Question

Let the binding energy per nucleon of nucleus be denoted by \( E_{bn} \) and radius of the nucleus by \( r \). If mass numbers of nuclei A and B are 64 and 125 respectively, then

• A. \( r_A < r_B \)
• B. \( r_A > r_B \)
• C. \( E_{bnA} > E_{bnB} \)
• D. \( E_{bnA} < E_{bnB} \)

Hint:

Radius of a nucleus grows as the cube root of the mass number. Binding energy per nucleon is highest near iron (A about 56) and falls slowly for heavier nuclei.

Answer 1

The Correct Option is A, C

Step 1: Understanding the Question:
Two things are compared for nuclei A and B: their radii and their binding energy per nucleon. More than one statement can be true, so each option is checked on its own.

Step 2: Key Formula or Approach:
The radius of a nucleus follows \( r = r_0 A^{1/3} \), where \( A \) is the mass number. Binding energy per nucleon rises fast up to about \( A = 56 \) (iron), then falls slowly for heavier nuclei.

Step 3: Compare the radii:
Here \( A_A = 64 \) and \( A_B = 125 \).
\[ r_A = r_0 (64)^{1/3} = 4r_0, \qquad r_B = r_0 (125)^{1/3} = 5r_0 \]
So \( r_A < r_B \). Option (A) is TRUE and option (B) is FALSE.

Step 4: Compare binding energy per nucleon:
A nucleus with \( A = 64 \) lies close to the peak of the curve, at about 8.7 MeV per nucleon. A nucleus with \( A = 125 \) lies on the falling side, at about 8.5 MeV per nucleon. So \( E_{bnA} > E_{bnB} \). Option (C) is TRUE and option (D) is FALSE.

Final Answer:
Statements (A) and (C) are correct. \[ \boxed{\text{(A) and (C)}} \]