Question

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 : $r=r_0{A}^{1/3}$
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.

Answer 1

Radius of nucleus r is given by the relation,
$r=r_0{A}^{1/3}$
$r_0=1.2f=1.2\times {10}^{-15}m$
Volume of nucleus, $V=\left(4/3\right)\pi r^3=\left(4/3\right)\pi (r_0{A}^{1/3}{)}^3=(4/3)\pi r_0^{3}A$ ..... (i)
Now, the mass of a nuclei M is equal to its mass number i.e., $M=A$ amu $=A\times 1.66\times {10}^{–27}$ kg.
Density of nucleus $\rho =\frac{\text{Mass of nucleus}}{\text{Volume of nucleus}}=\frac{A\times 1.66\times {10}^{-27}}{\left(4/3\right)\pi r_0^{3}A}$

Density of nucleus $=\frac{3\times 1.66\times {10}^{-27}}{4\pi r_0^{3}Kg{m}^{-3}}$
This relation shows that nuclear mass depends only on constant $r_0$. Hence, the nuclear mass densities of all nuclei are nearly the same.
Density of sodium nucleus is given by,
${\rho }_{Sodium}$ $=\frac{3\times 1.66\times {10}^{-27}}{4\times 3.14\times (1.2\times {10}^{-15}{)}^3}$

Density of sodium nucleus $=\frac{4.98\times {10}^{18}}{21.71}$

Density of sodium nucleus $=2.29\times {10}^{17}Kg{m}^{-3}$