An atomic emission spectrum of hydrogen shows three wavelengths: 1875 nm, 1282 nm, and 1093 nm. Assign these wavelengths to transitions in the hydrogen atom.

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Identify the series of transitions in the hydrogen atom: Lyman, Balmer, and Paschen series.

Recall that the Lyman series involves transitions to the n=1 level, the Balmer series to the n=2 level, and the Paschen series to the n=3 level.

Convert the given wavelengths from nanometers to meters for calculation purposes.

Use the Rydberg formula \( \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \) to calculate the initial and final energy levels (n1 and n2) for each wavelength.

Assign each wavelength to the appropriate series based on the calculated energy levels and the known series transitions.

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Key Concepts

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

Atomic Emission Spectrum

An atomic emission spectrum is a spectrum of the electromagnetic radiation emitted by a substance when its atoms transition from a higher energy state to a lower energy state. Each element has a unique emission spectrum, which can be used to identify the element and understand its electronic structure. The wavelengths observed correspond to specific energy differences between electron orbits in the atom.

Energy Levels in Hydrogen Atom

In a hydrogen atom, electrons occupy discrete energy levels, which are quantized. When an electron transitions between these levels, it either absorbs or emits a photon with energy equal to the difference between the two levels. The energy levels can be calculated using the Rydberg formula, which helps in determining the wavelengths of light emitted during these transitions.

Rydberg Formula

The Rydberg formula is a mathematical equation used to predict the wavelengths of spectral lines in hydrogen and other hydrogen-like atoms. It is expressed as 1/λ = R_H (1/n1² - 1/n2²), where λ is the wavelength, R_H is the Rydberg constant, and n1 and n2 are the principal quantum numbers of the lower and higher energy levels, respectively. This formula is essential for assigning observed wavelengths to specific electronic transitions.

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Textbook Question

An atomic emission spectrum of hydrogen shows three wavelengths: 121.5 nm, 102.6 nm, and 97.23 nm. Assign these wavelengths to transitions in the hydrogen atom.