1.2: The Zeroth Law of Thermodynamics
- Page ID
- 32011
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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
\[F(\vec{x_A},\vec{x_B})=0\]
if A and B are in thermal equilibrium. Thus the zeroth law states that
\[\left.
\begin{array}{c}
F(\vec{x_A},\vec{x_B})=0 \\
F(\vec{x_B},\vec{x_C})=0
\end{array} \right\} \Rightarrow F(\vec{x_A},\vec{x_C})=0\]
This is possible if and only if the relations are of the form
\[F(\vec{x_A},\vec{x_B})=t(\vec{x_A})-t(\vec{x_B})=0\]
This means that, for any body, there exists a function \(t(\vec{x})\) of the thermodynamic coordinates \(\vec{x}\), 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
\[t_A=t_B \Rightarrow T_A = T_B\]
The function \(t(\vec{x})\) 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 \((\vec{x},t)\) as the thermodynamic coordinates of the body, of which only \(n\) are independent. The relation \(t(\vec{x})\) is an equation of state.