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Temperature and Thermal Equilibrium

Level 1 - Physics topic page in Temperature and Heat.

Principle

Temperature is the state variable that decides whether two systems in thermal contact can remain unchanged.

If two systems can exchange energy because of a temperature difference, the interaction continues until they reach thermal equilibrium. At equilibrium there is no net heat transfer between them.

Notation

\(T\)
temperature
\(\mathrm{K}\)
\(T_A,T_B,T_C\)
temperatures of systems A, B, and C
\(\mathrm{K}\)
\(Q\)
energy transferred as heat
\(\mathrm{J}\)
\(H\)
heat current or rate of heat transfer
\(\mathrm{W}\)

Method

Temperature is defined operationally by equilibrium. A thermometer does not directly read the target system's hidden internal energy; it reaches equilibrium with the target, then reports its own calibrated state.

Thermal contact
\[T_A\ne T_B\Rightarrow H_{A\leftrightarrow B}\ne0\]
A temperature difference drives net energy transfer.
Equilibrium condition
\[H_{A\leftrightarrow B}=0\Rightarrow T_A=T_B\]
Thermometer reading
\[T_{\mathrm{thermometer}}=T_{\mathrm{object}}\]
This is valid only after the thermometer has settled into thermal equilibrium with the object.
Zeroth law
\[T_A=T_C\quad\text{and}\quad T_B=T_C\Rightarrow T_A=T_B\]

The zeroth law is what makes thermometers useful. A thermometer can compare systems that never touch each other, because equality of temperature is transitive.

Rules

These are the compact thermal-equilibrium statements.

Equilibrium
\[T_A=T_B\]
No net heat flow
\[T_A=T_B\Rightarrow H_{A\leftrightarrow B}=0\]
Direction of transfer
\[T_H>T_C\Rightarrow Q_{H\to C}>0\]
Zeroth law
\[T_A=T_C,\ T_B=T_C\Rightarrow T_A=T_B\]

Examples

Question
A thermometer initially at
\[18\,\mathrm{^\circ C}\]
is placed in a liquid. After a short time it reads
\[64\,\mathrm{^\circ C}\]
steadily. What temperature is being measured?
Answer
Once the reading is steady, the thermometer and liquid are in thermal equilibrium. The thermometer is measuring its own temperature, and at equilibrium that equals the liquid temperature:
\[T_{\mathrm{liquid}}=64\,\mathrm{^\circ C}\]

Checks

  • A thermometer must be allowed to reach equilibrium before its reading is meaningful.
  • Equal temperatures mean no net heat transfer, not no microscopic interaction.
  • Temperature is not the same thing as stored energy.
  • Thermal equilibrium is a state condition; it does not describe how fast equilibrium is reached.

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Temperature Measurement Scales