Question

Use the Bohr's first and second postulates to derive an expression for the radius of the nth orbit in a hydrogen atom.

Hint:

Bohr's model was pivotal in the development of quantum mechanics, introducing quantized orbital angular momenta, which was a significant departure from classical mechanics.

Answer 1

Bohr's First and Second Postulates:
- First Postulate: An electron in an atom revolves in certain stable orbits without the emission of radiant energy.
- Second Postulate: The electron revolves around the nucleus only in those orbits for which the angular momentum is an integral multiple of \( \frac{h}{2\pi} \).
Step 1: Electrostatic Force Provides Centripetal Force.
The centripetal force on the electron is given by:
\[ \frac{m v^2}{r_n} = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r_n^2} \] Step 2: Use of Bohr's Second Postulate.
The angular momentum of the electron is quantized: \[ m v r_n = n \hbar \] Solving for \( v \): \[ v = \frac{n \hbar}{m r_n} \] Step 3: Substitute into the Electrostatic Force Equation.
Substitute \( v \) into the equation for electrostatic force:
\[ \frac{m}{r_n} \left( \frac{n \hbar}{m r_n} \right)^2 = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r_n^2} \] Simplifying:
\[ n^2 \hbar^2 = \frac{m e^2}{4 \pi \epsilon_0} r_n \] Solving for \( r_n \):
\[ r_n = \frac{n^2 \hbar^2}{m e^2} \times 4 \pi \epsilon_0 \] Final Expression for the Radius:
\[ r_n = \frac{n^2 h^2}{4 \pi^2 m e^2 \epsilon_0} \] This is the expression for the radius of the \(n\)-th orbit in a hydrogen atom.