The fraction of volume occupied by the nucleus with respect to the total volume of an atom is.

Options

A. `10^-15`
B. `10^-5`
C. `10^-30`
D. `10^-10`

Explanation

To find the fraction of volume occupied by the nucleus with respect to the total volume of an atom, we can follow these steps:

### Step 1: Understand the Volume Formula
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 2: Define the Radii
Let:
- \( r_1 \) be the radius of the nucleus (approximately \( 10^{-13} \) m)
- \( r_2 \) be the radius of the atom (approximately \( 10^{-10} \) m)

### Step 3: Calculate the Volume of the Nucleus
Using the volume formula for the nucleus:
\[ V_{\text{nucleus}} = \frac{4}{3} \pi r_1^3 \]

### Step 4: Calculate the Volume of the Atom
Using the volume formula for the atom:
\[ V_{\text{atom}} = \frac{4}{3} \pi r_2^3 \]

### Step 5: Find the Fraction of Volume Occupied by the Nucleus
The fraction of volume occupied by the nucleus with respect to the total volume of the atom is given by:
\[ \text{Fraction} = \frac{V_{\text{nucleus}}}{V_{\text{atom}}} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} \]
The \( \frac{4}{3} \pi \) cancels out:
\[ \text{Fraction} = \frac{r_1^3}{r_2^3} \]

### Step 6: Substitute the Values of \( r_1 \) and \( r_2 \)
Substituting the values:
\[ \text{Fraction} = \left(\frac{10^{-13}}{10^{-10}}\right)^3 = \left(10^{-3}\right)^3 = 10^{-9} \]

### Step 7: Conclusion
Thus, the fraction of volume occupied by the nucleus with respect to the total volume of the atom is:
\[ 10^{-9} \]