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In the Gaussian (CGS) system of units, the unit of electric charge (statcoulomb) is derived from the units of length, mass and time. Using Coulomb's law, we find that the dimension of electric charge is $$\text{[mass]}^{1/2} \text{ [length]}^{3/2} \text{ [time]}^{−1}$$

According to this answer the Kelvin is the unit of (thermodynamic) temperature used with the Gaussian system of units. However, since the temperature is related to the average translational kinetic energy of particles, I would like to know if it is possible to derive a unit of temperature (let's call it statkelvin) from the units of length, mass and time (in a way similar to the statcoulomb).

Would such a unit of (thermodynamic) temperature be a usable alternative to the Kelvin for scientific purposes? (if we disregard the historic advantage of the Kelvin)

What would be the physical law used to derive this unit of temperature? And what would be the resulting dimension of that statkelvin? (in terms of mass, length and time)

RalphS
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1 Answers1

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Just change Boltzmann's constant.

The equation you're referring to is probably a version of the equipartition theorem:

$$\langle K\rangle =\frac{n}{2} k_BT$$

where $\langle K\rangle$ is the expectation value of kinetic energy of a particle, and $n$ is the number of degrees of freedom of the system in question. Since $K$ is expressed in units of ergs, and you want to create a unit "statkelvin" for $T$, then simply let $k_B$ equal some value, with units of ergs/statkelvin. The value chosen for $k_B$ will set the "size" of the statkelvin.

In CGS, the units of the Boltzmann constant are ergs/K, and its value is such that measurements of kinetic energy in ergs and of temperature in K are compatible.

You could also get rid of Boltzmann's constant entirely, making it dimensionless and setting its value to 1. This would mean that temperature has the same units of energy, meaning that the statkelvin would be equivalent to the erg.

probably_someone
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  • Making the Boltzmann constant dimensionless is appealing. Do you think that not having a dedicated unit of temperature could prevent some common theory of physics to be used with this system of units? – RalphS Jan 06 '20 at 00:50
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    While it's not formally part of the Gaussian system of units, the last part here is the practical right answer—you measure temperature in ergs. Note, moreover, that by setting $k=1$, you can actually use any unit of every for $T$. Thus, it is common to see temperature quoted in eV, for example. – Buzz Jan 06 '20 at 00:57
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    @RalphS As long as other relevant constants (e.g. the Stefan-Boltzmann constant, specific heats, etc.) have the correct values in terms of ergs rather than K, everything should work the same. – probably_someone Jan 06 '20 at 00:58
  • @Buzz there is a part missing in your interesting comment ("you can actually use any unit of every [?] for T") . @ probably_someone: Thanks for the clarification. – RalphS Jan 06 '20 at 01:13
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    @RalphS "every" should be "energy". – probably_someone Jan 06 '20 at 01:31
  • @RalphS Yes, it should be "energy." – Buzz Jan 06 '20 at 01:42
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    @RalphS Making the unit of temperature the same as the unit of energy also means that entropy and specific heat are both dimensionless, which is interesting. – probably_someone Jan 06 '20 at 01:58
  • Probably_someone, could you incorporate informations about entropy in your answer? The last paragraph of this answer suggests to replace K with J/bits (which makes things very clear and explicite). Maybe the best unit of temperature is erg/nat, with nat defined here? – RalphS Jan 07 '20 at 16:51