Draw an energy-level diagram, similar to Figure 38.21, for the He+ ion. On your diagram: Show the first five energy levels. Label each with the values of n and E_n.

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Understand the context: The He+ ion is a hydrogen-like ion with a single electron orbiting a nucleus of charge +2e. The energy levels for such ions are determined using the formula for the energy of hydrogen-like atoms:

E

_n = -13.6 imes Z² / n², where Z is the atomic number (Z = 2 for He+) and n is the principal quantum number.

Calculate the energy levels: Use the formula

E

_n = -13.6 imes Z² / n² to calculate the first five energy levels (n = 1, 2, 3, 4, 5). Substitute Z = 2 into the formula and compute the energy for each value of n.

Label the energy levels: For each energy level, write the corresponding value of n (e.g., n = 1, n = 2, etc.) and the calculated energy

E

_n. Ensure the energy values are negative, as they represent bound states of the electron.

Draw the energy-level diagram: On a vertical axis, represent energy (with more negative values lower on the axis). Plot horizontal lines for each energy level, spacing them according to the calculated energy differences. Label each line with its corresponding n value and energy

E

_n.

Add clarity to the diagram: Ensure the diagram is neat and clearly labeled. Indicate that the energy levels get closer together as n increases, reflecting the fact that the energy difference between successive levels decreases as n becomes larger.

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Energy Levels in Hydrogen-like Atoms

In hydrogen-like atoms, such as He+, energy levels are quantized and determined by the principal quantum number (n). The energy of each level can be calculated using the formula En = -Z² * 13.6 eV / n², where Z is the atomic number. For He+, Z = 2, leading to distinct energy values for each level as n increases.

Quantum Numbers

Quantum numbers are values that describe the unique quantum state of an electron in an atom. The principal quantum number (n) indicates the energy level and size of the orbital, while the azimuthal (l) and magnetic (m) quantum numbers describe the shape and orientation of the orbital. For energy-level diagrams, only n is typically labeled.

Energy-Level Diagrams

Energy-level diagrams visually represent the allowed energy states of electrons in an atom. Each horizontal line corresponds to a specific energy level, labeled with its quantum number (n) and energy (En). These diagrams help illustrate transitions between energy levels, which are fundamental to understanding atomic spectra and electron behavior.