The arrows in the figure depict the transition of an electron in the H atom. A photon is absorbed when the transition occurs. Which transition requires the absorption of the highest energy photon?

Question

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To compare the energy of the absorbed photon, use the hydrogen atom energy levels En = -R/n^2. For an absorption transition from ni to nf (nf > ni), the photon energy is ΔE = Ef - Ei = [-R/(nf^2)] - [-R/(ni^2)] = R(1/ni^2 - 1/nf^2).

Option 1: n = 1 to n = 2. Here ΔE = R(1/1^2 - 1/2^2) = R(1 - 1/4) = (3/4)R. This is a large energy change because you start from the ground state and move to the first excited state, yielding a substantial gap.

Option 2: n = 2 to n = 4. Here ΔE = R(1/2^2 - 1/4^2) = R(1/4 - 1/16) = R(3/16) ≈ 0.1875R. This is smaller than the 1→2 transition since you start from a higher initial level and the gap to the next levels is less pronounced.

Option 3: n = 3 to n = 7. Here ΔE = R(1/3^2 - 1/7^2) = R(1/9 - 1/49) = R((49 - 9)/441) = R(40/441) ≈ 0.0907R. This is even smaller because both the initial and final levels are higher, and the energy gap between these specific levels is relatively modest.

Comparing the three ΔE values, the largest energy difference—and thus the highest energy photon—is for the transition from n = 1 to n = 2. The other transitions involve smaller gaps due to larger n values and/or less favorable differences in 1/n^2, which reduces the photon energy required for absorption.

CHEM 110 Section 3 (PM) SP25 Pre-Lecture 6 Quiz C02-1-3

The arrows in the figure depict the transition of an electron in the H atom. A photon is absorbed when the transition occurs. Which transition requires the absorption of the highest energy photon?

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