Question:
The ratio of radii of \(3^{rd}\) and \(6^{th}\) Bohr's orbit in a hydrogen atom is
The ratio of radii of \(3^{rd}\) and \(6^{th}\) Bohr's orbit in a hydrogen atom is
Show Hint
In Bohr’s model of hydrogen atom,
\[
r_n \propto n^2
\]
Hence, ratios of orbital radii can be found directly using the squares of principal quantum numbers.
Updated On: Jun 24, 2026
- \(0.25\)
- \(0.33\)
- \(4\)
- \(3\)
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The Correct Option is A
Solution and Explanation
Step 1: Recall the formula for radius of Bohr orbit.
According to Bohr’s model, the radius of the \(n^{th}\) orbit is given by \[ r_n \propto n^2 \] For hydrogen atom, \[ r_n=n^2r_1 \] where \(r_1\) is the radius of the first Bohr orbit.
Step 2: Write the radii of the \(3^{rd}\) and \(6^{th}\) orbits.
For the \(3^{rd}\) orbit, \[ r_3=3^2r_1=9r_1 \] For the \(6^{th}\) orbit, \[ r_6=6^2r_1=36r_1 \]
Step 3: Find the required ratio.
The ratio of radii is \[ \frac{r_3}{r_6}=\frac{9r_1}{36r_1} \] \[ \frac{r_3}{r_6}=\frac{1}{4} \] \[ \frac{r_3}{r_6}=0.25 \]
Step 4: Final conclusion.
Therefore, the required ratio is \[ \boxed{0.25} \]
According to Bohr’s model, the radius of the \(n^{th}\) orbit is given by \[ r_n \propto n^2 \] For hydrogen atom, \[ r_n=n^2r_1 \] where \(r_1\) is the radius of the first Bohr orbit.
Step 2: Write the radii of the \(3^{rd}\) and \(6^{th}\) orbits.
For the \(3^{rd}\) orbit, \[ r_3=3^2r_1=9r_1 \] For the \(6^{th}\) orbit, \[ r_6=6^2r_1=36r_1 \]
Step 3: Find the required ratio.
The ratio of radii is \[ \frac{r_3}{r_6}=\frac{9r_1}{36r_1} \] \[ \frac{r_3}{r_6}=\frac{1}{4} \] \[ \frac{r_3}{r_6}=0.25 \]
Step 4: Final conclusion.
Therefore, the required ratio is \[ \boxed{0.25} \]
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