The volume occupied by an atom is greater than the volume of the nucleus by factor of about

Options

A. `10^1`
B. `10^5`
C. `10^10`
D. `10^15`

Explanation

To find the factor by which the volume occupied by an atom is greater than the volume of the nucleus, we can follow these steps:

### Step 1: Understand the volumes involved
The volume of an atom (V_atom) and the volume of its nucleus (V_nucleus) can be calculated using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius.

### Step 2: Define the radii
- The radius of an atom is approximately \( R_{atom} = 10^{-10} \) meters.
- The radius of a nucleus is approximately \( R_{nucleus} = 10^{-15} \) meters.

### Step 3: Calculate the volume of the atom
Using the formula for the volume of a sphere, we can calculate the volume of the atom:
\[ V_{atom} = \frac{4}{3} \pi (R_{atom})^3 = \frac{4}{3} \pi (10^{-10})^3 \]
\[ V_{atom} = \frac{4}{3} \pi (10^{-30}) \]

### Step 4: Calculate the volume of the nucleus
Similarly, we calculate the volume of the nucleus:
\[ V_{nucleus} = \frac{4}{3} \pi (R_{nucleus})^3 = \frac{4}{3} \pi (10^{-15})^3 \]
\[ V_{nucleus} = \frac{4}{3} \pi (10^{-45}) \]

### Step 5: Find the ratio of the volumes
Now, we can find the ratio of the volume of the atom to the volume of the nucleus:
\[ \text{Ratio} = \frac{V_{atom}}{V_{nucleus}} = \frac{\frac{4}{3} \pi (10^{-30})}{\frac{4}{3} \pi (10^{-45})} \]
The \( \frac{4}{3} \pi \) terms cancel out:
\[ \text{Ratio} = \frac{10^{-30}}{10^{-45}} \]

### Step 6: Simplify the ratio
Now, simplifying the ratio gives:
\[ \text{Ratio} = 10^{-30 + 45} = 10^{15} \]

### Conclusion
Thus, the volume occupied by an atom is greater than the volume of the nucleus by a factor of \( 10^{15} \).