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Assuming I got the below facts correct:

  • There are 7 SI units - mass (kg), length (m), time (s), current (A), temperature (K), luminous intensity (cd), amount of substance (mol).
  • Thermal energy is essentially nothing but kinetic energy.
  • Temperature is a measure of thermal energy (hence measure of kinetic energy).
  • Kinetic energy can be expressed in $[M^1 L^2 T^{-2}]$ (a function of mass, length and time).

Per these facts, is kelvin (SI unit for temperature) also a function of mass, length and time. If so, why do we need a separate SI unit specially for temperature? Am I missing something?

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    $\begingroup$ physics.stackexchange.com/questions/166568 and physics.stackexchange.com/questions/231017 may already cover your question. $\endgroup$
    – kaylimekay
    Jan 9, 2021 at 12:16
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    $\begingroup$ Does this answer your question? Is the Boltzmann constant really that important? $\endgroup$
    – Alchimista
    Jan 9, 2021 at 12:53
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    $\begingroup$ I don't think this question is a duplicate of the linked question, since on the surface one is asking about the units of $T$ and the other is asking about the value of $k_B$. (The OP might be able to answer their question by reading that duplicate, though.) $\endgroup$ Jan 11, 2021 at 16:54
  • $\begingroup$ Though the answer / reasoning might be the same, but the questions were different. I am a physics noob and am hearing Boltzman constant for the first time. I don't care about the constant, my question was more to understand the reason for choosing a separate SI unit for temperature. @MichaelSeifert $\endgroup$ Jan 17, 2021 at 16:56

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Good question. In any system of units the proportionality factor between the unit of temperature and the unit of energy is the Boltzmann constant. If we set the Boltzmann constant to $1$ - as it is in the system of Planck units - then in effect we are measuring temperature and energy in the same units.

However, using Planck units in everyday life would create practical difficulties, because the Planck temperature is so very large - the temperature of a cup of tea is approximately $2 \times 10^{-30}$ Planck units. You can think of this as being a measure of how small atoms are - an everyday object contains a large number of atoms, so its average thermal energy per atom is very small compared to its macroscopic kinetic energy, potential energy etc. So to produce a set of units in which both typical macroscopic energies and temperatures have reasonable values (not too large, not too small) you need to set the Boltzmann constant to a very small value.

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  • $\begingroup$ To be fair, you could use a system of units where $k = 1$ while not going all the way to Planck units. For example, if you used a version of SI where $k = 1$, the temperature of a cup of tea would be $5 \times 10^{-21}$ J. That's not nearly as numerically small as it is in Planck units, but your broader point of course remains true: it's inconveniently small. $\endgroup$ Jun 14, 2021 at 19:57
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Temperature is not energy. Temperature is energy per unit entropy. Entropy, most naturally measured in bits, or bytes, or digits, or nepers, or some other quantity that's basically a logarithm, is a measure of what we don't know about a system. In the context of thermodynamics, and therefore what we think of as temperature, this means all the ways the atoms and molecules and assorted what-not in a system can vibrate and still give us the system we observe. Temperature, therefore, is the amount of energy each of those "degrees of freedom" hold, averaged over the lot. The point of the matter is that, where temperature is concerned, what we really care about is the entropy, which is independent of the LTM system.

The problem is, logarithms, like counts, are fundamentally unitless. To properly keep track of entropy, we have to give it a unit. SI gives it the unit joule per kelvin. Or rather, it defines a unit for temperature (kelvin), then divides it into the unit for energy (joule - itself derived) to give units of entropy. SI goes this way because it is actually quite easy to measure temperature (thermometers exist) but almost impossible to measure entropy directly, and SI is fundamentally an engineer's system, not a scientist's. There do exist systems where it is easier to measure entropy - mostly in information science, google "Shannon entropy" when you have the time - and here we see entropy measured in bits (or whatever is most apropos) and temperature measured in units of energy per bit.

So, why do we care about temperature slash entropy in thermodynamics? Because objects in equilibrium have the same temperature, but not necessarily the same energy. A larger thing (which has more atoms, and therefore more ways those atoms can vibrate) will have more energy than a smaller thing at the same temperature. And since thermodynamics is all about how things attain and maintain equilibrium, it helps to know where that equilibrium point actually is, which energy can't help us with - at least not directly. Enter the thermometer and with it, temperature and entropy.

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