Question

Find the radius and energy of a He⁺ ion in the states (a) n = 1, (b) n = 4 and (c) n = 10.

Answer 1

For He⁺ ion,
Atomic number, Z = 2
For hydrogen like ions, radius $\left(r\right)$ of the nth state is given by
$r=\frac{0.53n^2}{Z}$Å
Here,
Z = Atomic number of ions
n = Quantum number of the state

Energy $\left(E\right)$ of the nth state is given by
E_n = $-\frac{13.6Z^2}{n^2}$
(a)
For n = 1,
Radius, $$\begin{gathered} r=\frac{0.53\times {\left(1\right)}^2}{2} \\ =0.265{\mathrm{A}}^0 \end{gathered}$$ Å

Energy, E_n = $\frac{-13.6\times 4}{1}$
= $-$ 54.4 eV

(b)
For n = 4,
Radius, $r=\frac{0.53\times 16}{2}=4.24\mathrm{\AA }$
Energy, $E=\frac{-13.6\times 4}{16}=-3.4e\mathrm{V}$
(c)
For n = 10,
Radius, $r=\frac{0.53\times 100}{2}$
= 26.5 Å
Energy, $E=\frac{-13.6\times 4}{100}=-0.544\mathrm{eV}$