Thermal Equilibrium & the Zeroth Law

The most trusted number in any hospital is produced by an instrument that never measures the patient. Slip a clinical thermometer under your tongue and wait: the device reports one thing only — its own temperature. Why should that number say anything about you? Because of two ideas so foundational that when physicists finally got around to naming the second one, the numbers one and two were already taken and it had to be called the zeroth law. The first idea is thermal equilibrium: leave two bodies in contact through a wall that lets heat pass, and the flow of energy dies away until nothing changes any more — they end up sharing a single temperature. The second is the zeroth law itself: if body A is in equilibrium with body C, and body B is in equilibrium with the same C, then A and B are in equilibrium with each other — even if they never touch. That innocent-looking sentence is the entire licence for thermometry: the thermometer is body C, and the law is what lets its reading speak for your body, for the lab's water bath, and for every calorimetry problem you will ever solve. Boards test this from every angle. CBSE and ICSE ask for one-mark definitions of thermal equilibrium and of diathermic versus adiabatic walls; NEET builds assertion–reason questions on the trap "equal temperature means equal heat"; JEE mixing problems silently assume one final common temperature and dare you to find it; AP Physics 2 and IB want the zeroth-law justification written in words. This article gives you the concept, the logic, and the small exact arithmetic of where the common temperature lands.

Key takeaways

Thermal equilibrium between two bodies means no net heat flows when they are connected by a diathermic wall — all macroscopic properties (temperature, pressure, length of a mercury column) have stopped changing.
A diathermic wall lets heat pass (thin copper); an adiabatic wall blocks it (an idealised vacuum flask). The enclosure of every calorimetry problem is adiabatic; the contact between the bodies is diathermic.
Zeroth law: if A is in equilibrium with C, and B is in equilibrium with C, then A is in equilibrium with B. Equilibrium is transitive — an experimental fact, not a logical freebie.
Transitivity is what licenses a thermometer: the thermometer is body C. It only ever measures itself; the zeroth law guarantees that, at equilibrium, its reading is also yours.
Temperature is the shared property at equilibrium — the one number that decides whether heat will flow (from higher to lower) and when it stops (equal).
Where does contact end up? At the capacity-weighted mean: $T_f = (C_1T_1 + C_2T_2)/(C_1+C_2)$ — for equal capacities the midpoint (80 °C and 20 °C meet at 50 °C), for the same liquid a mass-weighted mean (200 g at 90 °C plus 300 g at 40 °C lands at 60 °C).
Key idea: thermal equilibrium is transitive, and that transitivity is precisely what makes "temperature" a well-defined, measurable label — a thermometer reads itself, and the zeroth law makes that number yours.

The core insight

Two systems joined by a diathermic wall exchange heat until no net flow remains, and because equilibrium is transitive — A with C and B with C forces A with B — every body can be tagged with one number, temperature, that a thermometer (body C) can read on behalf of anything it touches.

Where this lives today

The zeroth law is the quiet infrastructure of every temperature ever quoted. Metrology runs on it: a national laboratory holds a triple-point-of-water cell — a sealed glass vessel where ice, water and vapour coexist at exactly 273.16 K — and calibrates a platinum resistance thermometer by letting it reach equilibrium with the cell. That thermometer calibrates others, which calibrate the ones in factories, clinics and weather stations: an unbroken chain of body-C comparisons, each link a zeroth-law equilibration, so that a fever measured in Chennai and one measured in Chicago mean the same thing. (Since 2019 the kelvin itself is defined through the Boltzmann constant, but the calibration chain is still equilibrium all the way down.) Your home thermostat is a body C wired to a switch: it sits in equilibrium with room air and acts on its own temperature. Even the infrared ear thermometer, which touches nothing and reads radiation, is calibrated against contact standards — its number inherits meaning from the same equilibrium logic.

The physical picture

Press a hot metal block against a cold one inside a perfectly insulated box. At the interface, fast-jiggling molecules on the hot side hammer their sluggish neighbours on the cold side; on average, energy leaks across from hot to cold. That leak is heat. As block A cools and block B warms, the difference in molecular agitation shrinks, the leak weakens, and eventually it dies altogether: energy still sloshes both ways across the boundary microscopically, but the two-way traffic balances exactly. Nothing macroscopic changes any more. The pair has reached thermal equilibrium, and — this is the kinetic-theory reading Maxwell taught us — the condition for the traffic to balance is that the average molecular kinetic energy is the same on both sides. Equal temperature is equal average agitation — exactly so for an ideal gas, and qualitatively for liquids and solids. The walls in this story deserve names, because boards ask for them. A diathermic wall permits heat to pass — a thin sheet of copper, the glass bulb of a thermometer. An adiabatic wall forbids it — thick styrofoam, a silvered vacuum flask, the idealised "insulated container" of every calorimetry problem. No real wall is perfectly adiabatic; the word is a promise that the leakage is negligible over the time you care about. With the vocabulary fixed, the definition compresses to one line:

Q_{\text{net}} = 0 \;\;\text{across a diathermic wall} \quad\Longleftrightarrow\quad T_A = T_B \qquad (1)

where: $Q_{\text{net}}$ — the net heat crossing the wall between the two bodies; $T_A, T_B$ — their temperatures. Read it in both directions: if the temperatures are equal, connecting the bodies changes nothing; and if connecting them changes nothing, their temperatures were equal. "Nothing happens" is real, usable data — it is exactly the observation a thermometer waits for.

Figure

Two bodies across a diathermic wall inside an adiabatic enclosure — heat flows hot to cold until one shared temperature remains

How do you see equilibrium arriving? Watch any measurable property: the pressure of a trapped gas, the length of a mercury thread, the resistance of a platinum wire. While heat flows, these drift; at equilibrium, they freeze. Note what the definition does not promise: it says nothing about how long the approach takes — copper contacts equilibrate in seconds, styrofoam-wrapped ones in hours, and that is the business of thermal conduction, a later article. Equilibrium physics only tells you where the system is heading and when the flow is zero.

The zeroth law: temperature by proxy

Now the real puzzle. You have a boiler bolted to the floor of one room and a vat of oil across the building, and you need to know whether bringing them into contact would move heat. You cannot press them together to run the test in equation (1). What you can do is carry a small third body — call it C — from one to the other. Hold C against the boiler and wait until nothing changes: C is in equilibrium with A. Carry C to the vat, touch it on, and watch. If, once again, nothing changes, experiment shows something remarkable you had no right to demand from pure logic:

T_A = T_C \;\;\text{and}\;\; T_B = T_C \quad\Longrightarrow\quad T_A = T_B \qquad (2)

where: the statement $T_X = T_Y$ is shorthand for "X and Y are in thermal equilibrium" in the operational sense of equation (1). This is the zeroth law of thermodynamics: two systems each in thermal equilibrium with a third are in thermal equilibrium with each other. Equilibrium is transitive — the property jumps across the middleman. Do not let the law's blandness fool you into thinking it is empty. Plenty of everyday relations are not transitive: "is a friend of" famously fails (your two friends may be strangers), while "is exactly as tall as" succeeds. Nothing in the definition of "no net heat flow" forces the transitive behaviour — it is an experimental fact about our universe, checked every time anyone uses a thermometer. In a universe where it failed, a single body could honestly return two different readings on two identical instruments, and the very word "temperature" would refuse to mean anything.

Figure

The transitivity triangle — A balances with C and B balances with C, so A and B must balance with each other even though they never touch

Here is what the law buys you, and it is enormous. Transitivity sorts every body in the world into families: all bodies mutually in equilibrium belong to one family, and each family can be tagged with a single number. That tag is temperature — not a vague sensation of hotness but the shared property at equilibrium, the one quantity guaranteed equal across a family. Heat then flows from a higher tag to a lower tag and stops when tags match. And the thermometer is simply body C wearing a scale: it never measures the patient, the tea, or the boiler — it measures itself, waits for equilibrium, and lets the zeroth law transfer the verdict. The concept is Victorian — Maxwell's Theory of Heat (1871) already builds temperature on exactly this equilibrium logic. The name came six decades later: Ralph H. Fowler observed that this postulate logically precedes the first and second laws, and with both numbers already taken, only "zeroth" remained; the name reached print in Fowler and Guggenheim's Statistical Thermodynamics (1939).

Where the common temperature settles

Equilibrium says the flow stops when temperatures are equal; energy conservation tells you at what temperature. Let two bodies with heat capacities $C_1$ and $C_2$ (recall $C = mc$ , the joules needed per kelvin of rise) start at $T_1 > T_2$ inside an adiabatic enclosure. Every joule that leaves the hot body enters the cold one:

C_1\,(T_1 - T_f) = C_2\,(T_f - T_2) \qquad (3)

where: $C_1, C_2$ — heat capacities in J K⁻¹; $T_1, T_2$ — initial temperatures; $T_f$ — the final common temperature. Both brackets are written hotter minus colder, so both sides are positive magnitudes: heat lost equals heat gained. Solving for $T_f$ gives the master result:

T_f = \frac{C_1 T_1 + C_2 T_2}{C_1 + C_2} \qquad (4)

where: the symbols are as in (3). Read equation (4) as a weighted average — the centre of mass of the initial temperatures, with heat capacities playing the role of masses. The final temperature always lies between the two starting values, pulled toward the body that takes more energy to shift. Two equal capacities meet exactly halfway: bodies at 80 °C and 20 °C settle at 50 °C. A teaspoon of tea dropped into a lake moves the lake not at all — the weighted mean with one giant weight simply returns the lake's own temperature.

Derivation AThe equilibrium temperature is a weighted average

From nothing more than "heat lost equals heat gained" inside an adiabatic enclosure, we derive the universal landing point — equation (4) — for two bodies brought into diathermic contact, and then check it in the two configurations every board exam recycles: equal heat capacities starting at 80 °C and 20 °C, and unequal masses of the same liquid, 200 g at 90 °C mixed with 300 g at 40 °C.

Step 1. Conservation. The enclosure is adiabatic, so no energy escapes; the wall between the bodies is diathermic, so energy redistributes until (1) holds — a single final temperature $T_f$ . The hot body releases $C_1(T_1 - T_f)$ ; the cold body absorbs $C_2(T_f - T_2)$ . These are the same joules, which is equation (3).

Step 2. Solve. Expand (3): $C_1T_1 - C_1T_f = C_2T_f - C_2T_2$ . Collect the $T_f$ terms: $C_1T_1 + C_2T_2 = (C_1 + C_2)\,T_f$ , and divide to get $T_f = (C_1T_1 + C_2T_2)/(C_1+C_2)$ — equation (4).

Step 3. Equal capacities. Put $C_1 = C_2 = C$ : the capacities cancel and $T_f = (T_1 + T_2)/2$ , the plain midpoint. For 80 °C and 20 °C, $T_f = (80+20)/2 = 50\ \mathrm{^\circ C}$ — the anchor case worth memorising.

Step 4. Same substance, different masses. Put $C_1 = m_1c$ and $C_2 = m_2c$ : the specific heat $c$ cancels before you ever look up its value, leaving $T_f = (m_1T_1 + m_2T_2)/(m_1+m_2)$ . For 200 g at 90 °C and 300 g at 40 °C: $T_f = (200\times90 + 300\times40)/500 = (18000+12000)/500 = 60\ \mathrm{^\circ C}$ .

Step 5. Sanity. Because the weights in (4) are positive, $T_f$ is always strapped between $T_2$ and $T_1$ — an answer of 95 °C or 35 °C from these inputs is arithmetic, not physics. And $T_f$ sits closer to the larger capacity: 60 °C is 20 degrees from the 300 g body but 30 degrees from the 200 g body, exactly the 3 : 2 mass ratio inverted. Verified: 50 °C, 60 °C and the weighted-mean identities recomputed in verify_hs2-thermal-equilibrium-zeroth-law.py.

Go deeper · Derivation B · 1 minWhy a thermometer must be small — the price of contact

Derivation BWhy a thermometer must be small — the price of contact

The zeroth law licenses body C to speak for body A, but only after C actually reaches equilibrium with A — and getting there costs energy that A itself must supply. We derive the reading of a real thermometer of heat capacity $C_{th}$ , initially at room temperature, dipped into a sample of capacity $C_x$ at true temperature $T_x$ , and show the reading is exact only in the limit $C_{th} \to 0$ . This is the honest fine print of thermometry: the act of measuring nudges the thing measured.

Step 1. The thermometer and the sample form an isolated pair, so equation (4) applies with the thermometer as body 2: the common temperature they reach — which is what the thermometer displays — is $T_{\text{read}} = (C_x T_x + C_{th} T_0)/(C_x + C_{th})$ , with $T_0$ the thermometer's starting temperature.

Step 2. Numbers. Take 100 g of water, $c = 4200\ \mathrm{J\,kg^{-1}\,K^{-1}}$ , so $C_x = 0.100\times4200 = 420\ \mathrm{J\,K^{-1}}$ , at a true $T_x = 60.00\ \mathrm{^\circ C}$ ; and a small glass thermometer with $C_{th} = 2\ \mathrm{J\,K^{-1}}$ starting at $T_0 = 25\ \mathrm{^\circ C}$ . Then $T_{\text{read}} = (420\times60 + 2\times25)/422 = 25250/422 = 59.83\ \mathrm{^\circ C}$ .

Step 3. The reading is low by about $0.17\ \mathrm{K}$ — precisely the warmth the water had to donate to drag the thermometer up to its own temperature. The instrument reports the joint equilibrium state, which the act of contact has slightly cooled.

Step 4. Push the limits. If the "thermometer" were as big as the sample ( $C_{th} = C_x$ ), the pair would meet at the midpoint $(60+25)/2 = 42.5\ \mathrm{^\circ C}$ — a reading off by a third of the gap, useless. As $C_{th} \to 0$ the reading slides up to the true 60 °C. That is the entire design brief of the clinical thermometer: tiny bulb, thin glass, smallest possible heat capacity, so that body C certifies your temperature without meaningfully changing it. Verified: 59.83 °C, the 0.17 K deficit and the 42.5 °C midpoint recomputed in verify_hs2-thermal-equilibrium-zeroth-law.py.

Limiting cases

case	result	meaning
$C_1 = C_2$	$T_f = (T_1+T_2)/2$	equal capacities meet at the midpoint — 80 °C and 20 °C settle at 50 °C
$C_2 \gg C_1$	$T_f \to T_2$	a reservoir imposes its temperature; a lake ignores the teaspoon
$T_1 = T_2$	zero net heat flow	already in equilibrium — contact changes nothing, which is the test in (1)
$C_{th} \to 0$	reading $\to$ true temperature	the ideal thermometer disturbs nothing; real ones read slightly low on hot samples
wall adiabatic	no equilibration, however long you wait	the vacuum-flask idealisation; "insulated" in problems means exactly this

Common trapSame temperature does not mean same energy

Thermal equilibrium equalises exactly one thing: temperature. It does not equalise internal energy, and nothing called "heat" is stored to be equalised at all. After the 200 g and 300 g of water settle at 60 °C, the larger mass holds one and a half times the internal energy of the smaller — same temperature, different totals, because energy is extensive while temperature is intensive. NEET assertion–reason items and CBSE true/false lines bait precisely this: "two bodies in thermal equilibrium have equal thermal energy" is false; "…equal temperature" is true. The same confusion in reverse decides flow direction: heat runs from higher temperature to lower, never from "more energy" to "less" — a spark leaks heat into a swimming pool that stores astronomically more energy than the spark ever did. When in doubt, say the safe sentence: equilibrium means equal temperature and zero net heat flow — nothing more.

Worked examples

Board level — one final temperature (CBSE/JEE-Main mixing pattern)

200 g of water at 90 °C is poured into 300 g of water at 40 °C in an insulated vessel. Find the final temperature and the heat exchanged. Take $c_w = 4200\ \mathrm{J\,kg^{-1}\,K^{-1}}$ .

Step 1. Name the physics before touching numbers: the vessel is adiabatic, so heat lost by the hot water equals heat gained by the cold, and both masses end at one common temperature $T_f$ — that single final temperature is thermal equilibrium. Write $0.200\,c_w\,(90 - T_f) = 0.300\,c_w\,(T_f - 40)$ .

Step 2. The specific heat cancels — same substance on both sides. Then $2(90 - T_f) = 3(T_f - 40)$ , so $180 - 2T_f = 3T_f - 120$ , giving $300 = 5T_f$ and $\boxed{T_f = 60\ \mathrm{^\circ C}}$ .

Step 3. Heat exchanged, computed from the hot side: $Q = 0.200\times4200\times(90-60) = 840\times30 = \boxed{25\,200\ \mathrm{J}}$ . Cross-check from the cold side: $0.300\times4200\times(60-40) = 1260\times20 = 25\,200\ \mathrm{J}$ . The two agree — that agreement is your built-in error detector, worth ten seconds in any exam.

Step 4. The mark-scheme moral: the answer is the mass-weighted mean, $(200\times90 + 300\times40)/500$ , sitting closer to 40 °C because the colder body carries the bigger mass. Quote the equilibrium logic in words — "both reach a common temperature; heat lost = heat gained" — CBSE explicitly allots a mark to that sentence.

Competitive level — three liquids and a hidden ratio (JEE classic)

Equal masses of three liquids A, B and C are at 12 °C, 19 °C and 28 °C. When A and B are mixed, the temperature is 16 °C; when B and C are mixed, it is 23 °C. What temperature results when A and C are mixed?

Step 1. Each mixing is an equilibrium event: one final temperature, heat lost = heat gained. From A + B (masses cancel throughout): $c_A(16-12) = c_B(19-16)$ , so $4c_A = 3c_B$ and $c_A = 0.75\,c_B$ .

Step 2. From B + C: $c_B(23-19) = c_C(28-23)$ , so $4c_B = 5c_C$ and $c_C = 0.80\,c_B$ . Do not solve for any specific heat individually — the problem never gives you enough for that, and you never need it. Everything rides on ratios.

Step 3. Mix A and C using the weighted mean, equation (4), with $c_B$ as the common yardstick: $T = \dfrac{c_A(12) + c_C(28)}{c_A + c_C} = \dfrac{0.75\times12 + 0.80\times28}{0.75 + 0.80} = \dfrac{9 + 22.4}{1.55} = \dfrac{31.4}{1.55} = \boxed{20.26\ \mathrm{^\circ C}}$ (exactly $628/31$ , so $\approx 20.3\ \mathrm{^\circ C}$ ).

Step 4. Sanity: the answer must land between 12 °C and 28 °C, and since $c_C$ slightly outweighs $c_A$ , it should sit just past the midpoint of 20 toward C's side — 20.26 °C, exactly as found. Timing note: about 4 minutes at full JEE pace. The entire problem is two capacity ratios; keep every capacity in units of $c_B$ and it collapses to one line. Sign discipline — writing each bracket as hotter minus colder — is where the marks usually die.

Exam traps

"Equilibrium means equal heat / equal internal energy." The NEET assertion–reason staple. Equilibrium equalises temperature only; internal energy stays unequal whenever masses or substances differ. Bodies never "contain heat" at all.

Double-signing the heat balance. Students write $Q = mc\,\Delta T$ with a signed $\Delta T$ and set "lost = gained", so the minus sign enters twice and the final temperature pops out beyond the starting range. Pick one convention: magnitudes on both sides, every bracket hotter minus colder. If your $T_f$ is not between the initial temperatures, you have made this exact error.

Wrong weights in the mean. $T_f = (m_1T_1 + m_2T_2)/(m_1+m_2)$ is legal only for the same substance, where $c$ cancels. Different substances demand the full capacity weights $C = mc$ in equation (4). JEE setters love a copper-block-into-water problem that punishes the mass-only shortcut.

Diathermic and adiabatic swapped. ICSE and CBSE one-markers ask for both definitions. Keep them straight with the root: dia = through (heat goes through); adiabatic = no heat crosses, the perfect insulator.

Treating the thermometer as instant and innocent. A thermometer reads its own temperature, needs time to reach equilibrium, and steals a little heat while doing so (0.17 K in our derivation). AP-2 and IB explanation questions specifically want the zeroth law named as the reason a thermometer works at all.

Scale mixing. Temperature differences are identical in kelvin and celsius, and the weighted mean (4) gives consistent answers in either scale — but only if every temperature in the equation uses the same scale. Feeding 90 °C and 313 K into one bracket is an instant zero.

Origins · 1 minJames Clerk Maxwell and how this was discovered

James Clerk Maxwell

1831–1879 · Edinburgh & Cambridge

Maxwell wrote the equations of light, but he also wrote the book on heat — literally. His textbook Theory of Heat (1871) defines temperature operationally, through equilibrium: bodies in thermal equilibrium share one temperature, and a body's temperature is its power to pass heat to others — the very logic later christened the zeroth law. Behind the definition sat his kinetic theory: the Maxwell velocity distribution of 1860 made "equal temperature" mean "equal average molecular kinetic energy," turning equilibrium from bookkeeping into molecular statistics. Appointed the first Cavendish Professor at Cambridge in 1871, he died in 1879, aged just forty-eight. Every thermometer you will ever read is running Maxwell's definition: the instrument measures itself, and his equilibrium logic is what makes that number yours.

History

1724

— Daniel Gabriel Fahrenheit's mercury thermometers make temperature comparison routine — two centuries before anyone states the law that justifies them.

1848

— William Thomson (later Lord Kelvin) proposes an absolute temperature scale, independent of any particular substance.

1871

— James Clerk Maxwell's Theory of Heat builds temperature on thermal equilibrium: bodies in equilibrium share one temperature.

1930

s — Ralph H. Fowler, noting that the logical foundation of thermodynamics was never numbered and one and two were taken, proposes the name "zeroth law".

1939

— Fowler and Guggenheim's Statistical Thermodynamics puts the name in print, and it sticks.

Going further

Texts. NCERT Physics Class 11, Ch. 12 (Thermodynamics) — thermal equilibrium and the zeroth law as CBSE/JEE state them; Ch. 11 (Thermal Properties of Matter) for the calorimetry arithmetic · Zemansky & Dittman, Heat and Thermodynamics — the classic careful treatment of equilibrium and temperature · Halliday, Resnick & Walker, "Temperature, Heat and the First Law of Thermodynamics" · Maxwell's own Theory of Heat (1871) is short, sharp and still readable.

Provenance. Standard results. A verifier should recompute: $(200\times90 + 300\times40)/500 = 60\ \mathrm{^\circ C}$ and the heat check $0.200\times4200\times30 = 25\,200\ \mathrm{J} = 0.300\times4200\times20$ ; the equal-capacity midpoint $(80+20)/2 = 50\ \mathrm{^\circ C}$ ; the thermometer reading $(420\times60 + 2\times25)/422 = 59.83\ \mathrm{^\circ C}$ with its $0.17\ \mathrm{K}$ deficit and the equal-capacity probe midpoint $(60+25)/2 = 42.5\ \mathrm{^\circ C}$ ; the three-liquid result $31.4/1.55 = 628/31 = 20.26\ \mathrm{^\circ C}$ ; the triple point of water at $273.16\ \mathrm{K}$ .

Prerequisites. Heat vs Temperature: What's the Difference · Thermometry & Temperature Scales. Leads to. Specific Heat & Heat Capacity — the quantitative machinery of $C = mc$ that this article's weighted mean already previews. Go deeper (UG). The rigorous construction of temperature from the equilibrium relation — equivalence classes, empirical temperature and all — is Zeroth Law of Thermodynamics and Empirical Temperature in the university Statistical Mechanics track.

Why is it called the "zeroth" law — did physicists just miscount?

No — they named things out of order. The first and second laws of thermodynamics were established and numbered in the nineteenth century. Only later did physicists articulate clearly that the whole framework silently assumes something more basic: that thermal equilibrium is transitive, so temperature is well defined in the first place. When Ralph H. Fowler proposed elevating this to a law in the 1930s, the numbers one and two were taken, and a statement more fundamental than the first law could hardly be the third. So it became the zeroth — a numbering joke that stuck because it is also logically honest.

Isn't the zeroth law obvious? Of course two things equal to a third are equal.

It feels obvious because you already believe in temperature — but the law is what entitles you to that belief, so the reasoning is circular. "Equal to" is transitive by arithmetic, but "in thermal equilibrium with" is a physical relationship between real objects, and physical relationships are not transitive for free: "is a friend of" fails all the time. It is an experimental fact — not a logical necessity — that no-net-heat-flow behaves transitively. Because it does, every body can be tagged with one number and thermometers give consistent verdicts. A universe without the zeroth law would be a universe where the question "how hot is this?" had no single answer.

Does a thermometer measure my temperature or its own?

Strictly, always its own. The mercury thread, the thermistor resistance, the digital display — each reports the state of the thermometer. The measurement works in two acts: first, contact and patience, while heat flows between you and the instrument until nothing changes any more; second, the zeroth law, which guarantees that at equilibrium the thermometer's temperature and yours are one number. That is also why an impatient reading is wrong — pull the thermometer out early and equilibrium was never reached, so its temperature is still partly the room's, not yours.

What is the difference between thermal equilibrium and a steady state?

Both look "settled" — nothing changes with time — but they differ in what the heat is doing. Hold one end of a metal spoon over a flame: after a while every point of the spoon sits at a constant temperature, hot end to cool end. That is a steady state: temperatures are constant, but heat is flowing continuously from flame to air through the spoon. Thermal equilibrium demands more — no net heat flow anywhere, which forces one uniform temperature. Every equilibrium is a steady state; a steady state with ongoing flow is not equilibrium. Exam questions about "insulated containers left for a long time" mean true equilibrium.

If two bodies are at the same temperature, do they have the same internal energy?

Almost never. Temperature is intensive — an average per molecule — while internal energy is extensive, a total over all molecules. After 200 g and 300 g of water equilibrate at 60 °C, the 300 g holds one and a half times the internal energy, and a swimming pool at the same 60 °C would hold millions of times more. Equilibrium equalises the average agitation per molecule, not the head-count of molecules. This is the single most profitable distinction in the whole chapter — see the sibling article on heat versus temperature for the full story.

Do perfectly adiabatic walls actually exist?

No. Every real wall leaks a little; "adiabatic" is an idealisation, like a frictionless plane. A silvered vacuum flask comes close — conduction and convection blocked by vacuum, radiation reflected by silvering — but your tea still cools by morning. In practice, "adiabatic" is a statement about timescales: a wall counts as adiabatic if the leakage during your experiment is negligible compared to the heat being shuffled inside. That is exactly the fine print of every calorimetry problem: mixing takes a minute, appreciable leakage takes hours, so "insulated vessel" is an honest approximation rather than a lie.

Why does heat flow stop exactly when the temperatures are equal?

Zoom in on the diathermic wall. Molecules on both sides constantly batter it and exchange energy through it — traffic in both directions that never stops. Which way does the net energy go? From the side whose molecules hit harder on average, which is the hotter side. As the hot side cools and the cold side warms, the two average agitations approach each other, and when the average kinetic energies match, each collision is as likely to pass energy left as right: the two-way traffic balances. Nothing dramatic happens at equality — the flow does not slam shut; it fades smoothly to zero. Equilibrium is dynamic at the molecular level and quiet only in the average.