An electron is in the third excited state in a hydrogen atom. It undergoes transitions to the lower energy states.

(a)

What is the maximum number of spectral lines that can be emitted ?

(b)

Calculate the minimum wavelength of the spectral lines emitted.

$\displaystyle 6$ (b)\begin{aligned} & h v=\frac{h c}{\lambda}=E_{$\displaystyle 4$}-E_{$\displaystyle 1$} & =-$\displaystyle 0.85$+$\displaystyle 13.6$ & =+$\displaystyle 12.75$ \mathrm{ev} & \frac{h c}{\lambda}=$\displaystyle 12.75$ \times $\displaystyle 1.6$ \times $\displaystyle 10$^{-$\displaystyle 19$} & \lambda=\frac{$\displaystyle 6.63$ \times $\displaystyle 10$^{-$\displaystyle 34$} \times $\displaystyle 3$ \times $\displaystyle 10$^{$\displaystyle 8$}}{$\displaystyle 12.75$ \times $\displaystyle 1.6$ \times $\displaystyle 10$^{-$\displaystyle 19$}} & \lambda=$\displaystyle 975$ \mathrm{~nm}=$\displaystyle 9.75$ \times $\displaystyle 10$^{-$\displaystyle 11$} \mathrm{~m} \end{aligned}