객관식
Which of the following best describes a continuous spectrum?
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9. Quantum Mechanics
Wavelength and Frequency
객관식
In the Balmer series of hydrogen, what is the wavelength (in nm) of the spectral line corresponding to the electron transition from n = 4 to n = 2?
A
486 nm
B
434 nm
C
410 nm
D
656 nm
검증된 단계별 안내1
Identify the initial and final energy levels for the electron transition. Here, the electron moves from the higher energy level \(n_i = 4\) to the lower energy level \(n_f = 2\) in the Balmer series.
Recall the Rydberg formula for the wavelength of emitted or absorbed light during an electron transition in a hydrogen atom:
\[\lambda = \frac{1}{R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)}\]
where \(\lambda\) is the wavelength, \(R\) is the Rydberg constant (\(1.097 \times 10^7 \ \text{m}^{-1}\)), \(n_f\) is the final energy level, and \(n_i\) is the initial energy level.
Substitute the known values into the formula:
\[\lambda = \frac{1}{1.097 \times 10^7 \left( \frac{1}{2^2} - \frac{1}{4^2} \right)}\]
Calculate the difference inside the parentheses:
\[\frac{1}{2^2} - \frac{1}{4^2} = \frac{1}{4} - \frac{1}{16}\]
After finding \(\lambda\) in meters, convert it to nanometers by multiplying by \$10^9$ (since \(1 \ \text{m} = 10^9 \ \text{nm}\)). This will give the wavelength of the spectral line in nanometers.
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