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1.7.2 · D5Thermodynamics

Question bank — Zeroth law — transitivity of thermal equilibrium

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True or false — justify

True/False: and together force .
True — this is the Zeroth law (transitivity). Sharing the same reference forces and onto the same equivalence class, so .
True/False: and force .
False — the law only acts on the positive case. Both and could be at (so ) while sits at ; both fail against yet agree with each other.
True/False: The Zeroth law tells you heat flows from hot to cold.
False — direction of flow is the Second law. The Zeroth law describes only the zero-flow (equilibrium) situation and says nothing about which way heat moves when systems disagree.
True/False: Two bodies in thermal equilibrium must hold equal internal energy.
False — equilibrium means equal temperature (intensive), not equal energy (extensive). A pool and a cup both at hold wildly different amounts of internal energy.
True/False: The relation "" is symmetric.
True — once heat flow has stopped there is no preferred direction, so immediately gives . Symmetry is one of the three properties making an equivalence relation.
True/False: A body is in thermal equilibrium with itself.
True — reflexivity, . A system placed against a copy of its own state exchanges no net heat; this is the "obvious" property that still must be listed.
True/False: The Zeroth law was discovered after the First and Second laws.
True — it was formalised later (Fowler, 1930s) even though it is logically prior; being more basic yet named last is exactly why it got the number zero.
True/False: If a thermometer reads the same mark in two cups, the cups are in equilibrium even if never touched.
True — the thermometer plays the shared system ; matching marks give and , and transitivity predicts without direct contact.
True/False: "Thermal contact" means the two systems are mixed together.
False — contact means they may exchange heat while staying separate (e.g. glass between mercury and tea). Mixing is a different process; equilibrium is judged with matter kept apart.

Spot the error

Find the flaw: "Since and , and equilibrium is transitive, heat flows from to ."
The transitivity conclusion is — meaning no net heat flow. Nothing about flow direction follows; that reasoning smuggles in the Second law.
Find the flaw: " and have the same internal energy, so ."
Equal energy does not imply equal temperature. Equilibrium is decided by , an intensive property; energy is extensive and can differ while temperatures still fail to match.
Find the flaw: "The Zeroth law defines temperature, so it must be a numerical formula for ."
It defines temperature's existence as a consistent label, not a formula. It guarantees a function exists such that ; assigning actual numbers is the job of scales.
Find the flaw: "Because is reflexive and symmetric, it's an equivalence relation."
You are missing transitivity — the Zeroth law itself. Reflexive + symmetric alone is not enough; all three are required to partition systems into classes.
Find the flaw: "A thermometer measures the tea's heat content."
A thermometer reaches equilibrium with the tea and reports its temperature, not its heat/energy content. A teaspoon and a bathtub at the same temperature give the same reading despite hugely different energy.
Find the flaw: " was verified by seeing the mercury stop, so and contain equal mercury temperature."
Mercury stopping shows the thermometer () reached equilibrium with each cup separately. The shared reading, not the mercury's presence in the cups, is what links and .

Why questions

Why is temperature called the "common label" of an equivalence class?
Because every member of a class is in mutual equilibrium, we may attach one number to the whole class; that number is , shared by all members and differing between classes.
Why does the Zeroth law let a thermometer work at all?
It guarantees that a matching reading against two objects (each an ) implies the objects agree with each other, so one instrument can compare bodies it never touches together.
Why can't exist as a single-valued label without transitivity?
Without transitivity, and could allow — then , , couldn't all carry the same , and the symbol would be ambiguous.
Why is temperature intensive rather than extensive?
Because it labels an equilibrium state, not a total amount: splitting a body in equilibrium into halves leaves each half at the same temperature, whereas its energy halves.
Why is the Zeroth law "logically prior" to the First and Second laws?
The First and Second laws speak about energy and entropy in terms of temperature, so temperature must first be defined — and that definition is exactly what the Zeroth law supplies.

Edge cases

Edge case: system is a thermometer with a broken (stuck) column — does the transitivity argument still hold?
No — a stuck column can't actually reach equilibrium with either body, so "" is never truly established and the transitivity chain has no valid link.
Edge case: three bodies pairwise at the same temperature — how many equivalence classes?
One class containing all three. Every pair satisfies , so they collapse into a single class carrying one temperature value.
Edge case: at , at , but at — what does the Zeroth law say?
Nothing about – through , since neither is in equilibrium with . Independently still gives ; the law simply isn't the tool that establishes it here.
Edge case: does the Zeroth law apply the instant two bodies touch?
No — the relation "" requires equilibrium to be reached (no net flow, properties settled). During transient heat exchange the bodies are not yet related by .
Edge case: a body compared with itself split into two portions — are they in equilibrium?
Yes, by reflexivity: identical states exchange no net heat, so each portion is in equilibrium with the other and with the whole.
Recall One-line self-test

If you can explain why " and " tells you nothing, you understand the single most missed point on this topic. The relation only forces conclusions in the positive case ::: Two bodies can both disagree with yet still be at identical temperatures, so "not-equilibrium with " carries no information about vs .

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