(a) Find the Bohr radius of an electron in the n=3 orbit of...

(a) Find the Bohr radius of an electron in the $n=3$ orbit of an H atom (see Problem 7.76) . (b) What is the energy (in J) of the atom in part (a)? (c) What is the energy of an $\mathrm{Li}^{2+}$ ion with its electron in the $n=3$ orbit? (d) Why are the answers to parts (b) and (c) different?

Key Concepts

Bohr Model

The Bohr model conceptualizes the atom as having electrons in fixed circular orbits around a nucleus, where only certain discrete orbits corresponding to specific energy levels are allowed. The model establishes that the radius of an electron’s orbit is proportional to the square of the principal quantum number n, and that the energy levels are inversely proportional to n², providing a framework for calculating atomic properties such as orbit radii and energy values.

Quantized Energy Levels

In quantum mechanics, electrons in an atom can only occupy specific energy levels rather than a continuous range. In the Bohr model, the energy associated with each orbit is quantized and given by an inverse square dependence on the principal quantum number n. For hydrogen-like atoms, this energy expression is further scaled by the square of the nuclear charge, leading to distinct energy values for different atomic species and energy levels.

Hydrogen-like Ion

A hydrogen-like ion is any ion that contains only a single electron, making its behavior similar to that of a hydrogen atom but with a different nuclear charge. The energy levels and orbital radii in such ions are influenced by this nuclear charge, Z, where the energy levels are proportional to -Z²/n², and the radius scales accordingly. This adjustment accounts for the stronger or weaker nuclear attraction relative to hydrogen.

Nuclear Charge

Nuclear charge refers to the total positive charge of the nucleus, determined by the number of protons. In the context of atomic structure and the Bohr model, a higher nuclear charge increases the electrostatic attraction between the nucleus and the electron, leading to smaller orbit radii and more tightly bound electrons. This factor is critical in explaining the differences in energy values for electrons in hydrogen versus hydrogen-like ions with greater nuclear charge.

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