12.1 Zeroth Law of Thermodynamics: Thermal Equilibrium

Thermal Equilibrium and the Zeroth Law of Thermodynamics

Thermal equilibrium is the state where objects reach the same temperature and heat stops flowing between them. The zeroth law of thermodynamics builds on this idea to give us a reliable way to define and measure temperature. Together, these concepts form the foundation for everything else in thermodynamics.

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Concept of Thermal Equilibrium

When two objects are in thermal contact, heat flows spontaneously from the hotter object to the cooler one. This continues until both objects reach the same temperature. At that point, the net heat transfer between them drops to zero, and they're in thermal equilibrium.

Think about a hot coffee mug sitting on a table. Heat flows from the mug into the table and the surrounding air. Over time, the mug cools and the surroundings warm slightly until everything settles at the same temperature. Unless something external changes (like you reheating the coffee), the system stays in equilibrium.

A few key details about this process:

• Heat always flows from higher temperature to lower temperature, never the other way around spontaneously.
• The rate of heat transfer depends on the temperature difference between the objects and their thermal properties (thermal conductivity, heat capacity, surface area). A metal pot on a stove heats much faster than a ceramic one because metal conducts heat more readily.
• During the approach to equilibrium, thermal energy is being redistributed between the objects. Once equilibrium is reached, energy is still exchanged at the microscopic level, but the net transfer is zero.

Zeroth Law of Thermodynamics

The zeroth law states: if system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then A and B are in thermal equilibrium with each other.

Mathematically: if $A \leftrightarrow C$ and $B \leftrightarrow C$, then $A \leftrightarrow B$, where $\leftrightarrow$ denotes thermal equilibrium.

This might sound obvious, but it's doing something important. It establishes that temperature is a transitive property: you can compare the temperatures of two objects without ever bringing them into direct contact, as long as you use a third object (like a thermometer) as a go-between.

This is exactly how a thermometer works:

1. You place the thermometer (system C) in contact with the first object (system A) and wait for thermal equilibrium. The thermometer now reads A's temperature.
2. You place the same thermometer in contact with a second object (system B) and wait for equilibrium again.
3. If the thermometer gives the same reading both times, the zeroth law guarantees that A and B are at the same temperature, even though they never touched each other.

The zeroth law also underpins thermometer calibration. A reference system with a precisely known temperature, such as a water triple point cell ($273.16 \, \text{K}$), can be used to calibrate multiple thermometers. Any two thermometers that agree with the reference will also agree with each other.

It was called the "zeroth" law because it was formalized after the first and second laws, but physicists realized it was logically more fundamental and needed to come before them.

Thermodynamic Systems and Equilibrium

A thermodynamic system is whatever region of space or collection of matter you've chosen to study. Everything outside it is the surroundings.

Full thermodynamic equilibrium is more demanding than just thermal equilibrium. It requires three conditions to be met simultaneously:

• Thermal equilibrium: uniform temperature throughout the system (no net heat flow).
• Mechanical equilibrium: uniform pressure (no net forces causing bulk motion).
• Chemical equilibrium: no net change in chemical composition (no ongoing reactions).

When a system is in thermodynamic equilibrium, its condition can be fully described by state variables like temperature, pressure, and volume. These variables depend only on the current state of the system, not on how it got there.

Applications of Thermal Equilibrium

Medical incubators for premature infants are a practical example of carefully controlled thermal equilibrium. The goal is to keep the infant at a stable, optimal temperature because premature babies can't regulate their own body temperature well.

Here's how an incubator maintains thermal equilibrium:

1. A heating element warms the air inside the incubator to a target temperature (typically around $36\text{–}37°\text{C}$).
2. A temperature sensor continuously monitors the internal air temperature and sends data to a control system.
3. The control system (often a PID controller) adjusts power to the heating element to keep the temperature within a narrow range of the setpoint.

Several design features help maintain this equilibrium:

• Insulation: Double-walled construction minimizes heat loss to the surrounding room.
• Humidity control: A humidifier regulates moisture levels inside the incubator. Without this, evaporation from the infant's skin would cause cooling and disrupt equilibrium.
• Accounting for other heat sources: The infant's own metabolic heat production and external devices like phototherapy lamps add thermal energy that the control system must compensate for.

Beyond medicine, thermal equilibrium shows up everywhere: cooking (an oven and its contents reaching the set temperature), climate control systems (thermostats maintaining room temperature), and industrial processes (tempering steel by allowing it to cool slowly to a uniform temperature).