Question

The radius \( r_n \) of the \( n^{th} \) orbit in the Bohr model of the hydrogen atom varies with \( n \) as:

• A. \( r_n \propto n \)
• B. \( r_n \propto \frac{1}{n^2} \)
• C. \( r_n \propto n^2 \)
• D. \( r_n \propto \frac{1}{n} \)

Hint:

In hydrogen-like atoms, the energy levels and orbital radii are quantized, following \( r_n \propto n^2 \) and \( E_n \propto -\frac{1}{n^2} \).

Answer 1

The Correct Option is C

Bohr’s Model and Orbital Radius:

- In Bohr’s atomic model, the radius of the \( n^{th} \) orbit is determined by the formula:

\[r_n = \frac{n^2 h^2 \epsilon_0}{\pi m e^2}\]

- The equation demonstrates that \( r_n \propto n^2 \). Consequently, the orbital radius escalates with the square of the principal quantum number \( n \).

- This essential finding stems from the quantization of angular momentum within the Bohr model.

Therefore, the radius of the \( n^{th} \) orbit is directly proportional to \( n^2 \), hence \( r_n \propto n^2 \) is the accurate representation.