#### 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_0A^(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

#### Solution 1

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

`r = r_0A^(1/3)` ....(i)

`r_0 = 1.2f = 1.2 xx 10^(-15) m`

Volume of nucleus V = `4/3 pir^3`

`=4/3pi(r_0A^(1/3))^3 = 4/3pir_0^3 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"

=`(Axx1.66xx10^(-27))/(4/3pir_^3A) = (3xx1.66xx10^(-27))/(4pir_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" = (3xx1.66xx10^(-27))/(4xx3.14xx(1.2xx10^(-15))^3`

=`4.98/21.71xx10^(18) = 2.29 xx 10^(17) "kg m" ^(-3)`

#### Solution 2

Assume that the nucleus is spherical. Volume of nucleus

= 4/3 πr^{3} = 4/3 π [r_{0} A^{1/3}]^{3} = 4/3 πr_{0}^{3}A

Mass of nucleus = A

∴ Nuclear mass density = Mass of nucleus/Volume of nucleus

= A/(4/3πr_{0}^{3}A) = 3/4πr_{0}^{3}

Since r0 is a constant therefore the right hand side is a constant. So, the nuclear mass density is independent of mass number. Thus, nuclear mass density is constant for different nuclei.

For sodium, A = 23

∴ radius of sodium nucleus,

r = 1.2 x 10^{-15} (23)^{1/3 }m = 1.2 x 2.844 x 10^{-15} m =3.4128 x 10^{-15}

Volume of nucleus = `4/3pir^3`

= `4/3xx22/7 (3.4128 xx 10^(15))^3 m^3`

= `1.66 xx 10^(-43) m^3`

if we neglect the mass of electron of a sodium atom.then the mass of its nucleus can be taken to be the mass of its atom

∴Mass of sodium nucleus = `3.82 xx 10^(-26)` kg

Mass density of sodium nucleus = `"Mass of nucleus"/"volume of nucleus"`

`= (3.82xx10^(-26))/(1.66xx10^(-43))` kg m^{-3}

= 2.3 x 10^{17} kg m^{-3}

Mass density of sodium atom = 4.67 x 10^{3} kg m^{-3}

The ratio of the mass density of sodium nucleus to the average mass density of a sodium atom is

`2.3 xx 10^(17)/4.67 xx 10^3` = 4.92 x 10^{13}

So, the nuclear mass density is nearly 50 million times more than the atomic mass density for a sodium atom