Multiple Choice
Which wavelengths appear in the emission spectrum of an atom?
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9. Quantum Mechanics
Emission Spectrum
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Among the following hydrogen atom transitions, which would emit a photon of light with the greatest energy?
A
n = 3 to n = 2
B
n = 2 to n = 1
C
n = 5 to n = 4
D
n = 4 to n = 1
Verified step by step guidance1
Recall that the energy of a photon emitted during an electron transition in a hydrogen atom corresponds to the difference in energy levels between the initial and final states.
Use the formula for the energy of an electron in the nth energy level of hydrogen: \(E_n = -13.6 \times \frac{1}{n^2}\) eV, where \(n\) is the principal quantum number.
Calculate the energy difference \(\Delta E\) for each transition using \(\Delta E = E_{final} - E_{initial} = -13.6 \left( \frac{1}{n_{final}^2} - \frac{1}{n_{initial}^2} \right)\) eV. Since the electron moves to a lower energy level, \(\Delta E\) will be positive and represents the photon energy emitted.
Compare the energy differences for each transition: \(n=3 \to 2\), \(n=2 \to 1\), and \(n=5 \to 4\). The transition with the largest \(\Delta E\) corresponds to the photon with the greatest energy.
Note that the problem's correct answer is \(n=4 \to 1\), which is not listed among the options. Calculate \(\Delta E\) for \(n=4 \to 1\) as well to confirm it has the greatest energy difference and thus emits the photon with the highest energy.
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