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Show that nuclear mass density is independent of the mass number.

Answer
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Hint To prove that nuclear mass density is independent of the mass number Awe will start with the understanding of what does nuclear mass density means. Nuclear mass density is the ratio of nuclear mass to nuclear volume and by using the formula of nuclear density we will compare it with mass number A.
Formula used
RA13
ρ=AV
Volume(V)=43πR3

Complete step by step answer:
We will start with the relation of the nuclear radius with the mass number which shows that the nuclear radius is directly proportional to the cube root of the mass number.
RA13
Now to remove the proportionality between the nuclear radius and mass number we will introduce a constant of proportionality.
R=R0A13 -------- Equation (1)
Where R0is constant of proportionality, Ris the radius of the nucleus and Ais the mass number.
Now the nuclear mass density of the nucleus can be given by the ratio of nuclear mass and its nuclear volume. Given as
Density(ρ)=massvolume
ρ=AV --------- Equation (2)
For nucleus mass = Aand Volume of the nucleus will be given by the formula of volume of a sphere, hence
Volume(V)=43πr3 (For sphere)
Volume(V)=43πR3 (For nucleus)
Now substituting the values of mass(A) and volume(V)of the nucleus in equation (2), we get
ρnucleus=A43πR3 -------- Equation (3)
Now from Equation (1) and Equation (3), we get
ρnucleus=A43πR03A
ρnucleus=34πR03
This shows that ρnucleusnuclear mass density is nearly constant so it is independent of mass numberA.

Note We can further deduce the value of nuclear mass density as we know that R0 is constant so by substituting the values of π=3.14and the value R0=1.25fm hence we can obtain the value of nuclear charge density as 2.3×1017kgm3, as an average.