Question

Write the mathematical forms of three postulates of Bohr’s theory of the hydrogen atom. Using them prove that, for an electron revolving in the \( n \)-th orbit,
(a) the radius of the orbit is proportional to \( n^2 \), and
(b) the total energy of the atom is proportional to \( \frac{1}{n^2} \).

Hint:

For Bohr’s model problems: - Use the quantization condition \( m v r = \frac{n h}{2\pi} \) to relate \( v \) and \( r \). - Total energy in the Bohr model is always negative, and its magnitude is proportional to \( \frac{1}{n^2} \).

Answer 1

Step 1: Enumerate the three postulates of Bohr's atomic theory.
1. Electrons orbit the nucleus in circular paths, where their angular momentum is quantized according to the equation: \[m v r = \frac{n h}{2\pi},\] with \( m \) representing the electron's mass, \( v \) its velocity, \( r \) the orbital radius, \( n \) the principal quantum number, and \( h \) Planck's constant.
2. Electrons maintain a constant energy while in these stable orbits, without radiating energy.
3. Electron transitions between orbits occur through the emission or absorption of a photon, with energy change given by: \[\Delta E = h u,\] where \( u \) is the photon's frequency. (a): Demonstrate that the radius \( r \) is proportional to \( n^2 \).
From the first postulate: \[m v r = \frac{n h}{2\pi} \quad \Rightarrow \quad v = \frac{n h}{2\pi m r}.\] Equating the centripetal force with the Coulomb force: \[\frac{m v^2}{r} = \frac{k e^2}{r^2}.\] Substituting the expression for \( v \): \[\frac{m}{r} \left( \frac{n h}{2\pi m r} \right)^2 = \frac{k e^2}{r^2} \quad \Rightarrow \quad \frac{n^2 h^2}{4\pi^2 m r^3} = \frac{k e^2}{r^2} \quad \Rightarrow \quad r = \frac{n^2 h^2}{4\pi^2 m k e^2}.\] Consequently, \( r \propto n^2 \). (b): Prove that the total energy \( E \) is proportional to \( \frac{1}{n^2} \).
The total energy is given by: \[E = \frac{1}{2} m v^2 - \frac{k e^2}{r}.\] From the force balance equation, \( \frac{m v^2}{r} = \frac{k e^2}{r^2} \), we derive: \[\frac{1}{2} m v^2 = \frac{1}{2} \frac{k e^2}{r}.\] Therefore: \[E = \frac{1}{2} \frac{k e^2}{r} - \frac{k e^2}{r} = -\frac{1}{2} \frac{k e^2}{r}.\] Substituting \( r = \frac{n^2 h^2}{4\pi^2 m k e^2} \): \[ E = -\frac{1}{2} \frac{k e^2}{\frac{n^2 h^2}{4\pi^2 m k e^2}}} = -\frac{2\pi^2 m k^2 e^4}{n^2 h^2}.\] Thus, \( E \propto -\frac{1}{n^2} \), indicating that the magnitude \( |E| \propto \frac{1}{n^2} \).