Nuclei - Result Question 3
3. Two nuclei have their mass numbers in the ratio of $1: 3$. The ratio of their nuclear densities would be
(a) $1: 3$
(b) $3: 1$
(c) $(3)^{1 / 3}: 1$
(d) $1: 1$
[2008]
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Answer:
Correct Answer: 3. (d)
Solution:
- (d) Requird ratio of nuclear densities $=\frac{r_1}{r_2}$
$ \begin{aligned} & =\frac{(\frac{M_1}{V_1})}{(\frac{M_2}{V_2})}=\frac{M_1}{M_2} \times \frac{V_2}{V_1}=\frac{1}{3} \times \frac{\frac{4}{3} \pi R_2^{3}}{\frac{4}{3} \pi R_1^{3}} \\ & =\frac{1}{3} \times(\frac{R_2}{R_1})^{3}=\frac{1}{3} \times(\frac{R_0 M_2^{1 / 3}}{R_0 M_1^{1 / 3}})^{3}[\therefore R=R_0 M^{1 / 3}] \end{aligned} $
$=\frac{1}{3} \times(\frac{M_2}{M_1})=\frac{1}{3} \times(\frac{3}{1})=1: 1$
Nuclear density, $\rho=\frac{3 m}{4 \pi R_0^{3}}$
Here, $R_0=1.2 \times 10^{-15} m m=$ Average of mass of a nuclean (mass of proton + mass of neutron) $=1.66 \times 10^{-27} kg$
This formula suggest that density of nuclear matter is same for all nuclei.
