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Found in: Page 165

### Matter & Interactions

Book edition 4th edition
Author(s) Ruth W. Chabay, Bruce A. Sherwood
Pages 1135 pages
ISBN 9781118875865

# Uranium-238 ( ${{\mathbf{U}}}^{{\mathbf{238}}}$) has three more neutrons than uranium-235 ( ${{\mathbf{U}}}^{{\mathbf{235}}}$). Compared to the speed of sound in a bar of ${{\mathbf{U}}}^{{\mathbf{235}}}$, is the speed of sound in a bar of ${{\mathbf{U}}}^{{\mathbf{238}}}$ higher, lower, or the same? Explain your choice, including justification for assumptions you make.

Speed of sound in ${\mathrm{U}}^{238}$ is lower than speed of sound in ${\mathrm{U}}^{235}$ because speed of sound is inversely proportional to the root of atomic mass.

See the step by step solution

## Step 1: Given data

Bars of ${\mathrm{U}}^{235}$ and ${\mathrm{U}}^{238}$.

It is given that the uranium ${\mathrm{U}}^{238}$ has three more neutrons as compared to uranium ${\mathrm{U}}^{235}$.

## Step 2: Speed of sound in solid

Speed of sound in a solid of stiffness constant $k$, atomic mass $m$ and interatomic bond length $d$ is given by,

${\mathbf{v}}{\mathbf{=}}\sqrt{\frac{\mathbf{k}}{\mathbf{m}}}{\mathbf{d}}{\mathbf{}}....\left(\mathrm{i}\right)$

## Step 3: Comparing speed of sound in U238 to that in U235

${\mathrm{U}}^{238}$ has three more neutrons in its nucleus than ${\mathrm{U}}^{235}$ . Thus the atomic mass of ${\mathrm{U}}^{238}$ is greater than that of ${\mathrm{U}}^{235}$. Their stiffness constant and interatomic distance are the same. From equation (i), it is seen that the speed of sound in a solid is inversely proportional to the root of its atomic mass. Thus a solid with greater atomic mass will have lower speed of sound in it. Hence speed of sound in ${\mathrm{U}}^{238}$ is lower than that of ${\mathrm{U}}^{235}$