De Broglie wavelength of nitrogen molecules

In summary, according to statistical mechanics, the average kinetic energy of a particle at temperature T is 3kT/2, where k is the Boltzmann constant. The average de Broglie wavelength of nitrogen molecules at room temperature can be calculated using the equation λ = h / (3kTm)1/2, where h is Planck's constant, k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of the molecule. When plugging in the values for h = 6.62607015 × 10-34 kg m2 / s, k = 1.3806488 × 10-23 m2 kg / s2 K, T =
  • #1
Kavorka
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Homework Statement


According to statistical mechanics, the average kinetic energy of a particle at temperature T is 3kT/2, where k is the Boltzmann constant. What is the average de Broglie wavelength of nitrogen molecules at room temperature?

Homework Equations


KE = 3kt/2 = p2/2m
λ = h / p

The Attempt at a Solution


So p = (2KEm)1/2, and inserting KE = 3kt/2
p = (3kTm)1/2

And thus λ = h/p = h / (3kTm)1/2

The only problem I have is that it seems like the units don't work out.
Inserting the units only into h / (3kTm)1/2 :

(kg m2 /s2)/((m2 kg/s2 K)(K)(kg))1/2 =
(kg m2 /s2)/(m kg/s) =
m/s, not m

Am I simplifying the units wrong or is the equation wrong?
 
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  • #2
Your units for h are not quite correct.
 
  • #3
Ah I see its kg m2 / s. Thanks!
 
  • #4
I'm having another problem. When I plug the values in where h = 1.602177x10-11 kg m2/s, k = 1.3806488 x 10-23 m2 kg/s2 K, T = 293.15 K, and m = 4.651734 x 10-26 kg/molecule

I get 6.741 x 1011 m when the answer in the book is 0.0276 nm. I'm obviously doing something very wrong.

EDIT: Oop wrong value for h, that explains why several of my answers to different questions were wrong hehe nevermind
 

1. What is the De Broglie wavelength of a nitrogen molecule?

The De Broglie wavelength of a nitrogen molecule can be calculated using the formula λ = h/mv, where λ is the De Broglie wavelength, h is Planck's constant, m is the mass of the nitrogen molecule, and v is its velocity. The value of the De Broglie wavelength for a nitrogen molecule is approximately 0.008 nanometers.

2. How does the De Broglie wavelength of nitrogen molecules compare to other molecules?

The De Broglie wavelength of nitrogen molecules is similar to other small molecules, such as oxygen and hydrogen. However, larger molecules, such as carbon dioxide and water, have longer De Broglie wavelengths due to their higher mass.

3. What is the significance of the De Broglie wavelength in understanding the behavior of nitrogen molecules?

The De Broglie wavelength helps us understand the wave-particle duality of matter, where particles, such as nitrogen molecules, can also exhibit wave-like properties. It also influences the behavior of these molecules, such as their diffraction patterns and interference effects.

4. Is the De Broglie wavelength of nitrogen molecules affected by temperature?

Yes, the De Broglie wavelength of nitrogen molecules is affected by temperature. As the temperature increases, the velocity of the molecules also increases, resulting in a shorter De Broglie wavelength.

5. Can the De Broglie wavelength of nitrogen molecules be measured experimentally?

Yes, the De Broglie wavelength of nitrogen molecules can be measured experimentally using techniques such as electron diffraction or neutron diffraction. These experiments involve passing a beam of electrons or neutrons through a sample of nitrogen molecules and measuring the resulting diffraction pattern.

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