Question

If the ratio of the radii of nuclei \( \mathbf{^{52}X_A} \) and \( \mathbf{^{13}Al^{27}} \) is 5:3, then the number of neutrons in the nucleus \( X \) is:
Choose the correct answer from the options given below:

• A. 52
• B. 63
• C. 27
• D. 73

Hint:

The relationship between the radius of a nucleus and its mass number can be used to solve problems involving the radii of nuclei.

Answer 1

The Correct Option is D

We are given that the ratio of the radii of the nuclei \( \frac{r_{\text{X}}}{r_{\text{Al}}} = \frac{5}{3} \). The radius of the nucleus is related to its mass number \( A \) as \( r \propto A^{1/3} \). Thus, \( \frac{r_{\text{X}}}{r_{\text{Al}}} = \left( \frac{A_{\text{X}}}{A_{\text{Al}}} \right)^{1/3} \). Substituting the given ratio, we get: \[ \frac{5}{3} = \left( \frac{A_{\text{X}}}{27} \right)^{1/3} \] Cubing both sides: \[ \left( \frac{5}{3} \right)^3 = \frac{A_{\text{X}}}{27} \] \[ \frac{125}{27} = \frac{A_{\text{X}}}{27} \] Thus, \( A_{\text{X}} = 125 \). Now, the number of neutrons is \( N = A - Z \), where \( A \) is the mass number and \( Z \) is the atomic number. Since \( X \) is an unknown element, we can assume the atomic number \( Z \) is the same as that of Aluminum (which is 13) for simplicity. Therefore: \[ N = 125 - 13 = 73 \] Thus, the number of neutrons in the nucleus \( X \) is 73.