The energy E of the electron in a hydrogen atom can be calculated from the Bohr formula:=E−Ryn2In this equation Ry stands for the Rydberg energy, and n stands for the principal quantum number of the orbital that holds the electron. (You can find the value of the Rydberg energy using the Data button on the ALEKS toolbar.)Calculate the wavelength of the line in the emission line spectrum of hydrogen caused by the transition of the electron from an orbital with =n5 to an orbital with =n3. Round your answer to 3 significant digits.μm
Question
The energy E of the electron in a hydrogen atom can be calculated from the Bohr formula:=E−Ryn2In this equation Ry stands for the Rydberg energy, and n stands for the principal quantum number of the orbital that holds the electron. (You can find the value of the Rydberg energy using the Data button on the ALEKS toolbar.)Calculate the wavelength of the line in the emission line spectrum of hydrogen caused by the transition of the electron from an orbital with =n5 to an orbital with =n3. Round your answer to 3 significant digits.μm
Solution
To solve this problem, we need to use the Rydberg formula for the wavelength of light emitted during electron transitions:
1/λ = R * (1/n1² - 1/n2²)
where:
- λ is the wavelength of the emitted light,
- R is the Rydberg constant (approximately 1.097373 x 10^7 m^-1),
- n1 and n2 are the principal quantum numbers of the initial and final energy levels (n1 < n2).
Given that the electron transitions from n2 = 5 to n1 = 3, we can substitute these values into the formula:
1/λ = R * (1/3² - 1/5²) 1/λ = R * (1/9 - 1/25) 1/λ = R * (16/225)
Solving for λ gives:
λ = 225/(16*R)
Substituting the value of R gives:
λ = 225/(16*1.097373 x 10^7) λ = 1.282 x 10^-6 m
Converting to micrometers (1 m = 10^6 μm) gives:
λ = 1.282 μm
So, the wavelength of the line in the emission line spectrum of hydrogen caused by the transition of the electron from an orbital with n = 5 to an orbital with n = 3 is approximately 1.282 μm.
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