GIVEN: A monochromatic laser is exciting hydrogen atoms from the n=2 state to the n=5 state.
A: What is the wavelength of the laser?
*435 nm
B: Eventually, all of the excited hydrogen atoms will emit photons until they fall back to the ground state. How many different wavelengths can be observed in this process?
*10
C: What is the longest wavelength that is observed?
* ?? nm
-I know that it occurs from n=4 to n=3, but I'm not sure how to put it into the equation
D: What is the shortest wavelength observed?
* ?? nm
I know C is 4059 nm
D is 95.1nm
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To find the longest and shortest wavelengths observed in the process of hydrogen atoms falling back to the ground state, we can use the formula for the wavelength of an emitted photon:
1/λ = R * (1/n₁² - 1/n₂²)
Where R is the Rydberg constant (approximately 1.097 × 10^7 m⁻¹), and n₁ and n₂ are the initial and final energy levels, respectively.
For the longest wavelength, we find the transition where n₁ is the highest and n₂ is the lowest. In this case, n₁ = 4 and n₂ = 3. Plugging these values into the equation:
1/λ = 1.097 × 10^7 m⁻¹ * (1/4² - 1/3²)
Simplifying and rearranging the equation, we find:
λ = 1/(1.097 × 10^7 m⁻¹ * (1/4² - 1/3²))
Evaluating this expression will give us the longest wavelength observed.
Similarly, for the shortest wavelength, we find the transition where n₁ is the lowest and n₂ is the highest. In this case, n₁ = 2 and n₂ = 5.
1/λ = 1.097 × 10^7 m⁻¹ * (1/2² - 1/5²)
Again, simplifying and rearranging the equation will give us the shortest wavelength observed.
Please note that the calculation requires the Rydberg constant, which is given for m⁻¹. To compare with the choices in nm, the wavelength can be converted using the relationship: 1 meter = 1 × 10^9 nm.