Determine the relative root mean square speeds of O2 and SF6 at a given temperature

The relative root mean square speed of a gas can be calculated using the formula:

vrms = √(3RT/M)

Where:
- vrms is the root mean square speed
- R is the ideal gas constant (8.314 J/(mol·K))
- T is the temperature in Kelvin
- M is the molar mass of the gas

Given that the temperature is the same for both O2 and SF6, we can compare their relative root mean square speeds by comparing their molar masses.

The molar mass of O2 is 32 g/mol (16 g/mol for each oxygen atom).
The molar mass of SF6 is 146 g/mol (32 g/mol for sulfur and 114 g/mol for six fluorine atoms).

Let's calculate the relative root mean square speeds:

For O2:
vrmsO2 = √(3RT/M)
vrmsO2 = √(3 * 8.314 J/(mol·K) * T / 0.032 kg/mol)
vrmsO2 = √(249.42 J/(K·kg) * T)

For SF6:
vrmsSF6 = √(3RT/M)
vrmsSF6 = √(3 * 8.314 J/(mol·K) * T / 0.146 kg/mol)
vrmsSF6 = √(48.641 J/(K·kg) * T)

Since the temperature is the same for both gases, we can compare their relative root mean square speeds using the ratio of their molar masses:

vrmsO2/vrmsSF6 = √(249.42 J/(K·kg) * T) / √(48.641 J/(K·kg) * T)
= √(32/146)
≈ 0.390

Therefore, the relative root mean square speed of O2 compared to SF6 at a given temperature is approximately 0.390.