The origin of spectral lines in the hydrogen atom (Hydrogen Spectrum) can be explained on the basis of Bohr’s theory. The hydrogen atom is said to be stable when the electron present in it revolves around the nucleus in the first orbit having the principal quantum number n = 1. This orbit is called the ground state.

The electron gains energy from the surrounding and jumps into a higher orbit with principal quantum number n = 2, 3, 4, 5, ….. These higher orbits are called excited states.  When electrons start revolving in the excited state the atom becomes unstable. To acquire stability the electron jumps from the higher orbit to lower orbit by the emission of the energy of value hν. Where ν is the frequency of radiation energy or radiation photon. This radiation is emitted in the form of spectral lines.

The Expression for the Wavelength of a line in the Hydrogen Spectrum:

Let E_n and E_p be the energies of an electron in the n^th and p^th orbits respectively (n > p) So when an electron takes a jump from the n^th orbit to the p^th orbit energy will be radiated in the form of a photon or quantum such that

E_n –  E_p = hν  ………… (1)

where ν is the frequency of radiation, h = Planck’s constant

This formula is called Bohr’s formula of spectral lines.

The wavelength λ obtained is characteristic wavelength due to jumping of the electron from n^th orbit to p^th orbit. We get different series of spectral lines due to the transition of the electron from different outer orbits to fixed inner orbit.

Energy Level Diagram for Hydrogen Atom:

Energy level diagrams indicate us the different series of lines observed in a spectrum of the hydrogen atom. The horizontal lines of the diagram indicate different energy levels. The vertical lines indicate the transition of an electron from a higher energy level to a lower energy level.

It is very important that as indicated in the diagram each transition corresponds to a definite characteristic wavelength. Thus different transitions give different series of lines. Different Series obtained are a) Lyman series, b)  Balmer series, c)  Paschen series, d)  Brackett series,  e)  Pfund series and f) Henry series

Different Series in Hydrogen Spectrum:

Lyman Series:

If the transition of electron takes place from any higher orbit (principal quantum number =  2, 3, 4,…….) to the first orbit (principal quantum number  = 1). We get a Lyman series of the hydrogen atom. It is obtained in the ultraviolet region.

This formula gives a wavelength of lines in the Lyman series of the hydrogen spectrum. Different lines of Lyman series are

• α line of Lyman series  p = 1 and n = 2
• α line of Lyman series  p = 1 and n = 3
• γ line of Lyman series  p = 1 and n = 4
• the longest line of Lyman series  p = 1 and n = 2
• the shortest line of Lyman series p = 1 and n = ∞

Balmer Series:

If the transition of electron takes place from any higher orbit (principal quantum number = 3, 4, 5, …) to the second orbit (principal quantum number = 2). We get Balmer series of the hydrogen atom. It is obtained in the visible region.

This formula gives a wavelength of lines in the Balmer series of the hydrogen spectrum. Different lines of Balmer series area l

• α line of Balmer series  p = 2 and n = 3
• β line of Balmer series  p = 2 and n = 4
• γ line of Balmer series  p = 2 and n = 5
• the longest line of Balmer series  p = 2 and n = 3
• the shortest line of Balmer series p = 2 and n = ∞

Paschen Series:

If the transition of electron takes place from any higher orbit (principal quantum number = 4, 5, 6, …) to the third orbit (principal quantum number = 3). We get Paschen series of the hydrogen atom. It is obtained in the infrared region.

This formula gives a wavelength of lines in the Paschen series of the hydrogen spectrum.

Brackett Series:

If the transition of electron takes place from any higher orbit (principal quantum number = 5, 6, 7, …) to the fourth orbit (principal quantum number = 4). We get the Brackett series of the hydrogen atom. It is obtained in the far-infrared region.

This formula gives a wavelength of lines in Brackett series of the hydrogen spectrum

Pfund Series:

If the transition of electron takes place from any higher orbit (principal quantum number = 6,7, 8, …….) to the fifth orbit (principal quantum number = 5). We get Pfund series of the hydrogen atom. It is obtained in the far-infrared region.

This formula gives a wavelength of lines in the Pfund series of the hydrogen spectrum

Notes:  Shortest wavelength is called series limit

Continuous or Characteristic X-rays:

Characteristic x-rays are emitted from heavy elements when their electrons make transitions between the lower atomic energy levels. The characteristic x-ray emission which is shown as two sharp peaks in the illustration at left occurs when vacancies are produced in the n=1 or K-shell of the atom and electrons drop down from above to fill the gap.

The x-rays produced by transitions from the n=2 to n=1 levels are called K-alpha x-rays, and those for the n=3 to n = 1 transition are called K-beta x-rays. For a particular material, the wavelength has definite value. Hence these x rays are called continuous or characteristic X-rays. The values of energy are different for different materials.

The frequencies of the characteristic x-rays can be predicted from the Bohr model. Moseley measured the frequencies of the characteristic x-rays from a large fraction of the elements of the periodic table and produced a plot of them which is now called a “Moseley plot”.

Characteristic x-rays are used for the investigation of crystal structure by x-ray diffraction. Crystal lattice dimensions may be determined with the use of Bragg’s law in a Bragg spectrometer.

Science > Physics > Atoms, Molecule, and Nuclei > Hydrogen Spectrum

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In the last article, we have studied Rutherford’s model of an atom, its merits, and demerits. In this article, we shall study Bohr’s Model of an atom, its merits, and demerits.

To overcome the limitations of Rutherford’s model of an atom, Neil Bohr put forward his theory of atom using Planck’s quantum theory. Bohr’s theory is applicable to the hydrogen atom.

Bohr’s Atomic Theory of Hydrogen Atom:

Postulate I (Postulate of Circular Orbit):

In a hydrogen atom, the electron revolves around a circular orbit around the nucleus.  The electrostatic force of attraction between the positively charged nucleus and the negativity charged electron provide necessary centripetal force for circular motion.

The Expression For the First Postulate:

Let m be the mass of an electron revolving around the nucleus in a circular orbit of radius r with a constant speed v round the nucleus.  Let – e and + e be the charges on the electron and the nucleus respectively.

By the first postulate,

Centripetal force =  Electrostatic force

Where ε_o is the electrical permittivity of free space

Postulate – II (Postulate of Selected Orbit):

The electron can revolve only in a certain selected orbit in which the angular momentum of the electron is equal to an integral multiple of nh/2π, where h is the Planck’s constant.  These orbits are called stationary or permissible orbits.  The electron does not radiate energy while revolving in these orbits.

The Expression for the Second Postulate:

Let m be the mass of an electron revolving around the nucleus in a circular orbit of radius r with a constant speed v round the nucleus.

Where, n   =    1, 2, 3………..

n = Principal quantum number

h = Planck’s constant

The integer n is called the principal quantum number and it denotes the number of the orbit.

Postulate –III (Postulate of The Origin of Spectral Lines):

When an electron takes a jump from a higher energy orbit to a lower energy orbit, energy is radiated in the form of a quantum or photon of energy hν, which is equal to the difference of energies of the electron in the two orbits.

Expression for the Third Postulate:

Let E_n and E_p be the energies of an electron in the n^th and p^th orbits respectively (n > p) So when an electron takes a jump from the n^th orbit to the p^th orbit energy will be radiated in the form of a photon or quantum such that

E_n –  E_p = hν

Where ν is the frequency of radiation.

Expression for Radius of Bohr’s Orbit:

Let m be the mass of an electron revolving in a circular orbit of radius r with a constant speed  v around the nucleus.  Let – e and + e be the charges on the electron and the nucleus, respectively.

By the first postulate,

Centripetal force =  Electrostatic force

Where ε_o is the electrical permittivity of free space

From the second postulate of Bohr’s theory

From equation (1) and (2)

This is the required expression for the radius of Bohr’s orbit. Since ε_o, h, π, m, e are constant

∴  r    ∝   n²

Thus the radius of the Bohr’s orbit of an atom is directly proportional to the square of the principal quantum number.

The Expression for Velocity of Electron in Bohr’s orbit:

From the second postulate of Bohr’s theory

Where, n = 1, 2, 3………..

n = Principal quantum number

h = Planck’s constant

This is the required expression for the velocity of the electron in Bohr’s orbit of an atom. Since ε_o, h, π, e are constant

∴   v  ∝ 1 / n

Thus the velocity of the electron in Bohr’s orbit of an atom is inversely proportional to the principal quantum number.

The Expression for Angular Velocity of Electron in Bohr’s Orbit:

Now, ε_o, m, h, π, e are constant

∴ ω  ∝ 1 / n³

Thus the angular velocity of the electron in Bohr’s orbit of an atom is inversely proportional to the cube of the principal quantum number.

Notes:

• The frequency of electron in Bohr’s orbit of an atom is inversely proportional to the cube of the principal quantum number.
• The time period of the electron in Bohr’s orbit of an atom is directly proportional to the cube of the principal quantum number.
• The centripetal acceleration of electron in Bohr’s orbit of an atom is inversely proportional to the fourth power of the principal quantum number

The Expression for Energy of Electron in Bohr’s Orbit:

Let m be the mass of an electron revolving in a circular orbit of radius r with a constant speed v around the nucleus.  Let – e  and + e be the charges on the electron and the nucleus, respectively.

The  potential energy of electron having charge,  – e is given by

The total energy of the electron is given by

The total energy of electron  =  Kinetic energy of electron + Potential energy of the electron

This is the required expression for the energy of the electron in Bohr’s orbit of an atom. Since ε_o, m, h, π, e are constant

∴   E ∝  1 / n²

Thus the energy of an electron in Bohr’s orbit of an atom is inversely proportional to the square of the principal quantum number.

The negative sign indicates that the electron is bound to the nucleus by attractive force and to remove the electron from the atom energy must be supplied to the electron to overcome the attractive force. This energy is called the binding energy of the electron.

Merits of Bohr’s Model of Hydrogen Atom:

• This theory explains the spectrum of hydrogen atom completely.
• The concept of electronic configuration i.e. the distribution of electrons in different orbits was introduced.
• This theory is capable of explaining the line spectra of elements in general.
• This theory can be used to find the ionization potential of an electron in an atom.
• The value of Rydberg can be calculated using this theory.

Demerits of Bohr’s Model of Hydrogen Atom

• Though spectra of a simple atom like hydrogen is explained by Bohr’s Theory, it fails to account for elements containing
  more than one electron.
• A line in an emission spectrum splits up into a number of closely spaced lines when the atomic source of radiation is placed in
  the magnetic field. This is known as the Zeeman Effect. Bohr Theory could not explain this.
• A line in an emission spectrum splits up into a number of closely spaced lines when the atomic source of radiation is placed in an electric field, which is known as the Stark effect. Bohr Theory has no explanation for it.

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