Among the following hydrogen atom transitions, which one emits a photon with the highest energy?

• A. n = 5 to n = 3
• B. n = 3 to n = 2
• C. n = 4 to n = 1
• D. n = 2 to n = 1

Answer: (C)

Verified step by step guidance

1. 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.
2. 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.
3. 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. The photon energy emitted equals the absolute value of this difference.
4. Compare the absolute values of \(\Delta E\) for each transition: \(n=5 \to n=3\), \(n=3 \to n=2\), \(n=4 \to n=1\), and \(n=2 \to n=1\) to determine which transition emits the photon with the highest energy.
5. Identify the transition with the largest energy difference as the one that emits the photon with the highest energy.