The H atom and the Be^3+ ion each have one electron. Would...

The $\mathrm{H}$ atom and the $\mathrm{Be}^{3+}$ ion each have one electron. Would
you expect the Bohr model to predict their spectra accurately? Would you expect their spectra to be identical? Explain.

Key Concepts

Bohr Model

The Bohr model is an early quantum model that describes the electron in atoms as orbiting the nucleus in fixed, quantized paths. It provides formulas for energy levels and predicts spectral lines by assuming that electrons can only occupy these discrete orbits and that light is emitted or absorbed when an electron transitions between them. While it successfully explains the hydrogen atom spectrum, its accuracy diminishes when applied to more complex atoms or interactions not accounted for by its simplified assumptions.

Hydrogen-like Ions

Hydrogen-like ions are systems that, like the hydrogen atom, contain a single electron, but they have a nucleus with a different charge. The Bohr model can be extended to these ions by modifying the energy levels with a factor that depends on the square of the nuclear charge (Z^2). Although the theoretical framework remains similar, the actual energy differences between levels increase with higher nuclear charge, resulting in distinct spectral wavelengths.

Spectral Differences Due to Nuclear Charge

While the Bohr model can be used to predict the spectra of one-electron systems, the resulting spectral lines are not identical across different atoms or ions. This is because the energy levels in the model are proportional to the square of the nuclear charge, meaning that ions with a higher charge will have larger energy separations between the orbits compared to hydrogen. Consequently, even though both the hydrogen atom and a hydrogen-like ion are described by similar equations under the Bohr model, their spectra are quantitatively different due to the varying nuclear charges.

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