Question

Determine the approximate ratio of the diameter of a uranium nucleus $(A=238)$to that of beryllium nucleus$(A=9)$

Short answer

The ratio of the diameter of uranium nucleus to that of beryllium nucleus is $2.98$.

Step-by-step solution

Given data

Mass number of Uranium nucleus$=238$.

Mass number of beryllium nucleus $=9$.

Formula of Radius of Uranium

The radius of uranium is given as, $r=A^{1/3}\left(R_0\right)$.

Here $A$is the mass number and$R_0$ is a constant its value is $1.2\times {10}^{-15}\text{m}$.

Calculation for the radius of uranium and beryllium

$9$Substitute$9$for mass number $A$ of uranium nucleus in the above equation.

$\begin{array}{l}{r}_U=A^{1/3}\left(R_0\right)\\ r_U={238}^{1/3}\left(R_0\right)\end{array}$

The radius of a beryllium nucleus is given as, $r_{Be}=A^{1/3}\left(R_0\right)$.

Substitutefor mass number $A$ of beryllium nucleus in the above equation.

$\begin{array}{l}{r}_{Be}=A^{1/3}\left(R_0\right)\\ r_{Be}=9^{1/3}\left(R_0\right)\end{array}$

Calculation of the ration of the diameter

The ratio of the diameter of uranium nucleus to that of beryllium nucleus is given as, .

$\frac{D_{UJ}}{D_{B_e}}=\frac{2\left(r_{UJ}\right)}{2\left(r_{B_{e}e}\right)}$

Substitute ${238}^{1/3}\left(R_0\right)$ for $r_U$ and $9^{1/3}\left(R_0\right)$ for$r_{Be}$ in the above equation.

$\begin{array}{c}\frac{D_u}{D_{ze}}=\frac{2\left(r_l\right)}{2\left(r_{R_k}\right)}\\ \frac{D_u}{D_{ze}}=\frac{2\left({238}^{1/3}{R}_0\right)}{2\left(9^{1/3}{R}_0\right)}\\ \frac{D_u}{D_{ze}}=2.98\end{array}$