A nucleus has mass number $\alpha$ and radius $R_\alpha$. Another nucleus has mass number $\beta$ an

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 :

Professor PC Thomas Classes · Answer

The radius of a nucleus is related to its mass number $A$ by the empirical formula $R = R_0 A^{1/3}$, where $R_0$ is a constant. For the two nuclei, $R_\alpha = R_0 \alpha^{1/3}$ and $R_\beta = R_0 \beta^{1/3}$. The ratio of their radii is $\frac{R_\alpha}{R_\beta} = \left(\frac{\alpha}{\beta}\right)^{1/3}$. Given that $\beta = 8\alpha$, we substitute to get $\frac{R_\alpha}{R_\beta} = \left(\frac{\alpha}{8\alpha}\right)^{1/3} = \left(\frac{1}{8}\right)^{1/3} = \frac{1}{2} = 0.5$.