Calculating Wavelength of Photons Emitted in H2 Molecule Transitions

In summary, to find the wavelength of the photons emitted in the n=2 to n=1 transition of the H2 molecule with a moment of inertia I = 0.5mr^2, you can use the equation λ = hc / ΔE, where ΔE is the energy difference between the two states, calculated using the equation E = L^2 / 2I and the values for I and m given.
  • #1
alfredbester
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A molecule with angular momentum L and moment of inertia I has a rotational energy [tex]E = L^2 / 2I[/tex]. Since angular momentum is quantized, find the wavelength of the photons emitted in n=2 to n=1 transition of the H2 molecule. This molecule has a moment of inertia [tex]I = [tex]0.5mr^2[/tex], where m = 938Mev/c^2 and r = 0.074nm.

My attempt is to say [tex]L = [[l(l+1)]^.5}\hbar[/tex] and use l =2 for n=2 state and l = 1 for n=1. Put these values for L into the equation for E.
Then E2 - E1 = [tex] \triangle E. [/tex]

[tex] \triangle E = hf, v = \lambda f. [/tex]

=> [tex] \lambda = hv / \triangle E = hc / \triangle E [/tex]


I've no idea if I'm on the right track, couldn't find anything similar in the textbook.
 
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  • #2
Yes, this is the correct approach. The equation you derived is:λ = hc / ΔEwhere h is Planck's constant, c is the speed of light, and ΔE is the energy difference between the two states. Plugging in the values for I and m, you can calculate the rotational energy for each state and find ΔE. Then, you can use the equation above to calculate the wavelength of the photons emitted in the transition.
 
  • #3


Your approach is correct. The equation for calculating the wavelength of a photon emitted during a transition is given by:

λ = hc / ΔE

Where h is Planck's constant, c is the speed of light, and ΔE is the difference in energy between the initial and final states of the transition.

In this case, the initial state is n = 2 and the final state is n = 1. The energy difference between these states can be calculated using the equation for rotational energy:

ΔE = L^2 / 2I = [l(l+1)] ℏ^2 / 2I

Substituting the values for l and I, we get:

ΔE = [2(2+1)] ℏ^2 / 2(0.5mr^2) = 3 ℏ^2 / mr^2

Now, we can substitute this value for ΔE into the equation for wavelength to get:

λ = hc / ΔE = hc / (3 ℏ^2 / mr^2) = 3hc / ℏ^2mr^2

Finally, we can substitute the values for h, c, m, and r to get the final result:

λ = (6.626 x 10^-34 Js)(3 x 10^8 m/s) / (3 x (1.055 x 10^-34 Js)^2 (938 x 10^6 eV/c^2)(0.074 x 10^-9 m)^2)

= 0.000065 nm

Therefore, the wavelength of the photon emitted during the n=2 to n=1 transition in the H2 molecule is approximately 0.000065 nm. This is in the ultraviolet region of the electromagnetic spectrum.
 

1. What is the H2 molecule and why is it important?

The H2 molecule, also known as dihydrogen or molecular hydrogen, is a molecule composed of two hydrogen atoms bonded together. It is the most abundant molecule in the universe and plays a crucial role in many chemical reactions and physical processes.

2. What is the relationship between wavelength and energy of a photon?

The wavelength and energy of a photon are inversely proportional to each other. This means that as the wavelength increases, the energy of the photon decreases, and vice versa. This relationship is described by the equation: E = hc/λ, where E is the energy of the photon, h is Planck's constant, c is the speed of light, and λ is the wavelength.

3. How do you calculate the wavelength of photons emitted in H2 molecule transitions?

The wavelength of photons emitted in H2 molecule transitions can be calculated using the Balmer-Rydberg equation: 1/λ = R(1/n1^2 - 1/n2^2), where R is the Rydberg constant and n1 and n2 are the initial and final quantum numbers, respectively. Additionally, the energy of the photon can also be calculated using the equation mentioned in question 2.

4. What is the significance of calculating the wavelength of photons emitted in H2 molecule transitions?

Calculating the wavelength of photons emitted in H2 molecule transitions allows us to understand the electronic structure and energy levels of the molecule. This information is important in various fields such as spectroscopy, quantum mechanics, and astrophysics.

5. Are there any practical applications of this calculation?

Yes, there are many practical applications of calculating the wavelength of photons emitted in H2 molecule transitions. For example, it is used in the development of new materials, understanding the properties of light and electromagnetic radiation, and in the study of chemical reactions and energy transfer processes.

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