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 \[ \Delta r = r_4 - r_3 \] \[ = 3.5 a_0 \] \[ = 3.5 \times 0.529 \times 10^{-10} \] \[ = 1.85 \times 10^{-10} \text{ m} \] Thus, the correct answer is \( 1.85 \times 10^{-10} \) m.

Answer 2

Given:
- Ion: \( He^+ \) (helium ion with one electron, similar to hydrogen but with nuclear charge \( Z = 2 \))
- We need to find the difference in radii between the fourth (\( n = 4 \)) and third (\( n = 3 \)) Bohr orbits.

Step 1: Formula for radius of the \( n^{th} \) orbit in hydrogen-like ions:
\[ r_n = \frac{n^2}{Z} \times a_0 \] where
\( r_n \) = radius of the \( n^{th} \) orbit,
\( n \) = principal quantum number,
\( Z \) = atomic number (nuclear charge),
\( a_0 = 0.529 \times 10^{-10} \, m \) is the Bohr radius.

Step 2: Calculate radius of 4th orbit (\( r_4 \)):
\[ r_4 = \frac{4^2}{2} \times a_0 = \frac{16}{2} \times 0.529 \times 10^{-10} = 8 \times 0.529 \times 10^{-10} = 4.232 \times 10^{-10} \, m \]

Step 3: Calculate radius of 3rd orbit (\( r_3 \)):
\[ r_3 = \frac{3^2}{2} \times a_0 = \frac{9}{2} \times 0.529 \times 10^{-10} = 4.5 \times 0.529 \times 10^{-10} = 2.3805 \times 10^{-10} \, m \]

Step 4: Find the difference \( \Delta r = r_4 - r_3 \):
\[ \Delta r = 4.232 \times 10^{-10} - 2.3805 \times 10^{-10} = 1.8515 \times 10^{-10} \, m \]

Rounding to two significant figures:
\[ \Delta r \approx 1.85 \times 10^{-10} \, m \]

Therefore, the difference in radii between the fourth and third Bohr orbits of \( He^+ \) is:
\[ \boxed{1.85 \times 10^{-10} \text{ m}} \]