(Chemistry Ch-2) 5. Bohr's Model for Hydrogen Atom

Postulates for Bohr’s model for hydrogen Atom

• The electron in the hydrogen atom moves around the nucleus in a circular path of fixed radius and energy. These circular paths are called orbits, stationary states, or allowed energy states.
• Energy is absorbed when electron jumps from lower orbit to a higher orbit and is emitted when electron jumps from higher orbit to a lower orbit.
• Frequency (ν) of absorbed or emitted radiation is given by,

 (Bohr’s frequency rule)

Where, E₁ and E₂ are the energies of lower and higher allowed energy states respectively

• Angular momentum (L) of an electron in a stationary state is given by,

On the Basis of These Postulates, Bohr’s Theory for Hydrogen Atom was Obtained

• The stationary states of electron are numbered as n = 1, 2, 3 … and these numbers are called principal quantum numbers.
• Radii of the stationary states (r_n) are given by,

r_n = n² a₀

Where, a₀ = 52.9 pm (called Bohr radius)

• Energy (E_n) of the stationary state is given by,

R_H is called Rydberg constant (= 2.18 × 10⁻¹⁸ J)

• When electron is free from the influence of nucleus, the energy will be zero.
• When the energy is zero, the electron has principal quantum number,

n = ∞ (It is called ionized hydrogen atom)

Thus,  E1 is the energy of the lower state (called as ground state). (Energy of different energy levels of hydrogen atom) This figure is called energy level diagram.

• Bohr’s theory can be applied to the ions, which are similar to hydrogen atom (containing only one electron). For example − He⁺, Li^2+, Be³⁺, etc.

• Energies and radii of the stationary states for hydrogen-like species is given by,

 pm = nm; Z is the atomic number

Example: Let us try to calculate the energy associated with the second orbit of Be3+ and the radius of this orbit. Energy,  For Be3+, n = 2, Z = 4

• Calculation of velocities of electrons moving in the orbits is possible by using Bohr’s theory.

• Magnitude of velocity of electron decreases with the decrease in positive charge on the nucleus and increase of principal quantum number.

Line Spectrum of Hydrogen

• Energy difference between two stationary states is given by,

(Where n_i and n_fstand for initial and final orbits respectively)

• The frequency (ν) for the absorption and emission of photon is given by,

• In terms of wave numbers ,

• When n_f > n_i, energy is absorbed (absorption spectra).
• When n_i > n_f, energy is released (emission spectra).
• Each spectral line is associated with particular transition in hydrogen atom.
• Intensity of spectral lines depends upon the number of photons of same wavelength or frequency absorbed or emitted.

Example: Let us try to calculate the frequency and wavelength of the photon emitted during transition from n = 5 to n = 3 state in hydrogen atom. Since n1 = 5 and n2 = 3, this transition gives rise to spectral lines in the infrared region of the Paschen series.  (It represents emission energy) Frequency (ν) Wavelength

Limitations of Bohr’s Model of Atom

• Unable to explain the spectrum of multi-electron atoms (For example − helium atom which contains two electrons)
• Unable to explain splitting of spectral lines in electric field (Stark effect) or in magnetic field (Zeeman effect)
• Fails to explain finer details (doublet − two closely spaced lines) of hydrogen atom spectrum
• Fails to explain the ability of atoms to form molecules by chemical bonds