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 volumes involved
We need to calculate the volume of the nucleus and the volume of the atom. The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 2: Calculate the volume of the nucleus
The radius of the nucleus is typically around \( 10^{-15} \) meters, which is equivalent to \( 10^{-13} \) centimeters.

Using the formula for the volume of a sphere:
\[ V_{\text{nucleus}} = \frac{4}{3} \pi (10^{-13} \, \text{cm})^3 \]
Calculating this gives:
\[ V_{\text{nucleus}} = \frac{4}{3} \pi (10^{-39} \, \text{cm}^3) = \frac{4\pi}{3} \times 10^{-39} \, \text{cm}^3 \]

### Step 3: Calculate the volume of the atom
The radius of an atom is typically around \( 10^{-8} \) centimeters.

Using the same volume formula:
\[ V_{\text{atom}} = \frac{4}{3} \pi (10^{-8} \, \text{cm})^3 \]
Calculating this gives:
\[ V_{\text{atom}} = \frac{4}{3} \pi (10^{-24} \, \text{cm}^3) = \frac{4\pi}{3} \times 10^{-24} \, \text{cm}^3 \]

### Step 4: Calculate the fraction of the volume
Now we can find the fraction of the volume occupied by the nucleus with respect to the total volume of the atom:
\[ \text{Fraction} = \frac{V_{\text{nucleus}}}{V_{\text{atom}}} \]
Substituting the volumes we calculated:
\[ \text{Fraction} = \frac{\frac{4\pi}{3} \times 10^{-39}}{\frac{4\pi}{3} \times 10^{-24}} = \frac{10^{-39}}{10^{-24}} = 10^{-15} \]

### Step 5: Conclusion
Thus, the fraction of volume occupied by the nucleus with respect to the total volume of an atom is:
\[ \text{Fraction} = 10^{-15} \]

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