Assuming the nuclei to be spherical in shape, how does the surface area of a nucleus of mass number `A_1` compare with that of a nucleus of mass number `A_2`?

Explanation

To compare the surface areas of two nuclei with mass numbers \( A_1 \) and \( A_2 \), we can follow these steps:

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

### Step 2: Express the radii for both nuclei
For the nucleus with mass number \( A_1 \):
\[ r_1 = r_0 A_1^{1/3} \]
For the nucleus with mass number \( A_2 \):
\[ r_2 = r_0 A_2^{1/3} \]

### Step 3: Write the formula for surface area
The surface area \( S \) of a spherical nucleus is given by:
\[ S = 4 \pi r^2 \]
Thus, the surface areas for both nuclei can be expressed as:
\[ S_1 = 4 \pi r_1^2 = 4 \pi (r_0 A_1^{1/3})^2 \]
\[ S_2 = 4 \pi r_2^2 = 4 \pi (r_0 A_2^{1/3})^2 \]

### Step 4: Calculate the ratio of the surface areas
To compare \( S_1 \) and \( S_2 \), we find the ratio:
\[ \frac{S_1}{S_2} = \frac{4 \pi (r_0 A_1^{1/3})^2}{4 \pi (r_0 A_2^{1/3})^2} \]
The \( 4 \pi \) and \( r_0^2 \) terms will cancel out:
\[ \frac{S_1}{S_2} = \frac{A_1^{2/3}}{A_2^{2/3}} = \left(\frac{A_1}{A_2}\right)^{2/3} \]

### Step 5: Conclusion
Thus, the ratio of the surface areas of the two nuclei is:
\[ \frac{S_1}{S_2} = \left(\frac{A_1}{A_2}\right)^{2/3} \]
This means that the surface area of a nucleus is proportional to the two-thirds power of its mass number.

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