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 Postulates:
- First Postulate: Electrons in an atom orbit stably without energy loss.
- Second Postulate: Electron orbits are restricted to those where angular momentum is a multiple of \( \frac{h}{2\pi} \).
Step 1: Centripetal Force from Electrostatics.
The centripetal force experienced by the electron is expressed as:
\[\frac{m v^2}{r_n} = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r_n^2}\]Step 2: Applying Bohr's Second Postulate.
Quantization of angular momentum implies:
\[m v r_n = n \hbar\]Solving for electron velocity \( v \):
\[v = \frac{n \hbar}{m r_n}\]Step 3: Substitution into the Force Equation.
Substituting the expression for \( v \) into the electrostatic force equation yields:
\[\frac{m}{r_n} \left( \frac{n \hbar}{m r_n} \right)^2 = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r_n^2}\]After simplification:
\[n^2 \hbar^2 = \frac{m e^2}{4 \pi \epsilon_0} r_n\]Rearranging to solve for the orbital radius \( r_n \):
\[r_n = \frac{n^2 \hbar^2}{m e^2} \times 4 \pi \epsilon_0\]Final Orbital Radius Formula:
\[r_n = \frac{n^2 h^2}{4 \pi^2 m e^2 \epsilon_0}\]This equation represents the radius of the \(n\)-th electron orbit in a hydrogen atom.