Question

Two nuclei have their mass numbers in the ratio of $1:3$. The ratio of their nuclear densities would be ____________.

• A. ${\left(3\right)}^{1/3}:1$
• B. $1:1$
• C. $1:3$
• D. $3:1$

Answer 1

Answer: (B)

$1:1$

Given, ratio of mass $=1:3$

Radius of nucleus $r=R_o[A{]}^{1/3}$

$A=$ mass number

$R_o=1.5\times {10}^{-15}m$ [fixed for all nuclei]

density(P) $=\frac{mass}{volume}$

Solution:- To find mass of any nucleus:- $A\times 1.66\times {10}^{-27}$kg

Radius of nucleus $r=R_o[A{]}^{1/3}$

where, $R_o=1.5\times {10}^{-15}m$

The volume of the nucleus $=\frac{4}{3}{\Pi }^3{r}^3=\frac{4}{3}\Pi (R_o{A}^{1/3}{)}^3$

$\Rightarrow \frac{4}{3}\Pi (1.5\times {10}^{-15}m{A}^{1/3}{)}^3=\frac{4}{3}\Pi (1.5\times {10}^{-15}m\left)\right(A^{1/3}{)}^3$

$V=\frac{4}{3}\Pi (1.5\times {10}^{-15}{)}^{3}A{m}^3$

Density$\left(\rho \right)=\frac{mass}{volume}=\frac{A\times 1.66\times {10}^{-27}kg}{4/3\Pi (1.5\times {10}^{-15}{)}^{3}A{m}^3}$

$=1.17\times {10}^{17}Kg{m}^{-3}$

As can be seen, the density of the nucleus is independent of the mass number. Therefore, the density of the nucleus is the same for every element and hence, the ratio will be $1:1$.