Question 78: Atoms & Nuclei

Question

A nucleus has mass number $\alpha$ and radius $R_{\alpha}$. Another nucleus has mass number $\beta$ and radius $R_{\beta}$.
If $\beta = 8\alpha$ then $R_{\alpha} / R_{\beta}$ is :

Accepted Answer

The nucleus is assumed as a sphere where the volume is proportional to the total number of nucleons (mass number, A).
Because the density of nuclear matter is approximately constant, the radius R of a nucleus is related to its mass number A by the following formula :
$$ \mathrm{R}=\mathrm{R}_0 \mathrm{~A}^{1 / 3} $$
Where R is the nuclear radius.
$R_0$ is a constant (approximately $1.2 	imes 10^{-15} \mathrm{~m}$ or 1.2 fm ).
A is the mass number.
For the first nucleus : $\mathrm{R}_\alpha=\mathrm{R}_0(\alpha)^{1 / 3}$
For the second nucleus : $R_\beta=R_0(\beta)^{1 / 3}$
Taking the ratio $\frac{\mathrm{R}_\alpha}{\mathrm{R}_\beta}$ :
$$ \frac{\mathrm{R}_\alpha}{\mathrm{R}_\beta}=\frac{\mathrm{R}_0 \alpha^{1 / 3}}{\mathrm{R}_0 \beta^{1 / 3}}=\left(\frac{\alpha}{\beta}ight)^{1 / 3} $$
It is given that $\beta=8 \alpha$.
$$ \frac{\mathrm{R}_\alpha}{\mathrm{R}_\beta}=\left(\frac{\alpha}{8 \alpha}ight)^{1 / 3}=\frac{1}{2}=0.5 $$
Therefore, the ratio of the radius of the first nucleus to the second is 0.5 . Hence, the correct option is (D).

Answer

1

Answer

2

Answer

8

Answer

0.5

Atoms & Nuclei

A nucleus has mass number \alpha and radius R_{\alpha}. Another nucleus has mass number \beta and radius R_{\beta}. If \beta = 8\alpha then R_{\alpha} / R_{\beta} is :

Choose the correct answer: