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

Verified step by step guidance

1. Understand that the wavelength of light emitted during an electron transition is inversely proportional to the energy difference between the two levels. This means that a smaller energy difference results in a longer wavelength.
2. 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 \( n_i \) and \( n_f \) are the initial and final principal quantum numbers, respectively.
3. For the transition 3p \( \rightarrow \) 2s, identify the principal quantum numbers: \( n_i = 3 \) and \( n_f = 2 \).
4. For the transition 4p \( \rightarrow \) 3p, identify the principal quantum numbers: \( n_i = 4 \) and \( n_f = 3 \).
5. Compare the energy differences for both transitions using the formula, and determine which transition has the smaller energy difference, indicating the longer wavelength of emitted light.

Key Concepts

Electron Transitions

Electron transitions refer to the movement of electrons between different energy levels or orbitals in 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 of the emitted light corresponds to the difference in energy between the two levels, influencing the wavelength of the light produced.

Wavelength and Energy Relationship

The wavelength of light is inversely related to its energy, as 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 wavelength. This means that transitions resulting in lower energy emissions will produce longer wavelengths of light. Understanding this relationship is crucial for determining which electron transition produces light of a longer wavelength.

Quantum Numbers and Energy Levels

Quantum numbers describe the properties of atomic orbitals and the electrons in those orbitals. The principal quantum number (n) indicates the energy level, while the angular momentum quantum number (l) indicates the shape of the orbital. In the hydrogen atom, transitions between different principal quantum levels (like 2s to 3p or 3p to 4p) determine the energy and wavelength of the emitted light, with higher n values generally corresponding to higher energy levels.