The ratio of mass densities of nuclei of `.^40Ca` and `.^16O` is close to:

Options

A. 1
B. `0.1`
C. `5`
D. `2`

Answer

To find the ratio of mass densities of the nuclei of \(^{40}\text{Ca}\) and \(^{16}\text{O}\), we can follow these steps:

### Step 1: Understand the formula for density
The density (\(\rho\)) of a nucleus is given by the formula:
\[ \rho = \frac{A}{V} \]
where \(A\) is the mass number (which can be approximated as the mass of the nucleus) and \(V\) is the volume of the nucleus.

### Step 2: Calculate the volume of the nucleus
The volume (\(V\)) of a spherical nucleus can be expressed as:
\[ V = \frac{4}{3} \pi r^3 \]
where \(r\) is the radius of the nucleus.

### Step 3: Relate radius to mass number
The radius of the nucleus can be approximated using the formula:
\[ r = r_0 A^{1/3} \]
where \(r_0\) is a constant (approximately \(1.2 \, \text{fm}\)) and \(A\) is the mass number.

### Step 4: Substitute the radius into the volume formula
Substituting the expression for \(r\) into the volume formula gives:
\[ V = \frac{4}{3} \pi (r_0 A^{1/3})^3 = \frac{4}{3} \pi r_0^3 A \]

### Step 5: Substitute volume back into the density formula
Now substituting \(V\) back into the density formula, we get:
\[ \rho = \frac{A}{\frac{4}{3} \pi r_0^3 A} = \frac{3}{4 \pi r_0^3} \]
Notice that the mass number \(A\) cancels out.

### Step 6: Calculate the density for \(^{40}\text{Ca}\) and \(^{16}\text{O}\)
For both \(^{40}\text{Ca}\) and \(^{16}\text{O}\), the density is the same:
\[ \rho_{Ca} = \frac{3}{4 \pi r_0^3} \]
\[ \rho_{O} = \frac{3}{4 \pi r_0^3} \]

### Step 7: Find the ratio of densities
Now, we can find the ratio of the densities:
\[ \frac{\rho_{Ca}}{\rho_{O}} = \frac{\frac{3}{4 \pi r_0^3}}{\frac{3}{4 \pi r_0^3}} = 1 \]

### Conclusion
The ratio of the mass densities of the nuclei of \(^{40}\text{Ca}\) and \(^{16}\text{O}\) is close to 1.