The radius of a nucleus (\(r\)) is often estimated using the formula:
\[ r = r_0 A^{1/3} \]
Where:
- \( r_0 \) is a constant approximately equal to 1.2 to 1.3 femtometers (fm)
- \( A \) is the mass number of the nucleus
For uranium (U), the mass number is approximately 238, and for hydrogen (H), the mass number is 1.
1. Comparing the radii of the nuclei:
The ratio of the radii of the uranium nucleus (\(r_{U}\)) to the proton nucleus (\(r_{H}\)) can be calculated as:
\[ \frac{r_{U}}{r_{H}} = \frac{r_0 \times (238)^{1/3}}{r_0 \times (1)^{1/3}} = \frac{238^{1/3}}{1} \]
\[ \frac{r_{U}}{r_{H}} = 6.51 \]
This means the radius of the uranium nucleus is approximately 6.51 times greater than that of the hydrogen nucleus.
2. Comparing the density of the atomic nucleus to granite:
The density of an atomic nucleus is extremely high, often expressed in g/cm³ or kg/m³. Typically, the density of a nucleus is around \(2.3 \times 10^{17}\) kg/m³.
The ratio of the density of the atomic nucleus to granite can be calculated as:
{Density of nucleus}}*{Density of granite}} = \frac{2.3 \times 10^{17} \, \text{kg/m³}}{2600 \, \text{kg/m³}} \]
\[ \frac{\text{Density of nucleus}{Density of granite}} = 8.8 \times 10^{13} \]
This means the atomic nucleus is approximately \(8.8 times 10^(13)times denser than a piece of granite.
