Bohr radius formula

For a hydrogen-like species, the radius of the $n$th Bohr orbit is $r_n = a_0\frac{n^2}{Z}$ where:

• $a_0$ = radius of first Bohr orbit of hydrogen
• $n$ = orbit number
• $Z$ = atomic number

Given data

• First Bohr orbit radius of hydrogen:
  $a_0 = 0.6\,\text{\AA}$
• For $\text{He}^+$, atomic number: $Z=2$
• Required orbit: third orbit, so $n=3$

Calculate radius of third orbit of $\text{He}^+$

$r_3 = a_0\frac{3^2}{2} = 0.6\times \frac{9}{2}\,\text{\AA}$

$r_3 = 0.6\times 4.5 = 2.7\,\text{\AA}$

Convert \AA to picometer

$1\,\text{\AA} = 100\,\text{pm}$

Therefore, $2.7\,\text{\AA} = 2.7\times 100 = 270\,\text{pm}$

Final answer

The radius of the third Bohr orbit of $\text{He}^+$ is $\boxed{270\,\text{pm}}$