Question

The surface areas of two nuclei are in the ratio \( 9:25 \). The mass numbers of the nuclei are in the ratio:

• A. \( 27:125 \)
• B. \( 9:25 \)
• C. \( 3:5 \)
• D. \( 1:1 \)

Hint:

For nuclei, the mass number ratio follows the cube of the radius ratio.

Answer 1

The Correct Option is A

Step 1: Relationship Between Surface Area and Mass Number The surface area of a nucleus is proportional to the square of its radius: \[ A \propto R^2 \] Since nuclear radius is related to mass number: \[ R \propto A^{1/3} \] Step 2: Compute the Mass Number Ratio \[ \frac{R_1}{R_2} = \sqrt{\frac{9}{25}} = \frac{3}{5} \] \[ \left(\frac{M_1}{M_2}\right)^{1/3} = \frac{3}{5} \] Cubing both sides: \[ \frac{M_1}{M_2} = \left(\frac{3}{5}\right)^3 = \frac{27}{125} \] Thus, the correct answer is \( 27:125 \).

Answer 2

Given:
- The surface areas of two nuclei are in the ratio \( 9 : 25 \).

We need to find the ratio of their mass numbers.

Step 1: Relation between surface area and radius of a nucleus:
The surface area \( S \) of a spherical nucleus is:
\[ S = 4 \pi r^2 \] where \( r \) is the radius of the nucleus.

Step 2: Using the ratio of surface areas:
\[ \frac{S_1}{S_2} = \frac{9}{25} = \frac{r_1^2}{r_2^2} \] Taking square root on both sides:
\[ \frac{r_1}{r_2} = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

Step 3: Relation between radius and mass number \( A \):
The radius of a nucleus is related to its mass number by:
\[ r = r_0 A^{1/3} \] where \( r_0 \) is a constant.

Step 4: Using the ratio of radii:
\[ \frac{r_1}{r_2} = \left( \frac{A_1}{A_2} \right)^{1/3} = \frac{3}{5} \] Cubing both sides:
\[ \frac{A_1}{A_2} = \left( \frac{3}{5} \right)^3 = \frac{27}{125} \]

Therefore, the ratio of the mass numbers of the two nuclei is:
\[ \boxed{27 : 125} \]