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.