Chapter 13, Problem 60P

To determine

To calculate:The ratio of the root mean square speed of the isotope of the uranium ${}^{235}U{,}^{238}U$ at constant temperature.

Answer to Problem 60P

The ratio of the root mean square speed of the isotope of the uranium ${}^{235}U{,}^{238}U$ at constant temperature is $1.004$ .

Explanation of Solution

Given:

  ${}^{235}U{,}^{238}U$

Formula used:

  $v_{rms}=\sqrt{\frac{3kT}{m}}$

Here,

  $k$ is boltzmann constant,

  $T$ is temperature in $\text{K}$ ,

  $m$ is mass of molecule.

Calculation:

  ${}^{235}U{,}^{238}U$

Rms speed is given by

  $\begin{array}{l}v=\sqrt{\frac{3kT}{m}}\\ \frac{{(v_{rms})}_{{}^{235}U}}{{(v_{rms})}_{{}^{238}U}}=\sqrt{\frac{3kT/m_{{}^{235}U}}{3kT/m_{{}^{238}U}}}\\ =\sqrt{\frac{m_{{}^{235}U}}{m_{{}^{238}U}}}=\sqrt{\frac{238+6(19)}{235+6(19)}}\\ =\sqrt{\frac{352}{349}}=1.004\end{array}$

Conclusion:

The ratio of the root mean square speed of the isotope of the uranium ${}^{235}U{,}^{238}U$ at constant temperature is $1.004$ .