According to the quantum-mechanical model for the hydrogen atom, which electron transition produces light with the longer wavelength: 2p to 1s or 3p to 1s?

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Identify the concept of electron transitions and how they relate to the emission of light. In the hydrogen atom, when an electron transitions from a higher energy level to a lower one, it emits a photon of light.

Recall that the energy of the emitted photon is related to the difference in energy levels between the initial and final states. The greater the energy difference, the shorter the wavelength of the emitted light.

Use the Rydberg formula to calculate the energy difference for each transition: \( \Delta E = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \), where \( R_H \) is the Rydberg constant, \( n_1 \) is the principal quantum number of the lower energy level, and \( n_2 \) is the principal quantum number of the higher energy level.

Calculate the energy difference for the 2p to 1s transition: \( n_1 = 1 \) and \( n_2 = 2 \).

Calculate the energy difference for the 3p to 1s transition: \( n_1 = 1 \) and \( n_2 = 3 \). Compare the energy differences to determine which transition emits light with a longer wavelength.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quantum Mechanical Model

The quantum mechanical model describes the behavior of electrons in atoms as wave functions rather than fixed orbits. It incorporates principles of quantum mechanics, such as quantization of energy levels, which dictate that electrons can only occupy certain energy states. This model is essential for understanding electron transitions and the emission of light in atoms.

Electron Transitions

Electron transitions refer to the movement of electrons between different energy levels within an atom. When an electron moves from a higher energy level to a lower one, it releases energy in the form of light. The energy difference between these levels determines the wavelength of the emitted light, with larger energy differences corresponding to shorter wavelengths.

Wavelength and Energy Relationship

The relationship between wavelength and energy is described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. This inverse relationship means that longer wavelengths correspond to lower energy transitions. Thus, to determine which transition produces light with a longer wavelength, one must compare the energy differences of the transitions.

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Related Practice

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Determine whether each transition in the hydrogen atom corresponds to absorption or emission of energy. a. n = 3 → n = 1 b. n = 2 → n = 4 c. n = 4 → n = 3

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Sketch the 3d orbitals. How do the 4d orbitals differ from the 3d orbitals?

Textbook Question

According to the quantum-mechanical model for the hydrogen atom, which electron transition produces light with the longer wavelength: 3p → 2s or 4p → 3p?

Textbook Question

Calculate the wavelength of the light emitted when an electron in a hydrogen atom makes each transition and indicate the region of the electromagnetic spectrum (infrared, visible, ultraviolet, etc.) where the light is found. a. n = 2 → n = 1 b. n = 3 → n = 1 c. n = 4 → n = 2 d. n = 5 → n = 2

Textbook Question

Calculate the frequency of the light emitted when an electron in a hydrogen atom makes each transition: a. n = 4 → n = 3 b. n = 5 → n = 1 c. n = 5 → n = 4 d. n = 6 → n = 5

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