Bohr's Theory

Bohr's Theory, also known as Bohr's model of the hydrogen atom, was proposed by Niels Bohr in 1913. This theory was a modification of the earlier Rutherford model and was a significant step in the development of quantum mechanics. Bohr's model provides an explanation for the stability of electron orbits in atoms and the emission spectra of hydrogen.

Postulates of Bohr's Theory

Bohr's model is based on a set of postulates that describe the behavior of electrons in an atom:

Electrons orbit the nucleus in certain stable orbits without emitting radiation. These orbits are called stationary states.
Only certain orbits are allowed. These orbits are quantized, meaning that the angular momentum of an electron in orbit is an integral multiple of the reduced Planck constant divided by 2π.
Electrons can move from one orbit to another. When an electron jumps from a higher energy orbit to a lower energy orbit, it emits a photon with energy equal to the difference in energy between the two orbits.

Mathematical Formulation

Bohr's theory introduces the concept of quantization of the electron's angular momentum. The angular momentum (L) of an electron in a quantized orbit is given by:

[ L = n\frac{h}{2\pi} ]

where:

( L ) is the angular momentum,
( n ) is the principal quantum number (an integer),
( h ) is Planck's constant.

The energy of an electron in a particular orbit is given by:

[ E_n = -\frac{Z^2 R_H}{n^2} ]

where:

( E_n ) is the energy of the nth orbit,
( Z ) is the atomic number (for hydrogen, ( Z = 1 )),
( R_H ) is the Rydberg constant for hydrogen.

The radius of the nth orbit can be calculated using:

[ r_n = n^2 \frac{h^2}{4\pi^2 k m_e Z e^2} ]

where:

( r_n ) is the radius of the nth orbit,
( k ) is Coulomb's constant,
( m_e ) is the electron mass,
( e ) is the elementary charge.

Differences Between Bohr's Theory and Classical Mechanics

Aspect	Bohr's Theory	Classical Mechanics
Electron Orbits	Electrons occupy fixed orbits without radiating energy.	Electrons would spiral into the nucleus due to radiation of energy.
Energy Levels	Energy levels are quantized.	Energy levels are continuous.
Radiation Emission	Occurs only when an electron jumps between orbits.	Continuous radiation as electrons move.
Angular Momentum	Quantized in units of ( \hbar = \frac{h}{2\pi} ).	No quantization of angular momentum.

Examples and Applications

Example 1: Energy of an Electron in Hydrogen

Calculate the energy of an electron in the second orbit (n=2) of a hydrogen atom.

[ E_2 = -\frac{Z^2 R_H}{2^2} = -\frac{1^2 \times 2.18 \times 10^{-18} \text{J}}{4} = -5.45 \times 10^{-19} \text{J} ]

Example 2: Wavelength of Emitted Photon

Calculate the wavelength of a photon emitted when an electron transitions from the third orbit (n=3) to the second orbit (n=2) in a hydrogen atom.

First, find the energy difference:

[ \Delta E = E_2 - E_3 = -\frac{R_H}{2^2} + \frac{R_H}{3^2} ]

[ \Delta E = R_H \left( \frac{1}{4} - \frac{1}{9} \right) ]

[ \Delta E = R_H \left( \frac{5}{36} \right) ]

Then, use the energy to find the wavelength:

[ \lambda = \frac{hc}{\Delta E} ]

[ \lambda = \frac{6.626 \times 10^{-34} \text{J s} \times 3 \times 10^8 \text{m/s}}{2.18 \times 10^{-18} \text{J} \times \frac{5}{36}} ]

[ \lambda \approx 6.56 \times 10^{-7} \text{m} ]

Limitations of Bohr's Theory

Bohr's theory was groundbreaking, but it had its limitations:

It could only accurately describe the hydrogen atom and ions with one electron (like He+, Li2+, etc.).
It did not explain the fine structure of spectral lines.
It could not account for the Zeeman effect (splitting of spectral lines in a magnetic field).
It was unable to explain the intensities of spectral lines.

Despite these limitations, Bohr's model was a crucial step towards the development of quantum mechanics and laid the foundation for the quantum description of atoms.