Q.Two bodies $A$ and $B$ are each separately found to be in thermal equilibrium with a third body $C$. State, with reasoning based on the Zeroth law of thermodynamics, what this tells you about the temperatures of $A$ and $B$.

Given: $A$ is in thermal equilibrium with $C$, and $B$ is in thermal equilibrium with $C$, separately.

The Zeroth law of thermodynamics states that if two systems are each separately in thermal equilibrium with a third system, then those two systems are also in thermal equilibrium with each other. Taking $C$ to play the role of the "third system" in this statement, the law directly guarantees that $A$ and $B$ must be in thermal equilibrium with each other as well, even though $A$ and $B$ were never placed in direct contact.

Since two bodies in thermal equilibrium are, by definition, at the same temperature, it follows that $A$ and $B$ are at the same temperature as each other -- and, in fact, at the same temperature as $C$ too.

Recognise $C$ as the common third system in the Zeroth law's statement, and apply the law directly.

• Thinking the Zeroth law requires $A$ and $B$ to be placed in direct contact before anything can be concluded about them -- the whole point of the law is that this is not necessary.
• Confusing the Zeroth law (about the transitivity of thermal equilibrium) with the first law (about energy conservation).