Textbook Question
The first three energy levels of the fictitious element X are shown in FIGURE P38.54. What is the ionization energy of element X?
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35. Special Relativity
Inertial Reference Frames
Problem 63b
Textbook Question
Draw an energy-level diagram, similar to Figure 38.21, for the He+ ion. On your diagram: Show the ionization limit.
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Understand the context: The He+ ion is a hydrogen-like ion with a single electron orbiting the nucleus. Its energy levels can be calculated using the formula for hydrogen-like atoms: E_n = -Z² * (13.6 eV) / n², where Z is the atomic number (Z = 2 for helium), n is the principal quantum number, and 13.6 eV is the ionization energy of hydrogen.
Determine the energy levels: For each principal quantum number n (e.g., n = 1, 2, 3, ...), calculate the energy using the formula E_n = -Z² * (13.6 eV) / n². For He+, Z = 2, so the formula becomes E_n = -4 * (13.6 eV) / n². These values represent the energy levels of the electron in the He+ ion.
Draw the energy-level diagram: On the vertical axis, represent energy in electron volts (eV). Start with the lowest energy level (n = 1) at the bottom and progressively add higher energy levels (n = 2, 3, 4, ...) above it. The spacing between levels decreases as n increases, reflecting the inverse square relationship in the formula.
Show the ionization limit: The ionization limit corresponds to the energy level where the electron is completely removed from the atom (n → ∞). In this case, the ionization limit is 0 eV, as the energy levels approach 0 eV as n increases indefinitely. Mark this limit clearly at the top of the diagram.
Label the diagram: Clearly label each energy level with its corresponding principal quantum number (n = 1, 2, 3, ...) and energy value (e.g., E_1, E_2, E_3). Also, label the ionization limit at 0 eV to indicate the point where the electron is no longer bound to the nucleus.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Energy-Level Diagram
An energy-level diagram visually represents the quantized energy states of an atom or ion. Each horizontal line corresponds to a specific energy level, with the lowest level at the bottom. For the He+ ion, these levels indicate the energy required for an electron to occupy various states, including ground and excited states, and help illustrate transitions between them.
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Ionization Energy
Ionization energy is the amount of energy required to remove an electron from an atom or ion in its gaseous state. For the He+ ion, this energy corresponds to the transition from the highest occupied energy level to a state where the electron is completely removed, reaching the ionization limit. This concept is crucial for understanding how energy levels relate to the stability and reactivity of ions.
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Ionization Limit
The ionization limit is the energy threshold at which an electron can be completely removed from an atom or ion, resulting in the formation of a positively charged ion. In an energy-level diagram, it is represented as a horizontal line above the highest energy level of the ion. For He+, this limit indicates the maximum energy state before ionization occurs, providing insight into the stability of the ion.
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