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

Assume that the nucleus is spherical. Volume of nucleus
`4//3 pi r^(3) = 4//3 pi [r_(0) A^(1//3)]^(3)=4//3 pi r_(0)^(3) A`
Mass of nuc,eus = A
`:.` Nuclear mass density = Mass of nucleus/Volume of nulceus
`=A//(4//3 pi r_(0)^(3)A)=3//4 pi r_(0)^(3)`
Since `r_(0)` is a constant therefore the right hand side is a constant. So, the nuclear nass density is independent of mass number. Thus, nuclear mass density is constant for different nuclei.
For sodium, `A=23`
`:.` radius of sodium nucleus,
`r=1.2xx10^(-15) (23)^(1//3) m=1.2xx2.844xx10^(-15) m=3.4128xx10^(-15)`
Volume of nucleus `=4/3 pi r^(3)`
`=4/3xx22/7 (3.4128xx10^(-15))^(3) m^(3)=1.66xx10^(-43) m^(3)`
If we neglect the mass of electrons of a sodium atom, then the mass of its nucleus can be taken to be the mass of its atom.
`:.` Mass of sodium nucleus `=3.82xx10^(-26) kg`
(Refer to Q. 2.27)
Mass density of sodium nucleus
`=("Mass of nucleus")/("Volume of nucleus")`
`=(3.82xx10^(-26))/(1.66xx10^(-43)) kg m^(-3)=2.3xx10^(17) kg m^(-3)`
Mass density of sodium atom `=4.67xx10^(3) kg m^(-3)`
(Refer to Q. 2.27)
The ratio of the mass density of sodium nucleus to the average mass density of a sodium atom is
`(2.3xx10^(17))/(4.67xx10^(3)) t.e., 4.92xx10^(13)`
So, the nuclear mass density is nearly 50 million times more than the atomic mass density for a sodium atom.