38.7.3 Hydrogen-Like Ions

An ion with a single electron orbiting $Z$ protons in the nucleus is called a hydrogen-like ion. $Z$ is the atomic number and describes the number of protons in the nucleus. ${He}^{+}$ , with one electron circling a $Z = 2$ nucleus, and ${Li}^{+ +}$ , with one electron and a $Z = 3$ nucleus, are hydrogen-like ions. So is $U^{+ 91}$ , with one lonely electron orbiting a $Z = 92$ uranium nucleus.

Any hydrogen-like ion is simply a variation on the Bohr hydrogen atom. The only difference between a hydrogen-like ion and neutral hydrogen is that the potential energy $- e^{2} / 4 π ϵ_{0} r$ becomes, instead, $- Z e^{2} / 4 π ϵ_{0} r$ . Hydrogen itself is the $Z = 1$ case. If we repeat the analysis of the previous sections with this one change, we find:

\begin{array}{l} r_{n} = \frac{n^{2} a_{B}}{Z} & E_{n} = - \frac{13.60 Z^{2} eV}{n^{2}} \\ v_{n} = Z \frac{v_{1}}{n} & λ_{0} = \frac{91.18 nm}{Z^{2}} \end{array}

(38.41)

As the nuclear charge increases, the electron moves into a smaller-diameter, higher-speed orbit. Its ionization energy $| E_{1} |$ increases significantly, and its spectrum shifts to shorter wavelengths. Table 38.3 compares the ground-state atomic diameter $2 r_{1}$ , the ionization energy $| E_{1} |$ , and the first wavelength $3 \to 2$ in the Balmer series for hydrogen and the first two hydrogen-like ions.

TABLE 38.3 Comparison of hydrogen-like ions with $Z = 1, 2$ , and $3$

Ion	Diameter $2 r_{1}$	Ionization energy $\| E_{1} \|$	Wavelength of $3 \to 2$
H $(Z = 1)$	0.106 nm	13.6 eV	656 nm
${He}^{+} (Z = 2)$	0.053 nm	54.4 eV	164 nm
${Li}^{+ +} (Z = 3)$	0.035 nm	122.4 eV	73 nm