Zeroth Law And Thermal Equilibrium

The Zeroth Law of Thermodynamics states that if two systems are each separately in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. This law is what allows temperature to be defined as a consistent property: thermal equilibrium is a transitive relationship. Here A is in equilibrium with C and B is in equilibrium with C, so by the Zeroth Law A and B share the same temperature even though they were never touched together. Equal heat capacities is incorrect because heat capacity depends on mass and material, not on temperature alone. Equal internal energy is incorrect because internal energy also depends on the amount and nature of the substance, so two bodies at the same temperature can store very different energies. The claim that nothing can be concluded is wrong precisely because the Zeroth Law removes the need for direct contact. As a consistency check, temperature is the unique quantity that becomes equal at equilibrium, confirming the conclusion.