Is energy emitted or absorbed when the following electronic transitions occur in hydrogen? b. from an orbit of radius 2.12 Å to one of radius 8.46 Å

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Identify the initial and final orbits of the electron in the hydrogen atom. The initial orbit has a radius of 2.12 Å, and the final orbit has a radius of 8.46 Å.

Recall that the energy levels of an electron in a hydrogen atom are quantized and can be described by the formula: E_n = -13.6 \(\frac{1}{n^2}\) \(\text{ eV}\), where n is the principal quantum number.

Determine the principal quantum numbers (n) corresponding to the given radii. Use the formula for the radius of an orbit in a hydrogen atom: r_n = 0.529 \(\times\) n^2 \(\text{ Å}\). Solve for n for both radii.

Compare the initial and final energy levels. If the electron moves to a higher energy level (larger n), energy is absorbed. If it moves to a lower energy level (smaller n), energy is emitted.

Conclude whether energy is emitted or absorbed based on the direction of the transition between the initial and final energy levels.

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주요 개념

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Energy Levels in Hydrogen

In a hydrogen atom, electrons occupy specific energy levels, which correspond to distinct orbits around the nucleus. The energy of these levels is quantized, meaning electrons can only exist in certain states. The radius of these orbits increases with energy, and transitions between these levels involve the absorption or emission of energy.

Photon Emission and Absorption

When an electron transitions between energy levels, it either absorbs or emits a photon, which is a particle of light. If an electron moves to a higher energy level (further from the nucleus), it absorbs energy, while moving to a lower level results in the emission of energy. The energy of the photon corresponds to the difference in energy between the two levels.

Rydberg Formula

The Rydberg formula provides a way to calculate the wavelengths of the spectral lines of hydrogen. It relates the wavelengths of emitted or absorbed light to the principal quantum numbers of the initial and final energy levels. This formula is essential for determining the energy change associated with electronic transitions in hydrogen.