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

Let m be the average mass of a nucleon (neutron or proton)
As the nucles contains contains A nucleons, :. Mass of nucleus, M = mA
radius of nucleus, `r = r_0 A^(1//3)`
Nuclear density, `rho = (mass)/(volume) = (M)/((4)/(3)pi r^3) = (3mA)/(4pi(r_0A^(1//3))^3) = (3m)/(4pir_0^3)`
As m and `r_0` are constant, therefore nuclear density is constant for all nuclei.
using ` m = 1.66xx10^(-27)kg and r_0 = 1.2 f = 1.2xx10^(-15)m`
we get, `rho = (3m)/(4pi r_0^3) = (3xx1.66xx10^(-27))/(4xx3.14(1.2xx10^(-15))^2) = 2.29 xx10^(17) kg m^(-3)`
As `rho` is constant for all nuclei, this must be the density of sodium nucleus also.
Density of sodium atom` rho = 4.67xx10^3 kg m^(-3) :. (rho)/(rho) = (2.29xx10^(17))/(4.67xx10^3) = 4.9xx10^(13)`