Question

The difference in radii between fourth and third Bohr orbits of \( He^+ \) (in m) is:

• A. \( 2.64 \times 10^{-10} \)
• B. \( 1.85 \times 10^{-12} \)
• C. \( 1.85 \times 10^{-10} \)
• D. \( 1.85 \times 10^{-9} \)

Hint:

For hydrogen-like atoms, the radius follows \( r_n = \frac{n^2 a_0}{Z} \).

Answer 1

The Correct Option is C

Step 1: Apply Bohr Radius Formula For a hydrogen-like ion: \[ r_n = \frac{n^2 a_0}{Z} \] For \( He^+ \) (\( Z = 2 \)): \[ r_4 = \frac{16 a_0}{2} = 8 a_0 \] \[ r_3 = \frac{9 a_0}{2} = 4.5 a_0 \] Step 2: Compute the Difference \[ \Delta r = r_4 - r_3 \] \[ = 3.5 a_0 \] \[ = 3.5 \times 0.529 \times 10^{-10} \] \[ = 1.85 \times 10^{-10} { m} \] Thus, the correct answer is \( 1.85 \times 10^{-10} \) m.