Answer
$L = 3.5\times 10^{-10}~m$
Work Step by Step
We can find the energy difference between states when light of wavelength 80.78 nm is emitted:
$E = \frac{hc}{\lambda}$
$E = \frac{(6.626\times 10^{-34}~J~s)(3.0\times 10^8~m/s)}{80.78\times 10^{-9}~m}$
$E = 2.46076\times 10^{-18}~J$
Light with the longest wavelength is absorbed when the electron jumps from the first excited state where $n=2$ to the next state $n=3$.
We can use the expression for the energy difference to find $L$:
$E_3-E_2 = 2.46076\times 10^{-18}~J$
$(\frac{h^2}{8m~L^2})~(3)^2 - (\frac{h^2}{8m~L^2})~(2)^2 = 2.46076\times 10^{-18}~J$
$\frac{5~h^2}{8m~L^2} = 2.46076\times 10^{-18}~J$
$L^2 = \frac{5~h^2}{(8m)~(2.46076\times 10^{-18}~J)}$
$L = \sqrt{\frac{5~h^2}{(8m)~(2.46076\times 10^{-18}~J)}}$
$L = \sqrt{\frac{(5)~(6.626\times 10^{-34}~J~s)^2}{(8)(9.109\times 10^{-31}~kg)~(2.46076\times 10^{-18}~J)}}$
$L = 3.5\times 10^{-10}~m$
