In Higher Frequency, More Energy you met the idea that light comes in packets. Here we put numbers on it. Light is an electromagnetic wave, but it delivers its energy in lumps called photons, and the size of each lump depends only on the frequency.
Key idea
$$E = hf = \frac{hc}{\lambda}, \qquad h = 6.626 \times 10^{-34}\ \text{J·s}.$$ In atomic units: $1\ \text{eV} = 1.602 \times 10^{-19}\ \text{J}$ (the energy one electron gains crossing $1\ \text{V}$, as in Electric Potential and Voltage), and $hc = 1240\ \text{eV·nm}$, so $E\,(\text{eV}) = \dfrac{1240}{\lambda\,(\text{nm})}$.
Why it works
Why is $hc = 1240\ \text{eV·nm}$ so handy? Work it once: $$\begin{aligned} hc &= (6.626 \times 10^{-34})(3.00 \times 10^{8}) \\ &= 1.988 \times 10^{-25}\ \text{J·m}. \end{aligned}$$ Divide by $1.602 \times 10^{-19}\ \text{J/eV}$ and multiply by $10^{9}\ \text{nm/m}$: $1240\ \text{eV·nm}$. Visible light, $400$ to $700\ \text{nm}$, comes out to $3.1$ to $1.8\ \text{eV}$ per photon, the same size as the energies of electrons in atoms and molecules. That is why light can drive chemistry and vision.
Common mistake
Thinking brighter light has more energetic photons. Brightness is how many photons arrive each second; the energy of each one is set by the color (frequency). A dim blue light has more energetic photons than a bright red one. Check: brightness appears nowhere in $E = hf$. That is why old photo darkrooms could use red lamps: a red photon carries too little energy to change the light-sensitive chemicals in photo paper, however many of them arrive.
Worked example: A green photon
Find the energy of a $500\ \text{nm}$ photon in eV and in joules.
- $E = \dfrac{1240\ \text{eV·nm}}{500\ \text{nm}} = 2.48\ \text{eV}$.
- $$E = (2.48)(1.602 \times 10^{-19}) = 3.97 \times 10^{-19}\ \text{J}.$$
Answer: $2.48\ \text{eV} = 3.97 \times 10^{-19}\ \text{J}$
Worked example: Photons from a laser pointer
A red laser pointer puts out $1.0\ \text{mW}$ at $650\ \text{nm}$. How many photons leave it each second?
- One photon: $E = \dfrac{1240}{650} = 1.91\ \text{eV} = 3.06 \times 10^{-19}\ \text{J}$.
- Photons per second: $\dfrac{1.0 \times 10^{-3}\ \text{J/s}}{3.06 \times 10^{-19}\ \text{J}} = 3.3 \times 10^{15}$.
- So many that light looks smooth and continuous.
Answer: About $3.3 \times 10^{15}$ per second