Question 43.1: The Density of Nuclear Material Estimate the density of nucl......
The Density of Nuclear Material
Estimate the density of nuclear material.
INTERPRET and ANTICIPATE
Because both the volume of a nucleus and its mass are proportional to A, we expect to find that its density is a constant, independent of A.
SOLVE
Find the density from the mass and the volume.
The mass is approximately given by A in atomic mass units. Substitute R=r_0 A^{1 / 3} (Eq. 43.2) for r in the volume expression.
\begin{aligned}& \rho \approx\left(\frac{A}{\frac{4}{3} \pi R^3}\right)=\left(\frac{A}{\frac{4}{3} \pi r_0^3 A}\right) \\& \rho \approx \frac{3(1 u )}{4 \pi r_0^3} \approx \frac{3\left(1.66 \times 10^{-27} kg \right)}{4 \pi\left(1.2 \times 10^{-15} m \right)^3} \\& \rho \approx 2 \times 10^{17} kg / m ^3\end{aligned}CHECK and THINK
As expected, our answer does not depend on A, so it is constant for all nuclei. Our result is much denser than familiar objects such as the Earth \left(\rho_{\oplus} \approx 5500 kg / m ^3\right) or water \left(\rho_{\text {water }} \approx 1000 kg / m ^3\right), but this isn’t surprising because we know that atoms and therefore objects are mostly empty space.