### Q.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_{0} A^{1/3}

where r is the radius of the nucleus, A its mass number, and r_{0} 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.

**Answer:-**

Radius of nucleus *r* is given by the relation,

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

r_{0} = 1.2 f = 1.2 × 10^{-15} m

Volume of nucleus, *V* = **(**4 **/** 3**)** π r^{3}

= **(**4 **/** 3**)** π **(**r_{0 }A^{1/3}**)**^{3} = **(**4 **/** 3**)** π r_{0 }A ….. (i)

Now, the mass of a nuclei *M* is equal to its mass number i.e.,

*M* = *A* amu = *A* × 1.66 × 10^{–27} kg

Density of nucleus, ρ = Mass of nucleus **/** Volume of nucleus

= A X 1.66 × 10^{-27} **/** (4/3) π r_{0}^{3} A

= 3 X 1.66 × 10^{-27} **/** 4 π r_{0}^{3} Kg m^{-3}

his 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,

ρ_{Sodium} = 3 × 1.66 × 10^{-27} **/** 4 × 3.14 × (1.2 × 10^{-15})^{3}

= 4.98 × 10^{18} **/** 21.71 = 2.29 × 10^{17} Kg m^{-3}