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 = r0 A1/3
where,
r is the radius of the nucleus, A its mass number, and
ro 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.
Radius of nucleus r is given by the relation,
r = r0 A1/3
r0 = 1.2 f = 1.2 × 10-15 m
Volume of nucleus, V = (4 / 3) π r3
= (4 / 3) π (r0 A1/3)3
= (4 / 3) π r0 A ... (i)
Now, the mass of a nuclei M is equal to its mass number.
That is,
M = A amu = A × 1.66 × 10–27 kg
Density of nucleus, ρ = Mass of nucleus / Volume of nucleus
= A X 1.66 × 10-27 / (4/3) π r03 A
= 3 X 1.66 × 10-27 / 4 π r03 Kg m-3
Density of sodium nucleus is given by,
ρSodium = 3 × 1.66 × 10-27 / 4 × 3.14 × (1.2 × 10-15)3
= 4.98 × 1018 / 21.71
= 2.29 × 1017 Kg m-3
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