Final answer:
The de Broglie wavelength of a nitrogen molecule at 300 K can be calculated by first determining the rms speed of the molecule using the kinetic theory of gases and then applying de Broglie's equation. This requires converting the molecular mass of nitrogen from atomic mass units to kilograms and using physical constants.
Step-by-step explanation:
The question is asking about the de Broglie wavelength of a nitrogen molecule in air at 300 K, assuming it moves with the root-mean-square (rms) speed typical at this temperature. The de Broglie wavelength (λ) can be found using the formula λ = h / mv, where h is Planck's constant, m is the mass of the nitrogen molecule, and v is the rms speed of the molecule.
To find the rms speed of the nitrogen molecule, we can use the formula derived from the kinetic theory of gases, which states that the rms speed (v) is proportional to the square root of the temperature (T) divided by the molar mass (M), such that v = sqrt(3kT/M), where k is Boltzmann's constant. As an air molecule is approximately the molecular mass of nitrogen, we'll consider the molecular mass of nitrogen, which is 28.0134 u (since nitrogen is diatomic), and convert it to kilograms. The exact calculation of v will require the conversion of the atomic mass unit (u) to kilograms and inserting the values into the rms speed formula.
Once the velocity is calculated, we can insert the value along with the known constants into the de Broglie wavelength formula to find the wavelength of the nitrogen molecule. This calculation requires knowledge of physical constants and unit conversion from the atomic mass unit to kilograms.