“If a system A is in thermal equilibrium with system B and system B is in thermal equilibrium with system C, then system A and system C are also in thermal equilibrium with each other.” This transitive statement is an expression of which law of thermodynamics?

Introduction / Context:
This question is about the basic laws of thermodynamics. The statement describes a transitive property of thermal equilibrium between systems A, B and C. Recognising which law formalises this idea helps in understanding how thermometers work and why temperature is a meaningful, measurable quantity.

Given Data / Assumptions:

• Systems A, B and C are considered.
• A is in thermal equilibrium with B.
• B is in thermal equilibrium with C.
• We infer that A must be in thermal equilibrium with C.
• We must identify which thermodynamic law this statement represents.

Concept / Approach:
The zeroth law of thermodynamics states that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. This law formalises the transitive nature of thermal equilibrium and provides the conceptual basis for defining temperature as a measurable property. The first law deals with conservation of energy, and the second law concerns the direction of spontaneous processes and entropy, not this transitive relation.

Step-by-Step Solution:
1) Read the statement carefully: A is in thermal equilibrium with B, and B is in thermal equilibrium with C.2) From this, we conclude that A and C are also in thermal equilibrium.3) This transitive relation allows us to assign a common property, temperature, to all three systems when they are in equilibrium.4) The law that specifically states this transitive property is known as the zeroth law of thermodynamics.5) The first law deals with energy conservation and the equivalence of heat and work.6) The second law deals with entropy and the direction of heat flow, such as heat flowing spontaneously from hot to cold.7) Therefore, the correct choice for the given statement is the zeroth law of thermodynamics.

Verification / Alternative check:
Thermodynamics textbooks often introduce the zeroth law before the first law, even though it was formulated later historically. They usually phrase it similarly: if two systems are in thermal equilibrium with a third system, they are in equilibrium with each other. This forms the theoretical foundation for using thermometers; a thermometer in equilibrium with a body has the same temperature as that body. If two bodies are in equilibrium with the same thermometer, they are at the same temperature. This is exactly the content of the given statement.

Why Other Options Are Wrong:
Cyclic rule of thermodynamics: This term does not represent a fundamental law describing thermal equilibrium; it sometimes refers to relationships among partial derivatives in thermodynamics, not this statement.
First law of thermodynamics: Expresses conservation of energy and the relation between heat, work and internal energy, not the transitive property of thermal equilibrium.
Second law of thermodynamics: Concerns entropy and the direction of spontaneous processes, such as heat flowing from hot to cold, but does not state the above equilibrium relation.
Third law of thermodynamics: Deals with behaviour of entropy as temperature approaches absolute zero, not with transitive equilibrium of three systems.

Common Pitfalls:
Some students confuse the numbering of the laws and may mistakenly think the first law must come first conceptually. The term zeroth law can sound strange, but it was named later to reflect its more fundamental character. Another confusion is to associate any statement about equilibrium with the second law, since it often mentions equilibrium states. To avoid these mistakes, remember that the zeroth law is specifically about the transitivity of thermal equilibrium and underlies the definition of temperature and the use of thermometers.

Final Answer:
The given statement about thermal equilibrium of systems A, B and C is an expression of the zeroth law of thermodynamics.