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Chapter 8: Problem 104

Consider separate \(1.0\) -\(\mathrm{L}\) samples of \(\mathrm{He}(g)\) and \(\mathrm{UF}_{6}(g),\) both at \(1.00\) atm and containing the same number of moles. What ratio of temperatures for the two samples would produce the same root mean square velocity?

Short Answer

Expert verified
The ratio of temperatures for the two samples of He(g) and UF6(g) that would produce the same root mean square velocity is 88.01.

Step by step solution

01

Write the rms velocity equation for He and UF6

For He: \(v_{rms, He} = \sqrt{\frac{3R(T_{He})}{M_{He}}}\) For UF6: \(v_{rms, UF6} = \sqrt{\frac{3R(T_{UF6})}{M_{UF6}}}\)
02

Set the rms velocity equations equal to each other

\(\sqrt{\frac{3R(T_{He})}{M_{He}}} = \sqrt{\frac{3R(T_{UF6})}{M_{UF6}}}\)
03

Square both sides of the equation to remove the square roots

\(\frac{3R(T_{He})}{M_{He}} = \frac{3R(T_{UF6})}{M_{UF6}}\)
04

Solve for the ratio of temperatures

First, simplify by dividing both sides of the equation by 3R: \(\frac{T_{He}}{M_{He}} = \frac{T_{UF6}}{M_{UF6}}\) Now, solve for the ratio of temperatures (\(\frac{T_{He}}{T_{UF6}}\)): \(\frac{T_{He}}{T_{UF6}} = \frac{M_{UF6}}{M_{He}}\)
05

Calculate the molar masses

Find the molar masses of He and UF6 in kg/mol. Molar mass of He = 4.00 g/mol (This is a well-known value, or you can find using the periodic table.) To convert to kg/mol, divide by 1000: Molar mass of He = \(\frac{4.00}{1000}\) kg/mol = 0.004 kg/mol Molar mass of UF6: 1 Uranium atom = 238.03 g/mol 6 Fluorine atoms = 6 × 19.00 g/mol = 114.00 g/mol Total molar mass of UF6 = 238.03 g/mol + 114.00 g/mol = 352.03 g/mol To convert to kg/mol, divide by 1000: Molar mass of UF6 = \(\frac{352.03}{1000}\) kg/mol = 0.35203 kg/mol
06

Calculate the ratio of temperatures

Using the molar masses, we can find the ratio of temperatures: \(\frac{T_{He}}{T_{UF6}} = \frac{M_{UF6}}{M_{He}} = \frac{0.35203\, kg/mol}{0.004\, kg/mol} = 88.01\) The ratio of temperatures for the two samples that would produce the same root mean square velocity is 88.01.

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