series of lines in the visible part of the spectrum. This series was named after the Swiss teacher Johann Balmar and called the Balmer series. In 1885, Balmar found a way to describe the wavelengths of these lines by a trial and error method.

\[\frac{1}{\lambda} = R(\frac{1}{4} - \frac{1}{n})\]

Here n are integers starting from 3, 4, 5 and go till infinity. R is the rydberg constant. This article will look closely into rydberg or R constant, a unit of rydberg constant, and also discuss the rydberg constant derivation.

### What is Rydberg Constant and its Importance

The Rydberg constant holds high importance in atomic physics as it is connected to fundamental atomic constants, i.e. e, h, c, and me. The constant can be derived with a high level of accuracy. The Rydberg constant first came into existence in 1890 when Swedish physicist Johannes Rydberg analyzed several spectra. Rydberg found that many of the Balmer line series could be explained by the equation:

n = n0 - N0/(m + m’)2, where m is a natural number, m’ and n0 are quantum defects specific for a particular series. N0 is the Rydberg constant. Rydberg is used as a unit of energy

### Speed of light

It is the speed of light waves that propagate through different materials. In a vacuum, the speed of light is given as 3*108 meters per second. The speed of light is the fundamental concept of nature. In a famous equation E=mc2, the speed of light ( c ) serves as constant linking energy (E) and mass of the particles.

In terms of speed of light, the Rydberg constant (R) is given as Rhc, where h is Planck's constant and c is the speed of light. The dimensional formula of Rhc is equal to the dimensional formula for energy.

### Explanation of Rydberg Constant

When an electron changes its position from one atomic orbit to another, there is a change in the electron’s energy. If the electron shifts from a higher energy state to a lower energy state, then a photon of light gets created. If the electron goes from a low energy state to a higher one, the atom absorbs a photon of light. A distinct spectral fingerprint characterizes every element.

R∞ or RH denotes the rydberg constant, and it is a wavenumber associated with the atomic spectrum of each element. The value of rydberg constant in cm ranges from 109,678 cm−1 to 109,737 cm−1. The first value of the constant is the value of the rydberg constant for hydrogen, and the last value is for the heaviest element. The value of the rydberg constant is based on the fact that the nucleus of an atom that is emitting light is exceedingly huger than the single orbiting electron.

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### Dimensional Formula of Rydberg Constant

The Rydberg formula is expressed as a mathematical formula that denotes the wavelength of light emitted by an electron that moves between energy levels within an atom. Findings of Rydberg, along with Bohr’s atomic model, give in below formula:

\[\frac{1}{\lambda} = RZ^{2}(\frac{1}{n_1^2} - \frac{1}{n_2^2})\]

Here Z is the atomic number of the atom and \[n_1\] and \[n_2\] are integers where n2 > n1. Later it was established that n1 and n2 are related to the energy quantum number (principal quantum number). This formula works quite well for hydrogen atoms with only one electron. By changing the values of n1 and n2, we can get other spectral series as shown in the table below:

### Values of Rydberg's Constant

The accepted values of the Rydberg constant, R∞, as in 1998 are:

Rydberg Constant in m- 10 973 731.568 548(83) m-1.

Rydberg Constant in Joules - 2.179 871 90(17).10-18 J.

Rydberg Constant in Electron Volt - 13.605662285137 eV.

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

Rydberg Constant in ergs - 2.179872E-11 ergs.

Rydberg Constant in Foot Pound-Force - 1.6077910774693E-18 ft lbf.

### Rydberg Constant Derivation

In classical format, the Rydberg constant comes from the electron radius and fine structure constant. The transverse wavelength equation gives the rydberg constant in wave format. The wave format is based on K = 10 (i.e. 10 wave centers) and is used with hydrogen calculations. If a nucleus has more than two protons, it will have different amplitude factors. Hence the Rydberg constant works only with hydrogen.

There is another derivation which is based on Bohr’s radius. Bohr stated that electrons existed in orbits with discrete energies. To know the energy released as light, we need to take the energy of these distinct states of an electron and find out their difference.

Let us consider the hydrogen luminesce spectrum where the atom has energy E, which is the sum of kinetic energy K and potential energy P equal to the energy of the rotating electron.

E = K + P

\[K = \frac{1}{2}\times m_{electron} \times v^{2}\]

\[P = \frac{(q_{1}\times q_{2})}{4\pi\epsilon_{0}r}\] , where \[\epsilon_{0}\] is the constant of permittivity of space

= \[\frac{(Ze\times (-e))}{4\pi\epsilon_{0}r}\] , where Z is the number of protons

= \[\frac{-Ze^{2}}{4\pi\epsilon_{0}r}\]

Now we apply the Virial theorem which states that the total energy of a system is equal to the negative of its kinetic energy or half of its potential energy.

E = -K = P/2

K = \[\frac{M^{2}}{2m_{e}r^{2}}\], Where M is the angular momentum of the electron = \[m_{e}\] v r (radius of the orbit of the electron).

From these we get R∞ = αe/4πa0, where a0 is Bohr’s radius = 52.92 pm and αe is the angular momentum of the electron.

### Conclusion

Johannes Robert Rydberg, a Swedish physicist, used wavenumbers instead of the wavelengths in his calculations during his studies of spectral series. While doing the calculations, he arrived at the simple expression which contains a constant term. Later, this constant term came to be known as Rydberg Constant. This constant is used for calculating the wavelengths in the hydrogen spectrum. When the binding energy is transferred to the transverse form, a photon is created or released which can be expressed in the terms of the Rydberg constant. The Rydberg constant is expressed in the terms of wavelength and frequency. The wavelength of the different spectral series can be calculated using this constant.

## FAQs on Rydberg Constant

**1. What is Meant by the Fine Structure in Spectroscopy?**

Splitting of the main spectral line of an atom into multiple components is termed as fine structure. Each spectral line represents a slightly different wavelength from the other one. When an electron emits light while moving from one energy state to another, it produces a fine structure. The split lines are created by interacting with the orbital motion of an electron and its mechanical spin motion. An electron behaves like a tiny bar magnet and interacts with the magnetic field generated by the electron’s rotation around the nucleus to create the fine structure. The level of splitting is determined by the dimensionless fine structure constant. The equation of the fine structure constant is given by:

\[\alpha = Ke^{2}/hc\], where K is Coulomb’s constant, h is Planck’s constant, and e is the charge on the electron. α = 1/137.

**2. What is a Wave Number?**

A wavenumber is a unit of frequency in nuclear, atomic, and molecular spectroscopy. The wavenumber is obtained by dividing the true frequency by the speed of light. It is equal to the number of waves in a unit distance. The frequency of a wave is denoted by v and is equal to c (speed of light)/ƛ(wavelength). For a typical spectral line in the visible zone of the spectrum, the values are as following:

Wavelength = 5.8 * \[10^{-5}\]cm.

Frequency = 5.17 * \[10^{14}\] Hertz.

**3. Calculate the shortest wavelength in the Paschen series of spectral lines.**

According to Rydberg’s formula

\[\frac{hc}{\lambda} = \frac{hc}{\lambda} = 21.76 * 10^{-19} [\frac{1}{n_1^2} - \frac{1}{n_2^2}]\]

Where,

h = Planck’s constant = 6.6 × 10-34

c = Speed of light = 3 × 108 m/s

n1 and n2 are integers

Therefore, the shortest wavelength present in the Paschen series of the spectral lines is given for n1 = 3 and n2 = ∞

\[\frac{hc}{\lambda} = 21.76 * 10^{-19} [\frac{1}{3^{2}} - \frac{1}{\infty^{2}}]\]

\[\lambda = \frac{6.6 * 10^{-34}}{21.76 * 10^{-19}}\times 9 \times 3 *10^{8}\]

\[\lambda = 8.189\times 10^{7}\] m

Hence, the value of the wavelength is λ = 818.9 nm