The Boltzmann constant (k or kB) is a proportionality factor that relates the average kinetic energy of an ideal gas particle to its absolute temperature. It is one of the seven SI base units, with a defined value of exactly 1.380694 x10-23 J/K, which is the same as 1.380694 x10-23 m2⋅kg/(s2⋅K). The Boltzmann constant is the ratio of the ideal gas constant (R) to Avogadro’s number (NA). Several chemistry and physics formulas use the Boltzmann constant, including variations of the ideal gas law, the definitions of the gas constant and kelvin unit, Boltzmann’s entropy formula, and Planck’s law of black-body radiation. The unit gets its name from Austrian physicist Ludwig Boltzmann.
The Boltzmann Constant in the Ideal Gas Law
The ideal gas law relates the pressure and volume of an ideal gas to its number of moles and absolute temperature:
PV = nRT
Here, P is pressure, V is volume, n is number of moles, R is the ideal gas constant, and T is absolute temperature.
But, the ideal gas constant depends on the Boltzmann constant and Avogadro’s number:
R = kNA
So, another way of writing the ideal gas law includes the Boltzmann constant:
PV = NkT
Here, N is the number of molecules of an ideal gas.
All of the ideal gas laws relate to the Boltzmann constant:
| Gas Law | Relationship to k |
|---|---|
| Ideal gas law | PV = NkT |
| Combined gas law | PV/T = k |
| Boyle’s law | PV = k |
| Charles’s law | V/T = k |
| Gay-Lussac’s law | P/T = k |
| Avogadro’s law | V/N = k |
Boltzmann Constant Values in Different Units
This table gives values of the Boltzmann constant in different units.
| Value of k | Unit |
|---|---|
| 1.380649×10−23 | J/K |
| 1.380649×10−23 | m2⋅kg/(s2⋅K) |
| 8.617333262×10−5 | eV/K |
| 2.083661912×1010 | Hz/K |
| 1.380649×10−16 | erg/K |
| 3.297623483×10−24 | cal/K |
| 1.832013046×10−24 | cal/°R |
| 5.657302466×10−24 | ft lb/°R |
| 0.695034800 | cm-1/K |
| 3.1668 ×10−6 | Eh/K (Eh = Hartree) |
| 0.001985875 | kcal/(mol⋅K) |
| 0.008314463 | kJ/(mol⋅K) |
| −228.5991672 | dB(W/K/Hz) |
The Other Boltzmann Constant and Planck’s Constant
There are actually two different Boltzmann constants. One is the Boltzmann constant (k) and the other is the Stefan-Boltzmann constant or Stefan constant (σ). The Stefan-Boltzmann constant is a proportionality constant in the Stefan-Boltzmann law, which states that total radiated intensity of a black body increases as temperature increases.
There is a relationship between the Stefan-Boltzmann constant, Boltzmann constant, and Planck’s constant (h). The Stefan-Boltzmann constant’s definition relies on values of the other two constants, pi, and the speed of light:
σ = 2πk4/15h3c3
References
- Bureau International des Poids et Mesures (2019). The International System of Units (SI) (9th ed.).
- Feynman, Richard (1970). The Feynman Lectures on Physics Vol I. Addison Wesley Longman. ISBN 978-0-201-02115-8.
- Newell, D. B.; et al. (2018). “The CODATA 2017 values of h, e, k, and N A for the revision of the SI”. Metrologia. 55 (1): L13. doi:10.1088/1681-7575/aa950a
- Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002). General Chemistry: Principles and Modern Applications (8th ed.). Prentice Hall. ISBN 0-13-014329-4.
- Pitre, L; et al. (2017). “New measurement of the Boltzmann constant by acoustic thermometry of helium-4 gas”. Metrologia. 54 (6): 856–873. doi:10.1088/1681-7575/aa7bf5