Order the following transitions in the hydrogen atom from smallest to largest frequency of light absorbed: n = 3 to n = 6, n = 4 to n = 9, n = 2 to n = 3, and n = 1 to n = 2.

검증된 단계별 안내

Understand that the frequency of light absorbed during an electronic transition in a hydrogen atom is related to the energy difference between the two levels involved in the transition.

Recall the formula for the energy difference between two levels in a hydrogen atom: \( \Delta E = -13.6 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \) eV, where \( n_i \) is the initial energy level and \( n_f \) is the final energy level.

Calculate the energy difference for each transition: \( n = 3 \to n = 6 \), \( n = 4 \to n = 9 \), \( n = 2 \to n = 3 \), and \( n = 1 \to n = 2 \).

Recognize that the frequency of light absorbed is directly proportional to the energy difference \( \Delta E \). Therefore, the transition with the smallest energy difference will have the smallest frequency, and the transition with the largest energy difference will have the largest frequency.

Order the transitions based on the calculated energy differences from smallest to largest to determine the order of increasing frequency of light absorbed.

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주요 개념

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Energy Levels in Hydrogen Atom

In a hydrogen atom, electrons occupy discrete energy levels, denoted by quantum numbers (n). The energy associated with each level increases with n, meaning that transitions between levels involve specific energy changes. The difference in energy between two levels determines the frequency of light absorbed or emitted during electron transitions.

Frequency and Energy Relationship

The frequency of light absorbed or emitted during an electron transition is directly related to the energy difference between the two levels, as described by the equation E = hν, where E is energy, h is Planck's constant, and ν is frequency. Higher energy transitions correspond to higher frequencies of light, while lower energy transitions correspond to lower frequencies.

Calculating Energy Differences

To determine the frequency of light absorbed for each transition, one must calculate the energy difference using the formula ΔE = -13.6 eV (1/n_final² - 1/n_initial²). By calculating these energy differences for each transition, one can rank them from smallest to largest frequency, as the transition with the smallest energy difference will have the lowest frequency.