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⁺?

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Step 1: Understand the Balmer series. The Balmer series corresponds to electronic transitions in hydrogen-like atoms where the electron moves from a higher energy level (n > 2) to the n = 2 energy level. For He+, the nucleus has a charge of +2, so the energy levels are scaled by a factor of Z², where Z is the atomic number (Z = 2 for helium).

Step 2: Use the Rydberg formula to calculate the wavelengths of the transitions. The formula is:

1/λ = R × Z² × (1/n₁² - 1/n₂²)

, where λ is the wavelength, R is the Rydberg constant (approximately 1.097 × 10⁷ m⁻¹), Z is the atomic number, n₁ is the lower energy level (n₁ = 2 for the Balmer series), and n₂ is the higher energy level (n₂ > n₁).

Step 3: Identify the three longest wavelengths. The longest wavelengths correspond to the smallest energy transitions, which occur when n₂ = 3, n₂ = 4, and n₂ = 5. Substitute these values into the Rydberg formula one at a time to calculate the wavelengths.

Step 4: Convert the calculated wavelengths from meters to nanometers. Since 1 nm = 10⁻⁹ m, multiply the result of each calculation by 10⁹ to express the wavelengths in nanometers.

Step 5: Summarize the results. The three longest wavelengths in the Balmer series of He+ correspond to the transitions from n₂ = 3, n₂ = 4, and n₂ = 5 to n₁ = 2. These wavelengths are expressed in nanometers after conversion.

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Balmer Series

The Balmer series refers to the set of spectral lines corresponding to transitions of electrons in hydrogen atoms from higher energy levels to the second energy level. Although the question pertains to He+, the concept is similar, as it involves electron transitions that emit or absorb light at specific wavelengths. The Balmer series is crucial for understanding how light interacts with ions and can be used to identify the presence of elements in stellar atmospheres.

Ionization and Electron Transitions

Ionization occurs when an atom or molecule gains enough energy to remove an electron, resulting in a positively charged ion. In the case of He+, the ion has lost one electron, and its electron transitions between energy levels lead to the absorption or emission of light at specific wavelengths. Understanding these transitions is essential for determining the wavelengths in the Balmer series and analyzing the light spectrum from stars.

Wavelength and Spectroscopy

Wavelength is the distance between successive peaks of a wave, typically measured in nanometers (nm) for light. Spectroscopy is the study of how light interacts with matter, allowing scientists to analyze the composition and properties of celestial objects. By examining the wavelengths of light absorbed or emitted by ions like He+, astronomers can infer the physical conditions and elemental makeup of stars, including their temperature and size.