Askiitians Tutor Team
Last Activity: 16 Days ago
To find the de Broglie wavelength of a nitrogen molecule in air at 300 K, we first need to calculate the root-mean-square (rms) speed of the nitrogen molecules and then use that speed to determine the de Broglie wavelength. The de Broglie wavelength is given by the formula:
Calculating the Root-Mean-Square Speed
The root-mean-square speed () of gas molecules can be calculated using the equation:
Where:
- k is the Boltzmann constant, approximately .
- T is the absolute temperature in Kelvin (300 K in this case).
- m is the mass of a single nitrogen molecule in kilograms.
First, we need to convert the atomic mass of nitrogen from atomic mass units (u) to kilograms. The atomic mass of nitrogen is given as 14.0076 u. Since 1 u is approximately , we can calculate:
Now, substituting the values into the formula:
Calculating the numerator:
Now, dividing by the mass:
Finding the de Broglie Wavelength
Now that we have the rms speed, we can calculate the de Broglie wavelength () using the formula:
Where:
- h is Planck's constant, approximately .
- v is the speed of the molecule (which we found to be ).
Substituting the values into the de Broglie wavelength formula:
Calculating the denominator:
Now, we can find the wavelength:
Final Result
Thus, the de Broglie wavelength of a nitrogen molecule in air at 300 K is approximately or (picometers). This wavelength is extremely small, reflecting the quantum nature of particles at the molecular level.