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 $n=2$ to $n=1$ transition is $122$ nm (see Example $39.6$). What is the wavelength of the photon emitted when a Be³⁺ ion undergoes this transition?

Verified step by step guidance

1. Step 1: Understand the problem. The triply ionized beryllium ion (Be3+) behaves like a hydrogen atom, but its nuclear charge (Z) is 4 instead of 1. The energy levels of such an atom are scaled by Z² compared to hydrogen. We need to calculate the wavelength of the photon emitted during the transition from n = 2 to n = 1 for Be3+.
2. Step 2: Recall the formula for the energy levels of a hydrogen-like atom: Eₙ = - (Z² × 13.6 eV) / n², where Z is the nuclear charge and n is the principal quantum number. For Be3+, Z = 4. Use this formula to calculate the energy difference (ΔE) between the n = 2 and n = 1 levels.
3. Step 3: Use the relationship between energy and wavelength for a photon: E = h × c / λ, where E is the energy of the photon, h is Planck's constant (6.626 × 10⁻³⁴ J·s), c is the speed of light (3 × 10⁸ m/s), and λ is the wavelength. Rearrange this formula to solve for λ: λ = h × c / ΔE.
4. Step 4: Substitute the calculated energy difference (ΔE) from Step 2 into the formula for λ. Ensure that the units are consistent (e.g., convert energy from eV to joules if necessary, using 1 eV = 1.602 × 10⁻¹⁹ J).
5. Step 5: Compare the result to the hydrogen atom's wavelength for the same transition (122 nm). Since the energy levels scale with Z², the wavelength for Be3+ will be shorter by a factor of Z². Use this scaling factor to verify your result.

Key Concepts

Hydrogen-like Atoms

Hydrogen-like atoms are ions that have only one electron, similar to hydrogen, but with a different nuclear charge. The energy levels of these atoms can be calculated using the formula E_n = -Z² * 13.6 eV/n², where Z is the atomic number. For Be3+, Z = 4, which means its energy levels are influenced by a stronger nuclear charge compared to hydrogen, affecting the wavelengths of emitted photons during electronic transitions.

Energy Level Transitions

When an electron transitions between energy levels in an atom, it emits or absorbs a photon with energy equal to the difference between those levels. The energy of the emitted photon can be calculated using the formula E = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength. For the Be3+ ion, the transition from n = 2 to n = 1 will release a photon with a specific wavelength that can be determined using the energy difference.

Rydberg Formula

The Rydberg formula provides a way to calculate the wavelengths of spectral lines in hydrogen-like atoms. It is expressed as 1/λ = RZ²(1/n₁² - 1/n₂²), where R is the Rydberg constant, Z is the atomic number, and n₁ and n₂ are the principal quantum numbers of the initial and final states. This formula is essential for determining the wavelength of emitted photons during transitions in ions like Be3+, allowing for direct comparisons with hydrogen.