$(i)$ The energy associated with the first orbit in the hydrogen atom is $-2.18 \times 10^{-18} \, J \, atom^{-1}$. What is the energy associated with the fifth orbit?
$(ii)$ Calculate the radius of Bohr's fifth orbit for the hydrogen atom.

$(i)$ The energy associated with the $n^{th}$ orbit of the hydrogen atom is given by $E_n = \frac{-2.18 \times 10^{-18}}{n^2} \, J \, atom^{-1}$.
For the fifth orbit $(n = 5)$:
$E_5 = \frac{-2.18 \times 10^{-18}}{5^2} = \frac{-2.18 \times 10^{-18}}{25} = -8.72 \times 10^{-20} \, J \, atom^{-1}$.
$(ii)$ The radius of Bohr's $n^{th}$ orbit for the hydrogen atom is given by $r_n = (0.0529 \, nm) \times n^2$.
For the fifth orbit $(n = 5)$:
$r_5 = 0.0529 \, nm \times 5^2 = 0.0529 \times 25 \, nm = 1.3225 \, nm$.