Bohr’s Postulates
Niels Bohr proposed a model of the hydrogen atom in 1913 to explain the stability of atoms and the spectral lines of hydrogen. His theory is based on the following postulates:
1. Stationary Orbits
Electrons revolve around the nucleus only in certain permitted circular orbits called stationary orbits. While moving in these orbits, electrons do not radiate energy.
2. Quantization of Angular Momentum
The angular momentum of an electron revolving in an orbit is quantized. It is given by
\[
mvr = \frac{nh}{2\pi}
\]
where \(m\) is the mass of the electron, \(v\) is its velocity, \(r\) is the radius of the orbit, \(h\) is Planck’s constant, and \(n\) is a positive integer known as the principal quantum number.
3. Emission or Absorption of Energy
Radiation is emitted or absorbed when an electron jumps from one stationary orbit to another. The energy of the emitted or absorbed photon is equal to the difference between the energies of the two orbits.
Derivation of the Radius of the \(n^{\text{th}}\) Orbit
Consider an electron of charge \(e\) and mass \(m\) revolving around a nucleus of charge \(+Ze\) in a circular orbit of radius \(r\). The electrostatic force of attraction between the nucleus and the electron provides the necessary centripetal force.
According to Coulomb’s law, the electrostatic force between the nucleus and electron is
\[
F = \frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r^2}
\]
This force acts as the centripetal force required for circular motion:
\[
\frac{mv^2}{r} = \frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r^2}
\]
Multiplying both sides by \(r\):
\[
mv^2 = \frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r}
\]
From Bohr’s quantization condition,
\[
mvr = \frac{nh}{2\pi}
\]
Solving for \(v\):
\[
v = \frac{nh}{2\pi mr}
\]
Substituting this value of \(v\) into the previous equation:
\[
m\left(\frac{nh}{2\pi mr}\right)^2 = \frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r}
\]
Simplifying:
\[
\frac{n^2h^2}{4\pi^2mr^2} = \frac{Ze^2}{4\pi\varepsilon_0 r}
\]
Solving for \(r\):
\[
r_n = \frac{n^2h^2\varepsilon_0}{\pi mZe^2}
\]
For hydrogen atom where \(Z = 1\):
\[
r_n = \frac{n^2h^2\varepsilon_0}{\pi me^2}
\]
This can also be written as
\[
r_n = n^2 a_0
\]
where \(a_0\) is called the Bohr radius and its value is approximately \(0.529 \times 10^{-10}\) m.
Conclusion
According to Bohr’s theory, the radius of the \(n^{\text{th}}\) orbit of the hydrogen atom is proportional to \(n^2\). Thus the radius of successive orbits increases as the square of the principal quantum number.