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Q16E

Expert-verified
Found in: Page 518

Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

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

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

See the step by step solution

Step 1: Given data

Mass number of Uranium nucleus $=238$.

Mass number of beryllium nucleus $=9$.

Step 2: 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 ${1}{.2}{×}{{10}}^{-15}{}{\text{m}}$.

Step 3: 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, ${r}_{Be}={A}^{1/3}\left({R}_{0}\right)$ .

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

$\begin{array}{l}{r}_{Be}={A}^{1/3}\left({R}_{0}\right)\\ {r}_{Be}={9}^{1/3}\left({R}_{0}\right)\end{array}$

Step 4: 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.

$\begin{array}{c}\frac{{D}_{u}}{{D}_{ze}}=\frac{2\left({r}_{l}\right)}{2\left({r}_{{R}_{k}}\right)}\\ \frac{{D}_{u}}{{D}_{ze}}=\frac{2\left({238}^{1/3}{R}_{0}\right)}{2\left({9}^{1/3}{R}_{0}\right)}\\ \frac{{D}_{u}}{{D}_{ze}}=2.98\end{array}$