The unit of length convenient on nuclear scale is a fermi, `1f = 10^9-15)`m. Nuclear sizes obey rougholy the following empricial relation : `r = r_0 A^(1//3)`, where r is radius of the nucleus and `r_0` is a constant equal to 1.2 f. show that the rule implies that nuclear mass density in nearly constant for different neclei. Estimate the mass density of sodium nucleus. Compare it with avarge mass density of sodium atom is Q. 27 `(4.67xx10^3 kg//m^3)`.

Explanation

Radius of nucleus, `r= r_(0)A^(1/3)`
Volume of the nucleus, `V= (4)/(3)pi r^(3)= (4)/(3)pi r_(0)^(3)A`
According to Avogadro.s hypothesis,
Mass of `6.02xx10^(23)` atoms of sodium `= Ag = Axx10^(-3)kg`
Mass of 1 atom of sodium `=(Axx10^(-3))/(6.02xx10^(23)xx4"/"3pi r_(0)^(3)A)kgm^(-3)`
`=(10^(-3))/(6.02xx10^(23)xx4"/"3pi (1.2xx10^(-15))^(3))kgm^(-3)`
`=0.3xx10^(18)kgm^(-3)`.
This is independent of mass number, hence mass density is same for different nuclei.
Mass density of sodium nucleus `=0.3xx10^(18)kgm^(-3)`
mass density of sodium atom `=0.6xx10^(3)kgm^(-3)`
Mass density of sodium nucleus/mass density of sodium atom `=(0.3xx10^(18)kgm^(-3))/(0.6xx10^(3)kgm^(-3)) approx 0.510^(15)`.