Planck's Quantum Theory: MDCAT Chemistry notes
Planck's Quantum Theory for MDCAT: quanta and photons, E = hν, energy vs frequency and wavelength, the spectrum and Planck's constant.
Background
Classical wave theory could not explain the radiation emitted by hot (black) bodies. In 1900 Max Planck proposed that energy is not emitted or absorbed continuously but in small discrete packets. This quantum theory later helped explain the photoelectric effect and Bohr's model of the atom.
Postulates of Planck's quantum theory
- Energy is emitted or absorbed discontinuously, in the form of small packets called quanta. A quantum of light is called a photon.
- The energy of each quantum is directly proportional to the frequency of the radiation: $E\propto\nu$, so $E=h\nu$.
- A body can emit or absorb energy only in whole-number multiples of a quantum: $E=nh\nu$, where $n=1,2,3,\dots$ So energy is quantised.
Key formulas
$$E=h\nu=\frac{hc}{\lambda}=hc\bar{\nu}$$
where $h$ is Planck's constant $=6.626\times10^{-34}$ J s, $c=3.0\times10^8$ m s$^{-1}$, $\lambda$ is wavelength and $\bar{\nu}=1/\lambda$ is wave number.
- Energy is directly proportional to frequency and to wave number.
- Energy is inversely proportional to wavelength.
- If frequency is doubled, $h\nu$ is doubled; if wavelength is doubled, energy is halved.
Energy across the electromagnetic spectrum
| Radiation | Wavelength | Frequency and energy per quantum |
|---|---|---|
| Radio waves | Longest | Lowest |
| Microwaves | Long | Low |
| Infrared | Longer than visible | Lower than visible |
| Visible (red to violet) | About 400 to 750 nm | Red lowest, violet highest |
| Ultraviolet | Shorter than visible | Higher than visible |
| X-rays, gamma rays | Shortest | Highest |
So among microwave, infrared, visible and ultraviolet, a quantum of ultraviolet carries the most energy.
Worked examples
Example 1. Energy of a photon of frequency $5.0\times10^{14}$ Hz: $E=6.626\times10^{-34}\times5.0\times10^{14}=3.3\times10^{-19}$ J.
Example 2. Photon A has wavelength 300 nm, photon B 600 nm. Since $E\propto1/\lambda$, A has twice the energy of B.
Link to de Broglie
Combining Planck's $E=h\nu$ with Einstein's $E=mc^2$ led de Broglie to propose that moving particles also have wave nature, with wavelength
$$\lambda=\frac{h}{mv}=\frac{h}{p}$$
Wavelength is inversely proportional to momentum, so it is noticeable only for tiny particles such as electrons.
Common MDCAT traps
- Energy is proportional to frequency, not wavelength or velocity.
- All electromagnetic radiation travels at the same speed $c$ in vacuum, so velocity does not decide energy.
- Planck's constant is $6.626\times10^{-34}$ J s, not $6.02\times10^{-34}$ (do not confuse with Avogadro's 6.02).
- de Broglie: $\lambda=h/mv$, not $mv/h$.
Quick revision
- Energy is quantised; quanta of light are photons.
- $E=h\nu=hc/\lambda$.
- Doubling frequency doubles energy.
- Ultraviolet quanta are more energetic than visible or infrared.
- $h$ has units of J s.