What is the ground state energy, according to Bohr theory, for (a) He^+,(b) Li^2+,(c) deuterium

step 1 For this exercise, we have to calculate the ground state energy for the ionized helium, the doub

What is the ground state energy, according to Bohr theory,...

What is the ground state energy, according to Bohr theory, for (a) $\mathrm{He}^{+},(\mathrm{b}) \mathrm{Li}^{2+},(\mathrm{c})$ deuterium (an isotope of hydrogen whose nucleus contains a neutron as well as a proton)?

Key Concepts

Reduced Mass Effect

The concept of reduced mass is important when considering the dynamics of the electron-nucleus system. Instead of taking the electron mass alone, the reduced mass (which accounts for the motion of both the electron and the nucleus) should be used in the energy calculations. For isotopes like deuterium, where the nucleus is heavier than that of protium (ordinary hydrogen), the slight increase in reduced mass leads to a small adjustment in the calculated energy levels compared to a system with an infinitely massive nucleus.

Energy Quantization and Z² Scaling

Within the Bohr model, the energies of the quantized orbits depend on the nuclear charge (Z) and the principal quantum number (n). Specifically, the energy in the nth level is proportional to -Z²/n². Therefore, when comparing different hydrogen-like species, the ground state energy (n = 1) for species with larger Z will be significantly more negative than that of hydrogen, reflecting the stronger Coulombic attraction.

Bohr Model

The Bohr model is an early quantum mechanical model of the atom in which electrons move in circular orbits around the nucleus with quantized angular momentum. It provides a simple formula for the energy levels of an electron in a hydrogen-like atom. The model is useful for calculating the energy states of one-electron systems and explains the discrete spectral lines observed in atomic spectra.

Hydrogen-like Systems

In hydrogen-like systems, which possess only a single electron, the Bohr energy level formula can be directly applied. These systems include ions such as He? and Li²? where the nuclear charge increases the attractive force between the nucleus and the electron, leading to energy levels that scale as the square of the atomic number (Z²).

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