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I am reading a book regarding the work of Anders Celcius that states the following -
By definition, any object at 100ºC must be in thermal equilibrium with boiling water.
I am confused by this statement.
Surely in thermal equilibrium, it must be between two different objects with no net flow of thermal energy.
However, an object at 100ºC does not have to have boiling water to be at that temperature. I may just be confused.
Can someone explain on this statement please?
Thanks.
The book does not imply that boiling water is necessary to heat an object to a temperature of $100^{\circ}$ -- there are several other methods by which an object can be heated to a desired temperature.
Rather, the book merely states that should we place an object with a uniform temperature of $100^{\circ}$ C in a body of boiling water, that the object would be in thermal equilibrium with that water, as it should be.
System in equilibrium
It is a fact of experience that when two systems come into thermal contact (energy is transferred with a mechanism different from work, Heat) generally changes happen to both of them.
However if we isolate them we will see that eventually they reach a state in which no further change is perceptible, no matter how long one waits.
For simple systems meaning that they have no magnetic or electrical properties their equilibrium state can be completely described by their volume V and pressure P.
Systems in equilibrium with each other
If we have two systems $S_1$ and $S_2$ in thermal contact and we consider the two as one system $S_{12}$, then if the system $S_{12}$ is in equilibrium then they two systems $S_1$ and $S_2$ are said to be in equilibrium with each other.
The zeroth law states that if two systems A and C are in thermal equilibrium with a third one B(which we later will see that can be defined as the thermometer) then they are in equilibrium with one another as well.
Suppose we have three systems A,B and C that are in equilibrium states.
$$A(P_A,V_A),B(P_B,V_B),C(P_C,V_C)$$
The condition under which A and B are in equilibrium may be expressed bu the equation: $$F_1(P_A,V_A,P_B,V_B)=0 \Rightarrow P_B = f_1(P_A,V_A,V_B) $$
The same applies for the equilibrium of systems C and B.
$$P_B = f_2(P_C,V_C,V_B) $$
From the two above equations turns out that:
$$f_1(P_A,V_A,V_B)=f_2(P_C,V_C,V_B) \Rightarrow G(P_A,V_A,V_B,P_C,V_C)=0 $$
From the zeroth law turns out that if A and C are separately in equilibrium with B the they are with each other as well. Therefore:
$$ F_3(V_A,P_A,P_C,V_C) $$
Since the last two equations are to be equivalent, then the implicit function G is independent from $V_B$. Therefore going backwards:
$$ φ_1(P_A,V_A)=φ_2(P_C,V_C)$$
Therefore we can define a function $$ φ(P,V)=θ$$ Which we call empirical temperature and it has the property that systems that take the same value are in equilibrium with each other.
We have defined that water boils at $100^oC$. Therefore according to the property of the emperical temperature function every system at $100^oC$ is in equilibrium with boiling water.
