The energy-level scheme for the hypothetical one-electron element Searsium is shown in Fig. E39.25E39.25. The potential energy is taken to be zero for an electron at an infinite distance from the nucleus. An 1818-eV photon is absorbed by a Searsium atom in its ground level. As the atom returns to its ground level, what possible energies can the emitted photons have? Assume that there can be transitions between all pairs of levels.

Ch 39: Particles Behaving as Waves

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 39: Particles Behaving as Waves

Problem 25b

Chapter 39, Problem 25b

The energy-level scheme for the hypothetical one-electron element Searsium is shown in Fig. $E 39.25$ . The potential energy is taken to be zero for an electron at an infinite distance from the nucleus. An $18$ -eV photon is absorbed by a Searsium atom in its ground level. As the atom returns to its ground level, what possible energies can the emitted photons have? Assume that there can be transitions between all pairs of levels.

Verified step by step guidance

Step 1: Analyze the energy-level diagram provided. The energy levels for Searsium are given as follows: Fundamental state (n=1) at -18.0 eV, First excited state (n=2) at -4.50 eV, Second excited state (n=3) at -2.00 eV, and Third excited state (n=4) at -1.13 eV.

Step 2: Determine the initial excitation caused by the absorption of the 18-eV photon. Since the ground state energy is -18.0 eV, absorbing an 18-eV photon will move the electron to the zero potential energy level, which corresponds to an infinite distance from the nucleus.

Step 3: As the atom returns to its ground state, the electron can transition between any pair of energy levels. The possible transitions are: (n=4 to n=3), (n=4 to n=2), (n=4 to n=1), (n=3 to n=2), (n=3 to n=1), and (n=2 to n=1).

Step 4: Calculate the energy of the emitted photons for each transition using the formula ΔE = E_final - E_initial. For example, for the transition from n=4 to n=3, ΔE = (-2.00 eV) - (-1.13 eV). Repeat this calculation for all transitions.

Step 5: List all possible photon energies resulting from these transitions. These energies correspond to the differences between the energy levels as calculated in Step 4.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Energy Levels

Energy levels in an atom represent the specific energies that electrons can have when bound to the nucleus. Each level corresponds to a quantized state, with lower energy levels being closer to the nucleus. Electrons can transition between these levels by absorbing or emitting photons, with the energy of the photon equal to the difference in energy between the two levels.

Photon Absorption and Emission

Photon absorption occurs when an electron in an atom absorbs a photon and moves to a higher energy level. Conversely, photon emission happens when an electron drops from a higher energy level to a lower one, releasing energy in the form of a photon. The energy of the emitted or absorbed photon is directly related to the energy difference between the initial and final states of the electron.

Transition Between Energy Levels

Transitions between energy levels in an atom can occur in various ways, allowing electrons to move between any two levels. The possible energies of emitted photons depend on the specific transitions that occur as the atom returns to its ground state. In the case of Searsium, the transitions can involve any pair of energy levels, leading to a range of possible photon energies upon emission.

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In a set of experiments on a hypothetical one-electron atom, you measure the wavelengths of the photons emitted from transitions ending in the ground level ( $n equals 1$ ), as shown in the energy-level diagram in Fig. E $39.27$ . You also observe that it takes $17.50$ eV to ionize this atom. What is the energy of the atom in each of the levels ( $n equals 1$ , $n equals 2$ , etc.) shown in the figure?

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Use Balmer's formula to calculate (a) the wavelength, (b) the frequency, and (c) the photon energy for the Hg line of the Balmer series for hydrogen.

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A triply ionized beryllium ion, Be³⁺ (a beryllium atom with three electrons removed), behaves very much like a hydrogen atom except that the nuclear charge is four times as great. For the hydrogen atom, the wavelength of the photon emitted in the $n equals 2$ to $n equals 1$ transition is $122$ nm (see Example $39.6$ ). What is the wavelength of the photon emitted when a Be³⁺ ion undergoes this transition?

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