Zeroth law of thermodynamics confusion

Question

I quote from Zemansky's "Heat & Thermodynamics";

"Imagine two systems A and B separated from each other by an adiabatic wall but each in contact with a third system C through diathermic walls, the whole assembly being surrounded by an adiabatic wall as shown in Fig. 1-2a. Experiment shows that the two systems will come to thermal equilibrium with the third and that no further change will occur if the adiabatic wall separating A & B is then replaced by a diathermic wall (Fig. 1-2b). If, instead of allowing both systems A & B to come to equilibrium with C ar the same time, we first have equilibrium between A & C and then equilibrium between B & C (the state of system C being the same in both cases), then, when A & B are brought into communication through a diathermic wall, they will be found to be in thermal equilibrium."

My question is;

What does he exactly mean by "the state of system C being the same in both cases"? Does C get connected to A first and then after reaching thermal equilibrium with A, gets connected to B? Or do we have like 2 identical systems to C and we connect A to one and B to the other?
If it means that C is just one system and we connect A first and then to B ( without C being in its initial condition before it was connected to A), then what I understand is that A & C will reach thermal equilibrium and will have same "temperature" (I know we still didnt define temperature yet but at  least based on how it "feels") so afterwards when B is connected to C, C being at the same temp as A now, the temp of C will change to the equilibrium temp with B. So A and B will have different temperatures, so how come will they be at thermal equilibrium when connected? (No change will occur in either A or B).

This is the figure he refers to
(https://i.stack.imgur.com/iiAe5.jpg)

Mitchell · Accepted Answer

Your arguments are correct.

If $A$ and $C$ are brought in equilibrium initially, then for $A$ and $B$ to be in equilibrium, B has to be in equilibrium with that state of $C$ which was initially in equilibrium with $A$, without any transfer of heat energy.

In other words, $B$ and $C$ must be at the same temperature before they are brought in contact (Here, the system $C$ is the one in equilibrium with $A$).

Therefore, the state of system $C$ ,which the same in both cases, refers to that state of C which is in equilibrium with $A$ $space$ and with $B$ (without any heat transfer). Only then $A$ and $B$ will be in thermal equilibrium.

Becharaj · Answer

your analysis is correct
The way the sentence in Case2 is written,there will be no thermal equilibrium between A, B and C
The sentence in the second case should read instead:
If, instead of allowing both systems A & B to come to equilibrium with C at the same time, we first bring A in thermal contact with C and then  B  with C (while keeping A in contact with C)), then, when A & B are brought into communication through a diathermic wall, they will be found to be in thermal equilibrium

Zeroth law of thermodynamics confusion

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