Q. 1.20 Uranium has two common isotopes,... [FREE SOLUTION]

91影视

An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition 路 356 pages

ISBN: 9780201380279

Chapter 1: Q. 1.20 (page 13)

Uranium has two common isotopes, with atomic masses of 238 and 235. One way to separate these isotopes is to combine the uranium with fluorine to make uranium hexafluoride gas, UF₆, then exploit the difference in the average thermal speeds of molecules containing the different isotopes. Calculate the rms speed of each type of molecule at room temperature, and compare them.

Short Answer

Expert verified

The rms speed of uranium-235 and uranium-238 are

v₂₃₅=145.92 m/s

v₂₃₈=145.53 m/s

Step by step solution

Given information

Temp (room temp) T=25^oC = 298K
Mass of U₂₃₅= 235.04 u
Mass of U₂₃₈= 238.028 u
Mass of fluorine = 18.998 u

Explanation

Root mean square speed of molecules is calculated as

$v = \sqrt{\frac{3 k T}{m}} . . . . . . . . . . . . . . . . . . . . . . . . . . (1)$

Where,

m = mass

k = Boltzmann constant

T = temp

First find the mass of hexa- fluoride of both isotopes

$m_{235} = m_{U_{235}} + 6 脳 m_{F_{19}} = (235.04 + 6 脳 18.998) u = 5.794 脳 10^{- 25} kg similarly m_{238} = m_{U_{238}} + 6 脳 m_{F_{19}} = (238.02891 + 6 脳 18.998) u = 5.843 脳 10^{- 25} kg$

Now using equation (1) calculate rms speed for both isotopes.

$v_{235} = \sqrt{\frac{3 k T}{m_{235}}} = \sqrt{\frac{3 脳 (1.38 脳 10^{- 23} m^{2} k g s^{- 2} K^{- 1}) 脳 (298 K)}{5.794 脳 10^{- 25} k g}} v_{235} = 145.92 {ms}^{- 1} S i m i l a r l y v_{238} = \sqrt{\frac{3 k T}{m_{238}}} = \sqrt{\frac{3 脳 (1.38 脳 10^{- 23} m^{2} k g s^{- 2} K^{- 1}) 脳 (298 K)}{5.843 脳 10^{- 25} k g}} v_{238} = 145.53 {ms}^{- 1}$