Zeroth Law of Thermodynamics

The Zeroth Law of Thermodynamics states that if two thermodynamic systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. This principle establishes a transitive relationship between systems, which allows for the objective definition of temperature and the creation of thermometers. Without this law, it would be impossible to define temperature as a fundamental measurable property of a system.

Short Definition

The Zeroth Law of Thermodynamics is a foundational principle of physics asserting that thermal equilibrium is a transitive property. If system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then system A and system B are in thermal equilibrium with one another. This observation justifies the use of a third system (a thermometer) to compare the temperatures of two other systems without bringing them into direct contact.

Detailed Explanation

The Zeroth Law of Thermodynamics was formulated long after the First and Second Laws had been established. However, scientists realized that the concept of temperature—which the First and Second Laws rely upon—required its own formal definition. In the 1930s, Ralph H. Fowler suggested the name “Zeroth Law” because it logically precedes the other laws but was identified chronologically later.

The Concept of Thermal Equilibrium

Thermal equilibrium occurs when two systems are in thermal contact but there is no net flow of heat between them. For this to happen, the systems must possess a common physical property that has reached a state of balance. When a system is in thermal equilibrium, its macroscopic variables (such as pressure, volume, and mass) remain constant over time, provided the external environment does not change.

Diathermic and Adiabatic Walls

To understand the Zeroth Law, one must distinguish between different types of boundaries or “walls” between systems:

• Diathermic Walls: These are heat-conducting walls that allow energy to pass between systems. When two systems are separated by a diathermic wall, they will eventually reach thermal equilibrium.
• Adiabatic Walls: These are insulating walls that prevent the transfer of heat. Systems separated by an adiabatic wall may remain at different temperatures indefinitely.

The Transitive Property

The core of the Zeroth Law is the mathematical concept of transitivity. In logic, if ##x = y## and ##y = z##, then ##x = z##. The Zeroth Law applies this logic to the state of “being in thermal equilibrium.” This allows us to use a reference system (System C) to determine if System A and System B would be in equilibrium if they were brought together. If System A is in equilibrium with System C, it means they share a property value. If System B also shares that same property value with System C, then A and B must share that value with each other.

Key Formula / Principle

The Zeroth Law does not have a complex algebraic formula in the traditional sense, but it is expressed through the existence of a state function called temperature (##T##).

If we denote the state of thermal equilibrium as the relation (##\sim##):

###A \sim C \quad \text{and} \quad B \sim C \implies A \sim B###

This implies there exists a scalar function ##T## (temperature) such that for any system in a state of equilibrium, the value of ##T## is the same for all systems in equilibrium with one another:

###T_A = T_C \quad \text{and} \quad T_B = T_C \implies T_A = T_B###

The equation of state for a system can be written as:

###f(P, V, T) = 0###

where ##P## is pressure, ##V## is volume, and ##T## is temperature. The Zeroth Law ensures that ##T## is a well-defined variable that can be measured.

Important Characteristics

The Zeroth Law possesses several characteristics that make it essential for classical and statistical thermodynamics:

• Definition of Temperature: It provides a physical basis for the concept of temperature as a measurable property. It shifts temperature from a subjective feeling (hot vs. cold) to an objective numerical value.
• Equivalence Relation: In mathematical terms, the law establishes that thermal equilibrium is an equivalence relation, meaning it is reflexive (##A \sim A##), symmetric (##A \sim B \implies B \sim A##), and transitive (##A \sim B, B \sim C \implies A \sim C##).
• State Function: It identifies temperature as a state function, meaning the temperature of a system depends only on its current state and not on how it reached that state.
• Foundation for Thermometry: It validates the use of thermometers. A thermometer is essentially the “third system” (System C) mentioned in the law.

Practical Example

The most common practical application of the Zeroth Law is the use of a mercury thermometer.

Consider three systems:

If we place the thermometer (System C) into the warm water (System A), the mercury expands until it reaches thermal equilibrium with the water. We record the height of the mercury. We then place the same thermometer (System C) into the oil (System B). If the mercury reaches the exact same height as it did in the water, the Zeroth Law allows us to conclude that the water and the oil are at the same temperature, even though we never mixed the water and oil or brought the two beakers into contact.

Another example is a digital thermostat in a room. The sensor (System C) reaches equilibrium with the air (System A). If the sensor reading matches a calibrated reference (System B), we know the room is at the desired temperature.

Common Confusions or Misconceptions

Despite its simplicity, several misconceptions surround the Zeroth Law:

• “It’s just common sense”: While it seems intuitive, physics requires a formal postulate to treat temperature as a universal coordinate. Without this law, we could not prove that two objects that feel “equally hot” would not exchange heat if touched.
• Equilibrium vs. Identity: Thermal equilibrium does not mean the systems are identical. A gallon of water and a copper penny can be in thermal equilibrium (same temperature), even though their pressures, volumes, and chemical compositions are entirely different.
• Heat vs. Temperature: The Zeroth Law defines temperature, not heat. Heat is the transfer of energy due to a temperature difference. The Zeroth Law describes the state where that transfer stops.
• Speed of Equilibrium: The law does not state how fast equilibrium is reached. It only describes the state once it has been achieved. The rate of heat transfer is governed by the laws of heat conduction and convection, not the Zeroth Law itself.

Related Terms

• Temperature: A physical property of matter that quantitatively expresses hot and cold.
• Thermal Equilibrium: The state in which two or more systems have the same temperature and no net heat flows between them.
• Isotherm: A line on a graph representing states of a system at a constant temperature.
• Equation of State: A thermodynamic equation relating state variables, such as ##PV = nRT##.
• Thermometric Property: A physical property (like the volume of a liquid or electrical resistance) that changes predictably with temperature.

Why It Matters

The Zeroth Law of Thermodynamics is the “silent partner” of the more famous laws. It is the logical cornerstone of all thermal sciences.

In summary, while the Zeroth Law may seem like a simple observation of nature, it provides the rigorous framework necessary for the entire field of thermodynamics to exist as a quantitative science.