Question

Calculate the average speed of a (a) chlorine molecule at \(-32^{\circ} \mathrm{C}\). (b) \(\mathrm{UF}_{6}\) molecule at room temperature \(\left(25^{\circ} \mathrm{C}\right)\).

Short answer

Question: Calculate the average speed of a chlorine molecule at -32°C and a UF6 molecule at 25°C (room temperature). Answer: The average speed of a chlorine molecule at -32°C is approximately 422.79 m/s, and the average speed of a UF6 molecule at 25°C (room temperature) is approximately 324.05 m/s.

Step-by-step solution

Convert the temperature from Celsius to Kelvin

To convert the temperature from Celsius to Kelvin, add 273.15 to the Celsius temperature T: $$K = T + 273.15$$ For Chlorine at -32°C: $$K = -32 + 273.15 = 241.15 \,\mathrm K$$

Determine the molar mass of a chlorine molecule

A chlorine molecule is made up of two chlorine atoms, and the molar mass of a chlorine atom is 35.45 g/mol. Therefore, the molar mass of a chlorine molecule is: $$M = 2 \times 35.45 = 70.90\,\mathrm{g/mol}$$ We need the molar mass in kg/mol, so we will convert: $$M = 70.90\,\mathrm{g/mol} \times \frac{1\,\mathrm{kg}}{1000\,\mathrm{g}} = 0.0709\,\mathrm{kg/mol}$$

Calculate the average speed of a chlorine molecule

Using the formula for the average speed of a molecule, plug in the values for R (8.314 J/mol·K), T (241.15 K), and M (0.0709 kg/mol), and calculate the average speed: $$ v_{avg} = \sqrt{\frac{8 \times 8.314\,\mathrm{J/mol\cdot K} \times 241.15\,\mathrm K}{\pi \times 0.0709\,\mathrm{kg/mol}}} = 422.79\,\mathrm{m/s} $$ The average speed of a chlorine molecule at -32°C is approximately 422.79 m/s. For (b) UF6:

Convert the temperature from Celsius to Kelvin

For UF6 at 25°C: $$K = 25 + 273.15 = 298.15\,\mathrm K$$

Determine the molar mass of a UF6 molecule

A UF6 molecule is composed of 1 uranium atom and 6 fluorine atoms. The molar mass of uranium is 238.03 g/mol, and that of fluorine is 19.00 g/mol. Therefore, the molar mass of a UF6 molecule is: $$M = 238.03 + 6 \times 19.00 = 351.03\,\mathrm{g/mol}$$ And we convert it to kg/mol: $$M = 351.03\,\mathrm{g/mol} \times \frac{1\,\mathrm{kg}}{1000\,\mathrm{g}} = 0.35103\,\mathrm{kg/mol}$$

Calculate the average speed of a UF6 molecule

Again, using the formula for the average speed of a molecule, plug in the values for R (8.314 J/mol·K), T (298.15 K), and M (0.35103 kg/mol), and calculate the average speed: $$ v_{avg} = \sqrt{\frac{8 \times 8.314\,\mathrm{J/mol\cdot K} \times 298.15\,\mathrm K}{\pi \times 0.35103\,\mathrm{kg/mol}}} = 324.05\,\mathrm{m/s} $$ The average speed of a UF6 molecule at 25°C (room temperature) is approximately 324.05 m/s.