The ratio of volume of nuclei (assumed ot be in spherical shape) with respective mass numbers 8 and 64 is

Options

A. `0.5`
B. 2
C. `0.125`
D. `0.25`

Explanation

To find the ratio of the volumes of nuclei with mass numbers 8 and 64, we can follow these steps:

### Step 1: Understand the relationship between mass number and radius
The radius \( r \) of a nucleus is given by the formula:
\[ r = r_0 A^{1/3} \]
where \( r_0 \) is a constant (approximately \( 1.2 \times 10^{-15} \) meters) and \( A \) is the mass number.

### Step 2: Calculate the radii for mass numbers 8 and 64
For mass number \( A_1 = 8 \):
\[ r_8 = r_0 \cdot 8^{1/3} \]

For mass number \( A_2 = 64 \):
\[ r_{64} = r_0 \cdot 64^{1/3} \]

### Step 3: Find the ratio of the radii
Now, we can find the ratio of the radii:
\[ \frac{r_8}{r_{64}} = \frac{r_0 \cdot 8^{1/3}}{r_0 \cdot 64^{1/3}} = \frac{8^{1/3}}{64^{1/3}} = \left(\frac{8}{64}\right)^{1/3} \]
Calculating \( \frac{8}{64} \):
\[ \frac{8}{64} = \frac{1}{8} \]
Thus,
\[ \frac{r_8}{r_{64}} = \left(\frac{1}{8}\right)^{1/3} \]

### Step 4: Calculate the ratio of the volumes
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
To find the ratio of the volumes \( \frac{V_8}{V_{64}} \):
\[ \frac{V_8}{V_{64}} = \frac{\frac{4}{3} \pi r_8^3}{\frac{4}{3} \pi r_{64}^3} = \frac{r_8^3}{r_{64}^3} \]
Substituting the ratio of the radii:
\[ \frac{V_8}{V_{64}} = \left(\frac{r_8}{r_{64}}\right)^3 = \left(\left(\frac{1}{8}\right)^{1/3}\right)^3 = \frac{1}{8} \]

### Step 5: Final result
Thus, the ratio of the volumes of the nuclei with mass numbers 8 and 64 is:
\[ \frac{V_8}{V_{64}} = \frac{1}{8} \]
This can also be expressed as:
\[ 1 : 8 \]