Thermal Equilibrium

What Does "Thermal Equilibrium" Mean?

The idea of equilibrium is central to thermodynamics. A system is said to be in thermal equilibrium when its macroscopic properties — temperature, pressure, volume — do not change with time, and there is no net flow of heat across its boundary. But this definition hides a deeper, more fundamental idea: the Zeroth Law of Thermodynamics.

The textbook introduces thermal equilibrium not by a single definition, but by building it from the ground up, starting with the concept of temperature itself.

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The Zeroth Law of Thermodynamics

Before we can say two bodies are in thermal equilibrium, we need a way to compare their "hotness" or "coldness" — that is, their temperature. The Zeroth Law provides the logical foundation for temperature measurement.

This statement seems obvious, but it is not a logical consequence of anything else — it is an empirical fact, and it is the reason we can use a thermometer. The third body C is the thermometer. When you place a thermometer in contact with a cup of coffee, the thermometer's mercury (or alcohol) expands or contracts until it reaches thermal equilibrium with the coffee. The thermometer now reads the coffee's temperature. If you then place the same thermometer in contact with a glass of milk, and it reaches equilibrium at the same reading, the Zeroth Law tells you that the coffee and the milk are in thermal equilibrium with each other — they are at the same temperature.

Without the Zeroth Law, you could not be sure that two objects at the same thermometer reading are actually at the same temperature. The law guarantees that temperature is a well-defined, transitive property.

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Defining Thermal Equilibrium Properly

With the Zeroth Law in hand, we can now define thermal equilibrium precisely.

A system is in thermal equilibrium if:

1. Its macroscopic variables (pressure $P$, volume $V$, temperature $T$, etc.) are constant in time.
2. There is no net heat transfer between any two parts of the system, or between the system and its surroundings.

But condition (2) is really a consequence of condition (1) when the system is isolated. The key point is that thermal equilibrium is a state — a specific set of values for the state variables — that the system settles into when left undisturbed for a long time.

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The Concept of Temperature

Temperature is the property that determines whether two bodies in thermal contact will be in thermal equilibrium. If two bodies have the same temperature, they are in thermal equilibrium. If they have different temperatures, heat will flow from the hotter to the colder body until their temperatures equalize.

The Zeroth Law allows us to assign a numerical value to temperature. We choose a thermometric property — a physical quantity that changes measurably with temperature, such as:

• The length of a mercury column in a capillary tube.
• The electrical resistance of a platinum wire.
• The pressure of a fixed volume of gas.
• The volume of a fixed mass of gas at constant pressure.

We then define a temperature scale by choosing a reference point (like the triple point of water, 273.16 K) and assuming a linear relationship between the thermometric property and temperature.

The most fundamental temperature scale is the ideal gas temperature scale, which uses the pressure of a gas at constant volume (or the volume at constant pressure) as the thermometric property. As the gas pressure approaches zero, all gases behave ideally, and the scale becomes independent of the gas used — this is the Kelvin scale.

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Key Properties of Thermal Equilibrium (as listed in the textbook)

The textbook lists three essential properties that follow from the Zeroth Law and the definition of thermal equilibrium. Each one is proved or justified in the text.

Property (I): Transitivity of Thermal Equilibrium

If body A is in thermal equilibrium with body B, and body B is in thermal equilibrium with body C, then body A is in thermal equilibrium with body C.

This is a direct restatement of the Zeroth Law. It is the property that makes temperature a well-defined quantity. Without transitivity, you could have A in equilibrium with B, B in equilibrium with C, but A not in equilibrium with C — which would mean that "same temperature" is not a transitive relation, and you could not define a single temperature scale.

›Proof

Proof (from the Zeroth Law):

Let body C be a thermometer. If A is in thermal equilibrium with C, then the thermometer reads the same value when in contact with A as it does when in contact with B (since B is also in equilibrium with C). By the Zeroth Law, A and B must therefore be in thermal equilibrium with each other. This argument can be repeated for any pair of bodies that are each in equilibrium with the same third body. Hence, thermal equilibrium is transitive.

Property (II): Existence of a Temperature Function

For every thermodynamic system in equilibrium, there exists a scalar quantity called temperature, such that equality of temperature is a necessary and sufficient condition for thermal equilibrium.

This property is the definition of temperature. It says that temperature is not just a number we assign arbitrarily — it is a physical property that determines whether two systems will be in thermal equilibrium. If two systems have the same temperature, they are in thermal equilibrium. If they have different temperatures, they are not.

Property (III): Temperature is a State Variable

The temperature of a system in thermal equilibrium is a function of its state variables (like pressure and volume) and is independent of the path taken to reach that state.

This means that for a given equilibrium state (say, a specific $P$ and $V$ for a fixed amount of gas), the temperature is uniquely determined. You cannot have two different equilibrium states with the same $P$ and $V$ but different temperatures. This is what allows us to write equations of state, like the ideal gas law $PV = nRT$.

›Proof

Proof (conceptual):

Suppose a system can be in two different equilibrium states, A and B, that have the same pressure and volume but different temperatures. Place a thermometer in contact with the system in state A. It reads $T_A$. Now change the system to state B without changing $P$ or $V$ — but if $P$ and $V$ are fixed, the system's state is completely determined (for a given amount of substance). Therefore, state B cannot have different $P$ and $V$ from state A — they are the same state. Hence, temperature must be a single-valued function of $P$ and $V$ (and amount of substance). This is the basis for the equation of state.

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The Equation of State

The textbook introduces the equation of state as the mathematical relationship between the state variables of a system in thermal equilibrium. For a fixed amount of a pure substance, the most common form is:

This is a general statement. For an ideal gas, the specific form is:

where $n$ is the number of moles and $R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$ is the universal gas constant.

For real gases, more complex equations exist (van der Waals equation, etc.), but the principle is the same: in thermal equilibrium, $P$, $V$, and $T$ are not independent — they are linked by a definite relation.