Photons, Planck’s Equation, and Energy Quantization
Planck’s equation, E = hν = hc/λ, states that electromagnetic radiation is emitted and absorbed in discrete packets of energy called quanta or photons, where h = 6.626 × 10−34 J·s is Planck’s constant. In brief: a photon of green light (520 nm) has an energy of 3.82 × 10−19 J, and a mole of such photons carries 230 kJ, which is enough to break many chemical bonds.
The idea that energy comes in quanta started quantum theory, and it explains the colors of heated objects, the photoelectric effect and the atomic spectra. Individual photons carry very little energy, but their energy depends on the frequency, and ultraviolet photons are energetic enough to damage molecules, whereas radio photons are not. This article explains Planck’s equation, shows how to calculate photon energies and photon numbers, gives a table across the spectrum and explains the origin of the quantum hypothesis.
What is Planck’s equation?
The energy of one photon is proportional to the frequency of the radiation:
E = hν = hc / λ
where h = 6.626 × 10−34 J·s, c = 3.00 × 108 m/s, and λ is in meters. A photon of short wavelength (high frequency) has more energy than one of long wavelength. The energy of one mole of photons is obtained by multiplying by Avogadro’s number: Em = NAhν = NAhc/λ. A useful shortcut: E (kJ/mol) ≈ 119,700 / λ (nm).
What does energy quantization mean?
Classical physics assumed that energy can change continuously. Planck’s hypothesis was that the energy exchanged between matter and radiation can only have values that are whole-number multiples of hν: E = nhν, with n = 1, 2, 3, … The same applies to the energy of electrons in atoms and molecules, which can have only certain values (energy levels). Because h is so small, the steps are not noticeable in everyday objects: a swing with a period of 1 s and an energy of 1 J has a quantum of only hν = 6.6 × 10−34 J, a step 10−33 of the total energy.
Photon energies across the spectrum
| Radiation | Wavelength | Frequency (Hz) | Energy per photon (J) | Energy per mole (kJ/mol) |
|---|---|---|---|---|
| FM radio | 3.00 m | 1.00 × 108 | 6.6 × 10−26 | 4.0 × 10−5 |
| Microwave oven | 12.2 cm | 2.45 × 109 | 1.6 × 10−24 | 9.8 × 10−4 |
| Infrared | 10 µm | 3.0 × 1013 | 2.0 × 10−20 | 12.0 |
| Red light | 700 nm | 4.3 × 1014 | 2.8 × 10−19 | 171 |
| Violet light | 400 nm | 7.5 × 1014 | 5.0 × 10−19 | 299 |
| Ultraviolet | 254 nm | 1.2 × 1015 | 7.8 × 10−19 | 471 |
| X-ray | 0.10 nm | 3.0 × 1018 | 2.0 × 10−15 | 1.2 × 106 |
Typical bond energies are 150–1000 kJ/mol (for example C–C 346, O–H 463 kJ/mol), so ultraviolet photons (about 300–1200 kJ/mol) can break bonds, which is why UV light damages DNA and causes sunburn, while infrared and microwave photons can only excite vibrations and rotations. Radio and microwave photons are far too weak to break bonds.
Worked examples
Example 1: energy of a photon of green light
E = hc/λ = (6.626 × 10−34 J·s)(3.00 × 108 m/s) / (520 × 10−9 m) = 1.988 × 10−25 / 5.20 × 10−7 = 3.82 × 10−19 J. Per mole: 3.82 × 10−19 × 6.022 × 1023 = 2.30 × 105 J/mol = 230 kJ/mol.
Example 2: number of photons from a laser
A green laser pointer emits 5.00 mW of light at 532 nm. The energy of one photon is E = 1.988 × 10−25 / 532 × 10−9 = 3.74 × 10−19 J. A power of 5.00 × 10−3 J/s corresponds to 5.00 × 10−3 / 3.74 × 10−19 = 1.34 × 1016 photons per second.
Example 3: wavelength needed to break a bond
What is the longest wavelength of light that can break the Cl–Cl bond (243 kJ/mol)? The energy per molecule is 243,000 / 6.022 × 1023 = 4.04 × 10−19 J. λ = hc/E = 1.988 × 10−25 / 4.04 × 10−19 = 4.93 × 10−7 m = 493 nm, which is in the blue-green region. This is why chlorine reacts with hydrogen in sunlight.
Example 4: energy of photons from a sodium lamp
The yellow sodium doublet is at 589 nm. E = 1.988 × 10−25 / 589 × 10−9 = 3.38 × 10−19 J, or 3.38 × 10−19 / 1.602 × 10−19 = 2.11 eV per photon.
Example 5: frequency from energy
A photon has an energy of 4.00 × 10−19 J. Then ν = E/h = 4.00 × 10−19 / 6.626 × 10−34 = 6.04 × 1014 Hz, and λ = c/ν = 497 nm.
Why is this important in chemistry?
- Photochemistry: reactions such as photosynthesis, the formation of ozone and the chlorination of methane are started by the absorption of photons.
- Spectroscopy: the lines in a spectrum correspond to energy differences ΔE = hν between quantized levels.
- Color: a substance has a color when it absorbs photons of a certain energy.
- Technology: LEDs, lasers, solar cells and photodetectors rely on the quantization of energy.
How did Planck discover the quantum?
At the end of the 19th century, the classical theory predicted that a hot body should emit an infinite amount of ultraviolet radiation, the “ultraviolet catastrophe”, in conflict with the measured spectrum of blackbody radiation. In December 1900 Max Planck found a formula that fitted the data, and he derived it by assuming that the oscillators in the walls of the body can only have energies that are multiples of hν. He regarded this as a mathematical trick, and not as a feature of light itself. In 1905 Albert Einstein proposed that light itself consists of quanta, which explained the photoelectric effect. Planck received the Nobel Prize in Physics in 1918.
Which mistakes are common?
- Using nanometers in E = hc/λ: convert to meters.
- Confusing energy per photon and per mole: multiply by NA and convert to kJ.
- Assuming that a brighter light has more energetic photons: brightness is the number of photons, and energy per photon depends on the frequency.
- Thinking that microwaves break bonds: their photons are far too weak.
- Mixing up h and ħ: ħ = h/2π = 1.055 × 10−34 J·s.