Use the definition of temperature to prove the zeroth law of thermodynamics, which says that if system A is in thermal equilibrium with system B, and system B is in thermal equilibrium with system C, then system A is in thermal equilibrium with system C. (If this exercise seems totally pointless to you, you’re in good company: Everyone considered this “law” to be completely obvious until 1931, when Ralph Fowler pointed out that it was an unstated assumption of classical thermodynamics.)

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The definition of temperature in terms of entropy is

System in thermal equilibrium have equal slopes in their entropy-versus-energy graph and the temperature is same. The zeroth law of thermodynamics states that if a system A is separately in thermal equilibrium with two other systems B and C then B and C are in thermal equilibrium with each other.From the definition of temperature above, since any two systems in thermal equilibrium have the same values of the system B and C must both have the same slope as system A. So they all have the same slopes each other.