Traffic on the German autobahns reaches speeds of up to 230 km/h.At what temperature (∘C) do oxygen molecules have this same average speed?

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5.2K

To find the temperature at which oxygen molecules have an average speed of 230 km/h, we can use the Maxwell-Boltzmann distribution equation. This equation relates the average speed of gas molecules to the temperature and molecular mass.

The equation is as follows:

v_avg = √((8 * k * T) / (π * m))

Where:
- v_avg is the average speed of the gas molecules
- k is the Boltzmann constant (1.380649 × 10^-23 J/K)
- T is the temperature in Kelvin
- m is the molecular mass of the gas

To convert the speed from km/h to m/s, we need to divide it by 3.6.

The molecular mass of oxygen (O2) is approximately 32 g/mol.

Now, let's rearrange the equation to solve for the temperature (T):

T = (v_avg^2 * π * m) / (8 * k)

First, convert the speed from km/h to m/s:

v_avg = 230 km/h / 3.6 = 63.89 m/s

Next, convert the molecular mass from grams to kilograms:

m = 32 g/mol / 1000 g/kg = 0.032 kg/mol

Now, we can plug these values into the equation and solve for T:

T = ((63.89^2) * π * 0.032) / (8 * 1.380649 × 10^-23)

T ≈ 408,964.2 K

To convert from Kelvin to Celsius, we subtract 273.15:

T ≈ 408,964.2 K - 273.15 = 408,691.05 °C (rounded to two decimal places)

Therefore, at a temperature of approximately 408,691.05 °C, oxygen molecules would have an average speed of 230 km/h.