What is temperature?
Feedback on the diagnostic test · Opening the thermodynamics unit
A room, four cubes
Which cube feels coldest to the touch?
- Glass
- Metal
- Wood
- Plastic
- They are all the same
Now the room is at 43 °C
Same four cubes. Which feels coldest now?
- Glass
- Metal
- Wood
- Plastic
- They are all the same
One more question
Both times, all four cubes had been sitting in that room for hours.
What are their temperatures — in the cold room, and in the hot room?
Say your answer to your neighbour. We are not voting on this one.
This stays open. We come back to it at the end of the hour.
So — what is temperature?
Let's actually find out, rigorously.
Fine — then let us just define it
Temperature is the quantity measured by a thermometer.
Does that work?
- And what is a thermometer? An instrument that measures temperature.
- Temperature → thermometer → temperature. The circle closes and we have defined nothing.
- This is called a tautology: a definition that quietly assumes what it was supposed to explain.
- To break out, we need something that comes BEFORE the thermometer.
Same room
A tiled floor feels colder than a rug — even in the same room, at the same temperature.
You have all seen it a thousand times
- You sit on a cold chair. The chair warms up, you cool down. Then it stops.
- You get into the swimming pool. Freezing at first. Ten minutes later, fine.
- Water goes into the freezer and ends up at the freezer's temperature.
What do all of these have in common?
Two bodies in contact. Something changes. Then it stops changing.
Thermal equilibrium
A system is in equilibrium when its observable properties stop changing.
Notice: we defined equilibrium without ever using the word temperature.
“Being in thermal equilibrium with” is a relation: give it two systems, it answers yes or no.
This relation has three properties
every system is in equilibrium with itself
obvious
if A is in equilibrium with B, then B is with A
obvious
if A~C and B~C, then A and B are in equilibrium too
not obvious
\forall a: a\sim a \qquad a\sim b \Rightarrow b\sim a \qquad a\sim b \wedge b\sim c \Rightarrow a\sim c
A relation with all three has a name in mathematics. We will get to it shortly.
The Zeroth Law of Thermodynamics
⇒ A and B are in equilibrium too (Zeroth Law)
T(A)=T(C)\ \text{and}\ T(B)=T(C)\ \Rightarrow\ T(A)=T(B)
- In practice: the thermometer reads 20 °C on the water and 20 °C on the metal, so they are in equilibrium — without ever putting them in contact.
- An experimental fact, not a logical necessity. It could have gone otherwise.
- Without it, a thermometer would only ever measure itself.
Why is it called the Zeroth Law?
- The First and Second Laws were established and numbered in the 1850s.
- Decades later, physicists noticed that both quietly assumed this more basic fact.
- It had to come first — but First and Second were taken.
- The only number left before one was zero. Ralph Fowler, 1930s.
The order of discovery and the order of logic are not the same thing.
Le tre insieme: una relazione di equivalenza
Una relazione di equivalenza si comporta come “essere uguali” — non identici in tutto, ma uguali per un aspetto specifico. Divide automaticamente il suo insieme in classi che non si sovrappongono mai.
Sono nato nel mio stesso anno — riflessiva. Se condivido il tuo anno, tu condividi il mio — simmetrica. Se io e te siamo del 2010, e tu e Marco siete del 2010, allora io e Marco lo siamo — transitiva.
Non sono più alto di me stesso — fallisce. Se sono più alto di te, tu non sei più alto di me — fallisce. È una relazione d'ordine. Mette le cose in fila. Non le raggruppa.
Un esempio
Una frazione è una COPPIA di interi: (1,2) e (2,4) sono coppie diverse.
(a,b)\sim(c,d) \iff a\cdot d = b\cdot c
Verifica: (1,2) ∼ (2,4) perché 1·4 = 2·2
un punto. Tre nomi.
Un numero razionale non è una coppia — è l'INTERA CLASSE: {(1,2), (2,4), (3,6), (50,100), ...}
“1/2” nomina la classe attraverso un rappresentante comodo.
Le classi di equivalenza sono ovunque
Fra 100 giorni è lo stesso giorno della settimana che fra 2: 100 e 2 danno lo stesso resto diviso 7. Sette classi.
Le 15:00 e le 3 sono la stessa classe. Un orologio è un insieme quoziente reso visibile.
“Stesso gruppo di” partiziona tutti in A, B, AB, 0. Quattro etichette, milioni di persone.
“Parallela a” è una relazione di equivalenza. Una direzione non è una retta — è una classe di rette.
Dunque: la temperatura è un'etichetta di classe
- “Essere in equilibrio termico con” è una relazione di equivalenza — riflessiva e simmetrica ovviamente, transitiva per il principio zero.
- Quindi partiziona ogni sistema termodinamico dell'universo in classi.
La temperatura è il nome della classe.
\tau: S/\sim \ \rightarrow\ \mathbb{R}
la temperatura è una FUNZIONE sull'insieme delle classi, e assegna un numero diverso a ogni classe.
Un numero razionale è una classe di frazioni. Una direzione è una classe di rette. Una temperatura è una classe di sistemi.
One more question, before the fix
Does a thermometer measure the object's temperature — or its own?
Hold that thought.
And now the thermometer comes back
- A thermometer is a system with one property influenced by its surroundings and easy to read: the volume of a liquid, the pressure of a gas, the resistance of a conductor.
- You put it in contact. It joins the same class as the object. It makes the label readable.
- Temperature was defined BEFORE the thermometer, not by it. That is what breaks the tautology.
- This is also why you must wait before reading a thermometer — you are waiting for equilibrium.
A real problem, in one sentence
Two thermometers — one mercury, one alcohol — calibrated to agree at 0 °C and 100 °C, read a few tenths of a degree apart in between: mercury and alcohol expand according to different laws.
They agree on WHICH systems are equivalent. They disagree only on the number in between — a scale problem, not a physics problem.
Quick check — answer it yourselves
How long should you leave a thermometer? Why can you not read it immediately?
You are not reading the object. You are waiting for the thermometer to join the object's equivalence class — to reach equilibrium with it.
Read it too early, and it has not “voted” yet: its number still belongs to its own earlier class, not the object's.
One last question: temperature of what?
A thermodynamic system is a system for which it makes sense to define a temperature — a macroscopic system with an “inside” the surroundings can influence.
Some systems hold their own temperature steady against the surroundings.
The human body. A cat. A working fridge.
They have a temperature. They just fight equilibrium instead of drifting into it.
A single atom has no temperature.
Neither does an idea, or a radio wave.
Temperature needs a system with many parts. We will see why on 2 October.
One list, three different questions
For which of these does it even make sense to give one temperature? (Hint: not everything is uniform inside.)
Which ones are heading toward equilibrium with the room — and which are actively resisting it?
Same list. Look again for each question — do not just sort once.
Pass 3 — could this be a thermometer?
- Battery, magnet, spring, resistor: each has a property that changes WITH temperature — voltage, magnetisation, elasticity, resistance.
- That is not a coincidence in the list. Each of them is a thermometer candidate.
- Which property would you read? Would it be easy to measure?
Give four more examples of your own, from ordinary objects around you — and say which pass they belong to.
Quick vote — no discussion, just choose
Regarding heat and temperature, we can say that:
- they are equivalent concepts
- a body receiving energy as heat must necessarily change temperature
- a body receiving energy as heat may change temperature
- temperature indicates the amount of heat a body contains
- none of the above
Count the votes. Write them down. We come back to this on 6 October. · Quiz from Vicentini Missoni, 1989.
Back to the four cubes
- Read both counts on the board. Glass, metal, wood, plastic — all four were at the SAME temperature both times: room temperature, then 43 °C.
- Your hand does not measure temperature. It measures how FAST energy leaves your finger.
Metal conducts energy away from your skin fastest. Wood and plastic conduct slowly, so less energy leaves you per second — they feel closer to your own temperature.
Above your skin's temperature the flow reverses — energy now moves INTO your finger. Metal still moves it fastest, so now it should feel hottest.
Why does the rate depend on the material? That one stays open — 30 October.
Three things to take away
Coming soon
The Eiffel Tower is taller in summer. By how much? Predict first.
Is the Sun's radiant heat actually heat? Does cold exist, or only less heat?
Can a body receive heat without changing temperature?
Coming soon
Why is 20 °C in air comfortable and 20 °C in water freezing? Why does blowing on soup cool it, but blowing on your hands warms them?
Hot water and cold water into the freezer — which freezes first? (Mpemba effect: genuinely disputed. We will hold it open, honestly, not resolve it for a cheap answer.)
For next time
1. Finish the three-pass classification from today.
2. Add four examples of your own, taken from ordinary objects around you.
- Everyone's four will be different. That is the point — our first real data set.
- Short. No calculation. More than one page means you misunderstood the task.
- Next lesson, 11 September: how do you actually build a thermometer?