When a hydrogen atom is bombarded, the atom may be raised into a higher energy state. As the excited electron falls back to the lower energy levels, light is emitted. What are the three longestwavelength spectral lines emitted by the hydrogen atom as it returns to the $n=1$ state from higher energy states? Give your answers to three significant figures.
We are interested in the following transitions (see Fig. $43-1$ ):
$$
\begin{array}{ll}
n=2 \rightarrow n=1: & \Delta \mathrm{E}_{2,1}=-3.4-(-13.6)=10.2 \mathrm{eV} \\
n=3 \rightarrow n=1: & \Delta \mathrm{E}_{3,1}=-1.5-(-13.6)=12.1 \mathrm{eV} \\
n=4 \rightarrow n=1: & \Delta \mathrm{E}_{4,1}=-0.85-(-13.6)=12.8 \mathrm{eV}
\end{array}
$$
To find the corresponding wavelengths, proceed as in Problem 43.1, or use $\Delta \mathrm{E}=h f=h c / \lambda$. For example, for the $n=2$ to $n=1$ transition,
$$
\lambda=\frac{h c}{\Delta E_{2,1}}=\frac{\left(6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\right)\left(2.998 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)}{(10.2 \mathrm{eV})\left(1.60 \times 10^{-19} \mathrm{~J} / \mathrm{eV}\right)}=1.22 \mathrm{~nm}
$$
The other lines are found in the same way to be $102 \mathrm{~nm}$ and $96.9$ $\mathrm{nm}$. These are the first three lines of the Lyman series.