In Bright-Line Spectra you saw that each element glows in its own set of colors. In 1913 Niels Bohr explained hydrogen's colors: the electron can only have certain energies, and light comes out only when it drops from one to another, as a photon.
Key idea
Hydrogen's allowed energies are $$E_n = -\frac{13.6\ \text{eV}}{n^{2}}, \quad n = 1, 2, 3, \dots$$ A jump between levels emits or absorbs one photon with $E_{\text{photon}} = E_{\text{upper}} - E_{\text{lower}}$, so $\lambda = \dfrac{1240\ \text{eV·nm}}{E_{\text{photon}}}$.
Why it works
Why negative energies? Zero is set for an electron just barely free of the proton. A bound electron has less energy than that, so its energy is negative, and $13.6\ \text{eV}$ is what it takes to free it from the lowest level (the ionization energy). Bohr's rule that only certain orbits are allowed was later explained by the electron's wave nature: only orbits that fit whole numbers of its wavelength survive.
Common mistake
Two slips. (1) Thinking the electron gives off light the whole time it circles. In an allowed level it does not radiate; light comes only from a jump. (2) Counting only the jumps that end on $n = 1$. From $n = 4$ the electron can go to 3, 2 or 1, and from each of those down again: $4\to3$, $4\to2$, $4\to1$, $3\to2$, $3\to1$, $2\to1$, six lines in all. In general $\frac{n(n-1)}{2}$.
Worked example: The red line
Find the photon energy and wavelength for hydrogen's jump from $n = 3$ to $n = 2$.
- $$\begin{aligned} E_3 &= -\frac{13.6}{9} = -1.51\ \text{eV} \\ E_2 &= -\frac{13.6}{4} = -3.40\ \text{eV} \end{aligned}$$
- $$E_{\text{photon}} = -1.51 - (-3.40) = 1.89\ \text{eV}.$$
- $\lambda = \dfrac{1240}{1.89} = 656\ \text{nm}$: red, the brightest line in hydrogen's visible spectrum.
Answer: $1.89\ \text{eV}$, $656\ \text{nm}$
Worked example: Absorbing a photon
Hydrogen atoms in the ground state are hit by photons of $10.2\ \text{eV}$ and of $11.0\ \text{eV}$. Which are absorbed?
- From $n = 1$: $$E_2 - E_1 = -3.40 - (-13.6) = 10.2\ \text{eV}.$$ A $10.2\ \text{eV}$ photon fits exactly and is absorbed.
- $E_3 - E_1 = 12.1\ \text{eV}$. $11.0\ \text{eV}$ matches no jump, and it is below the $13.6\ \text{eV}$ needed to ionize, so it passes by.
- An atom takes whole photons only, and only at the exact energy of a jump.
Answer: Only the $10.2\ \text{eV}$ photon
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