Question

The ratio of areas of electron orbits for the second excited state to the first excited state in hydrogen atom, is

• A. $\dfrac{4}{9}$
• B. $\dfrac{16}{81}$
• C. $\dfrac{81}{16}$
• D. $\dfrac{9}{4}$

Hint:

In Bohr model, area of orbit varies as the fourth power of principal quantum number.

Answer 1

The Correct Option is C

Step 1: Recall radius of Bohr orbit.
In hydrogen atom, radius of $n^{\text{th}}$ orbit is given by: \[ r_n = n^2 r_1 \] Step 2: Write expression for area of orbit.
Area of orbit: \[ A_n = \pi r_n^2 \propto n^4 \] Step 3: Identify excited states.
First excited state: $n = 2$
Second excited state: $n = 3$
Step 4: Find ratio of areas.
\[ \dfrac{A_3}{A_2} = \dfrac{3^4}{2^4} = \dfrac{81}{16} \]