Question

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

• A. \( r_A<r_B \)
• B. \( r_A>r_B \)
• C. \( E_{\text{b/n}}^A>E_{\text{b/n}}^B \)
• D. \( E_{\text{b/n}}^A<E_{\text{b/n}}^B \)

Hint:

The radius of a nucleus increases with the cube root of its mass number, and the binding energy per nucleon decreases with increasing mass number. Use these relationships to solve problems related to nuclear properties.

Answer 1

The Correct Option is A

We are analyzing nuclei A and B, with mass numbers 64 and 125 respectively. The goal is to determine the relationship between their binding energy per nucleon and their radii. Step 1: Radius and Mass Number Relationship The nuclear radius \( r \) is related to the mass number \( A \) by: \[ r \propto A^{1/3} \] This indicates that the radius increases with the cube root of the mass number. For nuclei A and B: - \( A_A = 64 \) - \( A_B = 125 \) The ratio of their radii is: \[ \frac{r_A}{r_B} = \left( \frac{A_A}{A_B} \right)^{1/3} = \left( \frac{64}{125} \right)^{1/3} \] Calculating the value: \[ \frac{64}{125} = 0.512 \] \[ \left( 0.512 \right)^{1/3} \approx 0.799 \] Therefore: \[ r_A<r_B \] Step 2: Binding Energy per Nucleon and Mass Number Relationship The binding energy per nucleon, \( E_{\text{b/n}} \), is related to the mass number \( A \) by: \[ E_{\text{b/n}} \propto \frac{1}{A} \] Hence, the binding energy per nucleon decreases as the mass number increases. For nuclei A and B, the ratio of their binding energies per nucleon is: \[ \frac{E_{\text{b/n}}^A}{E_{\text{b/n}}^B} = \frac{A_B}{A_A} = \frac{125}{64} \approx 1.953 \] Thus, nucleus A's binding energy per nucleon is greater than B's, as \( E_{\text{b/n}}^A>E_{\text{b/n}}^B \). Step 3: Conclusion From the analysis: - \( r_A<r_B \) - \( E_{\text{b/n}}^A>E_{\text{b/n}}^B \) Therefore, the correct answer is: \[ \boxed{(A) \, r_A<r_B} \]