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Has the zeroth law of thermodynamics ever been proven? I know the zeroth law of thermodynamics probably follows from the second law of thermodynamics but according to The statistical nature of the 2nd Law of Thermodynamics, the second law of thermodynamics probably hasn't been proven to be absolute. I see why it could be possible to show that two substances in contact will form a solubility and thermal equilibrium with each other but I don't see how to show that when ever 3 immiscible liquids are in contact with one another, they will all form a thermal and solubility equilibrium with one another.

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It depends on what temperature is. A simple statistical argument could go like this.

  • the multiplicity of two systems is the product of their multiplicities
  • in thermodynamic equilibrium the multiplicity is at its maximum
  • $\frac{d}{dE}(\Omega_A \Omega_B) = 0$ for an infinitesimal transfer of internal energy from A to B
  • the fractional change of multiplicity with internal energies ($\beta = \frac{1}{\Omega} \frac{d\Omega}{dE}$) must be equal for both systems (if $\Omega_A$ goes down by 1 ‰, $\Omega_B$ goes up by 1 ‰).
  • identify $\beta = 1/kT$

If this is the case for systems A and B and for B and C, the fractional change of multiplicity with energy ($\beta$) should also be the same for A and C. At room temperature, it is about 4 % per meV, for any system.

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