Chemistry

Problem

An ion consisting of a helium nucleus (charge

+2e

) with a single orbiting electron behaves similarly to hydrogen but with a larger nuclear charge. (a) Generalize the expression for the allowed radii of a one-electron atom from hydrogen (

Z=1

) to an arbitrary nuclear charge

Z

. (b) Use your result from part (a) to obtain the formula for the allowed radii

r_{n,\text{He}^+}

of the singly ionized helium atom (

Z=2

). (c) Calculate the numerical value of the ground-state radius (

n=1

) for He

^+

and compare it to that of hydrogen by computing the ratio

r_{1,\text{He}^+}/r_{1,\text{H}}

. Express the ratio to two significant figures. (d) Based on your findings, discuss qualitatively how the increased nuclear charge affects the spacing between energy levels compared with hydrogen.

Intuitive explanation

This is about scaling the hydrogen atom model to a nucleus with different charge. The key insight is that all the energy and radius formulas for hydrogen have a $Z$ factor (nuclear charge in units of $e$ ) built in—so you can directly substitute $Z = 2$ for helium instead of rederiving everything. This lets you predict how ionization energy, orbital radii, and spectral lines change when you increase nuclear pull on the electron.

Intuitive explanation

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