Q 2.28: The unit of length convenient on the nuclear scale is a fermi : $1 f = 10^{-15} m$. Nuclear sizes obey roughly the following empirical relation :$r = r_{o} A^{1/3}$ where r is the radius of the nucleus, A its mass number, and $r_{o}$ is a constant equal to about, 1.2 f. Show that the rule implies that nuclear mass density is nearly constant for different nuclei. Estimate the mass density of sodium nucleus. Compare it with the average mass density of a sodium atom obtained in Exercise. 2.27.

The equation for the radius of the nucleus is given by,

$r = r_{0} A^{1/3}$

The volume of the nucleus using the above relation, $\small V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (r_{0}A^{1/3})^3 = \frac{4}{3}\pi Ar_{0}^3$

We know,

Mass = Mass number× Mass of single Nucleus

= $\small A\times1.67\times10^{-27} kg$ (given)

$\small \therefore$ Nuclear mass Density = Mass of nucleus/ Volume of nucleus =

$\small \frac{A\times1.67\times10^{-27} kg}{\frac{4}{3}\pi Ar_{0}^3\ m^3} =\frac{3\times1.67\times10^{-27}}{4\pi r_{0}^3\ }\ kgm^{-3}$

The derived density formula contains only one variable,$\small r_{0}$ and is independent of mass number A. Since $\small r_{0}$ is constant, hence nuclear mass density is nearly constant for different nuclei.

∴ The density of the sodium atom nucleus = $\small 2.29\times 10^{17} kgm^{-3} \approx 0.3\times 10^{18} kgm^{-3}$               $\small (Putting\ r_{0} = 1.2 f = 1.2\times 10^{-15} m )$

Comparing it with the average mass density of a sodium atom obtained in Q 2.27. (Density of the order $\small 10^3 kgm^{-3}$)

Nuclear density is typically $\small 10^{15}$ times the atomic density of matter!

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