The figure below represents part of the emission spectrum...

The figure below represents part of the emission spectrum for a one-electron ion in the gas phase. All the lines result from electronic transitions from excited states to the $n=3$ state. (FIGURE CANNOT COPY) a. What electronic transitions correspond to lines $A$ and B? b. If the wavelength of line $B$ is $142.5 \mathrm{nm}$, calculate the wavelength of line A.

The figure below represents part of the emission spectrum for a one-electron ion in the gas phase. All the lines result from electronic transitions from excited states to the $n=3$ state.
(FIGURE CANNOT COPY)
a. What electronic transitions correspond to lines $A$ and B?
b. If the wavelength of line $B$ is $142.5 \mathrm{nm}$, calculate the wavelength of line A.

Key Concepts

Emission Spectrum

An emission spectrum is a set of discrete wavelengths of light emitted when electrons in atoms transition from higher energy levels to lower ones. For one?electron ions, these transitions are quantized, leading to sharp spectral lines that are characteristic of the element and its electronic structure.

Electronic Transitions

Electronic transitions refer to the movement of an electron between fixed energy levels in an atom or ion. In a one?electron ion, when an electron drops from a higher energy state (n > 3 in this case) to a lower energy state (n = 3), it releases energy in the form of light. The specific initial and final energy levels determine the energy, and thus the wavelength, of the emitted photon.

Energy Levels and Quantum Numbers

Energy levels in atoms are quantized and are labeled by the principal quantum number n. In hydrogen-like ions, the energy of each level is inversely proportional to the square of n. Understanding these levels is essential for analyzing the transitions that result in emission lines, as each jump between levels corresponds to a specific amount of energy being released or absorbed.

Rydberg Formula

The Rydberg formula is a key equation used to predict the wavelengths of spectral lines produced by electron transitions in hydrogen-like atoms. It relates the inverse wavelength to the difference of the reciprocals of the squares of the initial and final quantum numbers, typically expressed as 1/? = RZ²(1/n_final² - 1/n_initial²), where R is the Rydberg constant. This formula is crucial for calculating unknown wavelengths when a related wavelength is known.

Wavelength Calculation Based on Energy Difference

The calculation of wavelengths for different electronic transitions involves understanding the relationship between the energy difference of the states and the wavelength of the emitted photon. Since the energy difference corresponds to the difference in the reciprocal squares of the quantum numbers, minor changes in the quantum numbers can result in significant variations in the wavelength. This concept underlies the process of correlating one measured spectral line with another.

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