Final answer:
The rms speed of an oxygen molecule in air at 0 degrees Celsius is calculated using the kinetic theory of gases formula, taking into account the Boltzmann constant, Kelvin temperature, and molar mass in kilograms. After conversion and simplification, the rms speed is approximately 461 m/s.
Step-by-step explanation:
To calculate the root mean square (rms) speed of an oxygen molecule at 0 degrees Celsius, we will apply the kinetic theory of gases. The rms speed is derived from the equation:
Urms = √(3kBT/M)
where:
- kB is the Boltzmann constant (1.38 × 10-23 J/K)
- T is the temperature in Kelvin (for 0 degrees Celsius, T = 273.15 K)
- M is the molar mass of the gas in kilograms per mole (for oxygen, M = 32.0 g/mol or 0.032 kg/mol)
First, we convert the temperature to Kelvin and the molar mass to kg:
T = 0°C + 273.15 = 273.15 K
M = 32 g/mol = 0.032 kg/mol
Next, plug the values into the rms speed formula:
Urms = √((3 * 1.38 × 10-23 J/K * 273.15 K) / 0.032 kg/mol)
After simplifying, you get the value for the rms speed:
Urms ≈ 461 m/s (rounded to three significant digits)
This result is not explicitly one of the options provided (a-d), so one may need to check for any rounding or unit conversion errors in those choices. However, it's important to note that we rounded to the necessary significant digits, and the calculation is accurate based on the information provided.