Example

Question:

Uranium has two isotopes of masses 235 and 238 units. If both are present in uranium hexafluoride gas, which would have the larger average speed? If atomic mass of fluorine is 19 units, estimate the percentage difference in speeds at any temperature.

Solution:

At fixed temperature, the average kinetic energy \(\frac{1}{2} m \langle v^2 \rangle\) is constant. So the lighter molecule moves faster.
The ratio of speeds is inversely proportional to the square root of their masses:
Masses: \[ m_{\text{UF}_6,235} = 235 + 6 \times 19 = 349 \] \[ m_{\text{UF}_6,238} = 238 + 6 \times 19 = 352 \]
So, \[ \frac{v_{349}}{v_{352}} = \left(\frac{352}{349}\right)^{1/2} = 1.0044 \]
The percentage difference: \[ \frac{\Delta v}{v} = 0.44\% \] Thus, the molecule with smaller mass (UF$_6$ with $^{235}$U) has the larger average speed, and the difference is 0.44%.