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.)
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.