1.2: The Zeroth Law of Thermodynamics

Last updated
Save as PDF

Page ID: 32011

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

This can be stated as follows.

Zeroth Law of Thermodynamics:

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.