Question

The species having identical radii according to the Bohr's theory are:
A. H (first orbit)
B. \( He^+ \) (first orbit)
C. \( He^+ \) (Second orbit)
D. \( Li^{2+} \) (first orbit)
E. \( Be^{3+} \) (Second orbit)
Choose the correct answer from the options given below:

• A. A and C Only
• B. A and E Only
• C. B and E Only
• D. C and D Only

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Bohr's theory provides a formula for the radius of an orbit in H-like species. The radius depends on the square of the principle quantum number \( n \) and inversely on the atomic number \( Z \).

Step 2: Key Formula or Approach:
Radius of \( n^{\text{th}} \) orbit: \( r_n = 0.529 \frac{n^2}{Z} \text{ \AA} \).
For identical radii, the ratio \( \frac{n^2}{Z} \) must be the same.

Step 3: Detailed Explanation:
Calculate \( \frac{n^2}{Z} \) for each:
A. H (first orbit): \( n=1, Z=1 \implies 1^2/1 = 1 \).
B. \( He^+ \) (first orbit): \( n=1, Z=2 \implies 1^2/2 = 0.5 \).
C. \( He^+ \) (second orbit): \( n=2, Z=2 \implies 2^2/2 = 2 \).
D. \( Li^{2+} \) (first orbit): \( n=1, Z=3 \implies 1^2/3 = 0.33 \).
E. \( Be^{3+} \) (second orbit): \( n=2, Z=4 \implies 2^2/4 = 1 \).
Comparing values: A and E have the same value (1).

Step 4: Final Answer:
Species A and E have identical radii.