Question

Prove that, in Bohr model of hydrogen atom as principal quantum number \(n\) becomes large, the energy levels get closer and closer.

Answer 1

Energy levels become closely spaced for large \(n\)

Step 1: Energy of hydrogen atom in Bohr model
The energy of the electron in the \(n^{th}\) orbit is given by: \[ E_n = -\frac{13.6}{n^2} \ \text{eV} \]

Step 2: Energy difference between successive levels
Consider spacing between adjacent levels: \[ \Delta E = E_{n+1} - E_n \] Substitute values: \[ \Delta E = -\frac{13.6}{(n+1)^2} + \frac{13.6}{n^2} \] \[ \Delta E = 13.6 \left( \frac{1}{n^2} - \frac{1}{(n+1)^2} \right) \]

Step 3: Simplify expression
\[ \frac{1}{n^2} - \frac{1}{(n+1)^2} = \frac{(n+1)^2 - n^2}{n^2 (n+1)^2} \] \[ = \frac{n^2 + 2n + 1 - n^2}{n^2 (n+1)^2} = \frac{2n + 1}{n^2 (n+1)^2} \] So: \[ \Delta E = 13.6 \cdot \frac{2n + 1}{n^2 (n+1)^2} \]

Step 4: Behaviour for large \(n\)
For very large \(n\): \[ 2n+1 \approx 2n, \quad (n+1)^2 \approx n^2 \] So: \[ \Delta E \approx 13.6 \cdot \frac{2n}{n^4} = \frac{27.2}{n^3} \]

Step 5: Final conclusion
As: \[ n \to \infty \Rightarrow \Delta E \to 0 \] Thus, energy levels become more and more closely spaced at higher quantum numbers and eventually form a continuum. Final Answer:
• Bohr’s second postulate: \(mvr = \frac{nh}{2\pi}\)
• Energy spacing decreases as \( \frac{1}{n^3} \)