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 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 the sodium nucleus. Compare it with the average mass density of a sodium atom obtained
Radius of the nucleus
r = r0 A1/3
ro = 1.2 f = 1.2 x 10-15 m
Considering the nucleus is spherical. Volume of nucleus
= \( \dfrac{4}{3}\) πr3 = \( \dfrac{4}{3}\) π [r0 A1/3]3 = \( \dfrac{4}{3}\) πr03A
Mass of nucleus = mA
m is the average mass of the nucleon
A is the number of nucleons
Nuclear mass density = Mass of nucleus/Volume of nucleus
= \(( \dfrac{mA}{\dfrac{4}{3πr^3}} )\) = \(
\dfrac{ 3mA}{4πr^3}\) = \(
\dfrac{ 3mA}{4πr_0^3A}\)
= \( \dfrac{ 3m}{4πr_0^3}\)
Using m = 1.66 x 10-27 kg and ro = 1.2 f
= 1.2 x 10-15 m in the above equation
= \( \dfrac{3 \times 1.66 \times 10^{-27}}{4 \times 3.14 \times ( 1.2 \times 10^{-15})^3}\)
= \( \dfrac{4.98 \times 10^{-27}}{21. 703 \times 10^{-45}}= 2.29 \times 10^{17} kg/m^3\)
So, the nuclear mass density is much larger than atomic mass density for a sodium atom we got in 2.27.
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