#### Answer

\[\underline{6.8\times {{10}^{-15}}}\]

#### Work Step by Step

Radius of hydrogen atom is \[52.9\text{ pm}\].
\[\text{1 pm}=\text{1}{{\text{0}}^{-10}}\text{ cm}\]
Thus, radius of hydrogen atom in centimeters is written as follows:
\[r=52.9\times {{10}^{-10}}\text{ cm}\]
Now, volume of an atom (sphere) is as follows:
\[V=\frac{4}{3}\pi {{r}^{3}}\]
Thus,
\[V=\frac{4}{3}\pi {{\left( 52.9\times {{10}^{-10}}\text{ cm} \right)}^{3}}\]
Radius of proton is \[1.0\times {{10}^{-13}}cm\]. Thus, volume of a proton is as follows:
\[V=\frac{4}{3}\pi {{\left( 1.0\times {{10}^{-13}}\ \text{cm} \right)}^{3}}\]
Now, fraction of the space within the atom occupied by the nucleus is as follows:
\[\begin{align}
& F=\frac{\frac{4}{3}\pi {{\left( 1.0\times {{10}^{-13}}\ \text{cm} \right)}^{3}}}{\frac{4}{3}\pi {{\left( 52.9\times {{10}^{-10}}\text{ cm} \right)}^{3}}} \\
& =6.8\times {{10}^{-15}}
\end{align}\]
The fraction of the space within the atom occupied by the nucleus is \[\underline{6.8\times {{10}^{-15}}}\].