An electron in a rigid box absorbs light. The longest wavelength in the absorption spectrum is$600nm$. How long is the box?

We have to given that the longest wavelength in the absorption spectrum is $600nm$.

We need to find the length of the box.

Absorbing the photon with the longest wavelength will result in the transition of the electron from the ground state to the $(n=2)$state, and that's because the photon with the longest wavelength is the one with the smallest energy, and the smallest energy difference between the two states happens between $n=1$and $n=2$states. The energy of levels of an electron in a rigid box is given by:

$E_n=n^2\frac{h^2}{8m{L}^2}$

in order for the transition to take place, the incident photon must have an energy equal to the energy difference between $n=1$and $n=2$. Since,

$\begin{array}{c}hf=\frac{hc}{\lambda }=E_2-E_1\\ \frac{hc}{\lambda }=\frac{4h^2}{8m{L}^2}-\frac{h^2}{8m{L}^2}\\ \frac{hc}{\lambda }=\frac{3h^2}{8m{L}^2}\end{array}$

the equation to isolate and substitute the numerical values of the different variables.

$\begin{array}{rr}L=\sqrt{\frac{3h\lambda }{8mc}}& =\sqrt{\frac{3\left(6.626\times {10}^{-34}\text{J}\cdot \text{s}\right)\left(600\times {10}^{-9}\text{m}\right)}{8\left(9.11\times {10}^{-31}\text{kg}\right)\left(3.0\times {10}^8\text{m}/\text{s}\right)}}\\ L& =7.39\times {10}^{-10}\text{m}=0.739\text{nm}\end{array}$