1.2: The Zeroth Law of Thermodynamics
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This can be stated as follows.
If two bodies A and B are in thermal equilibrium with a third body C, then they are in thermal equilibrium with each other.
Consequences of the Zeroth Law
Thermal equilibrium of two bodies will mean a restrictive relation between the thermodynamic coordinates of the first body and those of the second body. In other words, thermal equilibrium means that
F(→xA,→xB)=0
if A and B are in thermal equilibrium. Thus the zeroth law states that
F(→xA,→xB)=0F(→xB,→xC)=0}⇒F(→xA,→xC)=0
This is possible if and only if the relations are of the form
F(→xA,→xB)=t(→xA)−t(→xB)=0
This means that, for any body, there exists a function t(→x) of the thermodynamic coordinates →x, such that equality of t for two bodies implies that the bodies are in thermal equilibrium. The function t is not uniquely defined. Any single-valued function of t, say, T(t) will also satisfy the conditions for equilibrium, since
tA=tB⇒TA=TB
The function t(→x) is called the empirical temperature. This is the temperature measured by gas thermometers.
The zeroth law defines the notion of temperature. Once it is defined, we can choose n+1 variables (→x,t) as the thermodynamic coordinates of the body, of which only n are independent. The relation t(→x) is an equation of state.