Units and Measurement

Question : 28

Total: 33

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=r0A1∕3
where r is the radius of the nucleus, A its mass number, and r0 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 question 27.

Solution:

Let m be the average mass of a nucleon (neutron or proton).As the nucleus contains A nucleons,
∴‌ mass of nucleus M=mA, radius of nucleus r=r0A1∕3
Nuclear density, ρ=

‌mass

‌volume

πr3

3mA

4π(r0A1∕3)‌3

4πr03

As m and r0 are constant, therefore, nuclear density is constant for allnuclei.
Using m=1.66×10−27 kg and r0=1.2 f=1.2×10−15 m
we get ,ρ=

4πr03

3×1.66×10−27

4×3.14 (1.2×10−15 )3

=2.29×1017 kg m−3.
As ρ is constant for all nuclei, this must be the density of sodium nucleusalso.
Density of sodium atom, ρ,=0.58×103 kg m−3
∴

ρ,

2.29×1017

0.58×103

=4×1014

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