Ch 38: Quantization

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 38: Quantization

Problema 63c

Capitolo 38, Problema 63c

Draw an energy-level diagram, similar to Figure 38.21, for the He+ ion. On your diagram: Show all possible emission transitions from the n = 4 energy level.

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Understand the context: The He+ ion is a hydrogen-like ion with a single electron. Its energy levels can be described using the Bohr model, where the energy of a level is given by the formula:

E = - \frac{Z_{2}}{R} \frac{1}{n^{2}}

, where Z is the atomic number (Z=2 for He+), R is the Rydberg constant, and n is the principal quantum number.

Draw the energy-level diagram: Start by sketching horizontal lines to represent the energy levels for n=1, n=2, n=3, and n=4. The energy levels get closer together as n increases because the energy difference between levels decreases with increasing n.

Label the energy levels: Indicate the principal quantum number (n) for each level on the diagram. For He+, the energy levels are more negative than for hydrogen because Z=2, so the energy values are scaled by Z².

Identify possible emission transitions: From the n=4 level, the electron can transition to any lower energy level (n=3, n=2, or n=1). Each transition corresponds to the emission of a photon with energy equal to the difference between the two levels:

E = | E_{initial} - E_{final} |

Indicate transitions on the diagram: Draw arrows pointing downward from the n=4 level to each of the lower levels (n=3, n=2, and n=1). Label each arrow with the corresponding transition (e.g., n=4 to n=3, n=4 to n=2, n=4 to n=1) to show all possible emission transitions.

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Energy Levels in Hydrogen-like Ions

In hydrogen-like ions, such as He+, electrons occupy discrete energy levels defined by quantum mechanics. The energy levels are quantized and can be calculated using the formula E_n = -Z² * 13.6 eV / n², where Z is the atomic number and n is the principal quantum number. For He+, Z=2, which means the energy levels are closer together compared to hydrogen, affecting the emission and absorption spectra.

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Emission Transitions

Emission transitions occur when an electron in an excited state drops to a lower energy level, releasing energy in the form of a photon. The energy of the emitted photon corresponds to the difference in energy between the two levels, given by ΔE = E_initial - E_final. For the n=4 level in He+, transitions can occur to n=3, n=2, and n=1, each producing photons of specific wavelengths.

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Energy-Level Diagrams

Energy-level diagrams visually represent the allowed energy states of an atom or ion and the transitions between them. Each horizontal line represents an energy level, with the vertical distance indicating energy differences. These diagrams help in understanding the possible transitions and the corresponding emissions, making it easier to predict the spectral lines produced by the ion.

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