Ch 38: Quantization

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 38: Quantization

Problem 63b

Chapter 38, Problem 63b

Draw an energy-level diagram, similar to Figure 38.21, for the He+ ion. On your diagram: Show the ionization limit.

Verified step by step guidance

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

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.

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.

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.

Recommended video:

Guided course

20:32

Mass Spectrometers

Related Practice

Textbook Question

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.

views

Textbook Question

Very large, hot stars—much hotter than our sun—can be identified by the way in which He⁺ ions in their atmosphere absorb light. What are the three longest wavelengths, in nm, in the Balmer series of He⁺?

views

Textbook Question

Consider a hydrogen atom in stationary state n. On average, an atom stays in the n = 2 state for 1.6 ns before undergoing a transition to the n = 1 state. On average, how many revolutions does the electron make before the transition?

views

Textbook Question

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.

views

Textbook Question

What is the difference in the wavelengths emitted in a 199→2 transition and a 200→2 transition?

views

Textbook Question

INT Two hydrogen atoms collide head-on. The collision brings both atoms to a halt. Immediately after the collision, both atoms emit a 121.6 nm photon. What was the speed of each atom just before the collision?

views