Chemistry

Problem

Spectral lines of hydrogen-like ions arise from electronic transitions between Bohr orbits. Consider the ions

C^{5+}

(Z = 6) and

He^{+}

(Z = 2). a. (a) Derive an expression for the difference in energy

\Delta E

between two arbitrary Bohr levels

n_{2}

(upper) and

n_{1}

(lower) of a hydrogen-like ion of atomic number

Z

. b. (b) Using your result from part (a), calculate the frequency of the photon emitted when an electron in

C^{5+}

jumps from

n = 4

n = 2

. c. (c) For the same transition (

n = 4 \rightarrow 2

) in

He^{+}

, determine the ratio of the frequencies of photons emitted by

C^{5+}

and

He^{+}

Intuitive explanation

The key to comparing spectral lines in hydrogen-like ions is recognizing that the energy of each Bohr orbit scales with $Z^2$ , so the transition energy—and thus the wavelength—depends directly on nuclear charge. When an electron jumps between the same pair of orbits in $C^{5+}$ versus $He^{+}$ , the carbon ion's much higher $Z$ produces a much larger energy gap, shifting its spectrum to shorter wavelengths. Understanding this $Z^2$ dependence lets you predict how spectra change across different ions without recalculating every transition from scratch.

Intuitive explanation

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