Bohr's Atom Orbit Radius
Question: In Bohr’s model of the atom, the radius of the electron’s third stationary orbit is \( R \). What is the radius of the fifth orbit?
- (1) \( \frac{25}{9} R \)
- (2) \( 9R \)
- (3) \( \frac{25}{16} R \)
- (4) \( 25R \)
Select the Correct Option
Step‑by‑Step Explanation
Step 1: Bohr’s Radius Formula
In Bohr’s model, the radius of the \(n^{th}\) stationary orbit is given by:
\[
r_n = a_0 n^2,
\]
where \(a_0\) is the Bohr radius (a constant) and \(n\) is the principal quantum number.
Step 2: Given Information
We are told that the third orbit (\(n = 3\)) has a radius:
\[
r_3 = a_0 \times 3^2 = 9a_0 = R.
\]
Thus, \(a_0 = \frac{R}{9}\).
Step 3: Find the Radius of the Fifth Orbit
The radius of the fifth orbit is:
\[
r_5 = a_0 \times 5^2 = a_0 \times 25.
\]
Substitute \(a_0 = \frac{R}{9}\):
\[
r_5 = \frac{R}{9} \times 25 = \frac{25}{9} R.
\]
Final Answer: The radius of the fifth orbit is \( \frac{25}{9} R \), corresponding to option (1).
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