The zeroth law states that if 2 systems are in thermodynamic equilibrium with a third system, then they are in equilibrium with each other. I assume that this is really important since it’s a law but it seems akin to saying that if 2 objects have the same mass as a third, then they have the same mass as each other, which doesn’t seem particularly profound.
What am I missing?
It may seem obvious, but it’s not something you can simply assume. For example, the Peano axioms for arithmetic contain the axiom:
For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
Same idea. In a way, this is just a way of establishing the meaning of equality (or here, equilibrium).
The 0th law of Thermodynamics is basically “thermometers are a thing”. That is, you know the temperature of object A because you used thermometer B which was calibrated against object C.
The hint is in the name: thermodynamics deals with dynamical systems, so equilibrium between them is not the same thing as asserting a mathematical identity.
Also, that all heat is the same kind. Perhaps 100 quatloos can be exchanged for 100 eurodollars, and 100 eurodollars can be exchanged for 100 whuffies, but 100 whuffies are only worth 50 quatloos.
So, no heat arbitrage. It’s all the same “currency”.
There’s no a priori reason to think that thermodynamic equilibrium must be a transitive relation. Not all relations are transitive. For example, if two people are both friends with a third person, it doesn’t automatically follow that they are friends with each other.
And it’s always good practice to be explicit about your assumptions, because sometimes, it turns out that something that you just assumed was obvious isn’t actually true. Like, everyone knows that when you add heat to something, its temperature goes up… except that sometimes, adding heat will make something’s temperature go down.
Interesting. Can you give an example of this?
As a simpler example than @Chronos alluded to, sometimes adding or subtracting heat causes no change in temperature.
Right at the 0C freeze point of ordinary water, quite a lot of energy must be injected to convert ice to liquid water and only then does the bulk temperature start to increase. Likewise quite a lot of energy must be extracted to convert 0C liquid water into ice. Only after all that excess energy is removed does the bulk temp continue decreasing below 0C.
Heh, that sounds like the notorious stall point when smoking a brisket.
Or how deep water can’t get colder than 4 degrees Celsius no matter how cold the surface is.
Kinda. I think the consensus is that the stall point is when evaporative cooling from water in the meat starts to dominate. So the temperature stays stable until enough water is lost that evaporation doesn’t help.
It’s a phase change, which is the same thing that happens in the water->steam transition, but it’s not kept at the temperature of the phase change (which would be 212 F here). Instead it’s a sort of intermediate temperature based on typical moisture content. The heat absorbed by the meat is the same as the cooling effect of the water. If it got a little hotter, the evaporation would increase and cool it back down again. So it’s fairly stable for a while.
That’s not what’s happening during a phase change of a pure substance, like water or CO2. In those cases, the temperature is kept absolutely stable, and all the entropy of the heat goes into the arrangement of the molecules–breaking up the crystalline lattice in the case of ice->water. Only once the phase change is complete (all the ice has turned to water, or all the water has turned to steam) does the temperature increase again.
The example I was thinking of was a black hole, but I’ve heard that it’s a property of gravitational systems in general (given appropriate definitions of “temperature”).
EXACTLY what I thought, too!
I thought it was a connective tissue/collagen thing.
I’ve seen both theories.
Could be both.
For example, as any sports-bettor (should) know, superiority is not transitive.
I was always told that thermometers on only measure the temperature of one thing – the thermometer itself. Everything else is an inference.
I read the zeroth law as provding a bootstrap definition of temperature. It (plus some more basic assumptions) says that “is in thermal equilibrium with” is an equivalence relation. And equivalence relations have equivalence classes, and if you put a label on each of those equivalence classes, those labels are “temperatures”.
Zeroth-law-only temperature doesn’t include an ordering, so there is no concept of hot or cold, just same or different, but it can be used for some things.
(where “this” = unusual heat/temperature relationship)
When two objects (one hotter than the other) come into thermal contact, energy will flow between them until the global entropy is maximized. Note that the directionality of flow cares about entropy and not energy content. Thermal equilibrium is reached when the change in entropy per unit change of internal energy is equal between the two subsystems. (Otherwise, the sum of the entropies could be made higher still by moving a bit of energy from one subsystem to the other.)
In “normal” systems, increasing the internal energy increases the entropy. In a discrete system (often realized with spin states) or a cleverly constrained system where there is a maximum internal energy, you can have the number of available internal states start to decrease as internal energy increases, and thus so too does the entropy – which is the logarithm of the number of available internal states – decrease as the internal energy increases. In those systems, energy needs to flow out to increase the entropy. And that’s what matters.
The italicized text two paragraphs above is the fundamental definition of the inverse of temperature. So, 1/(temperature) is defined as “change in entropy per unit change of internal energy”, and so the latter being equal between two systems in thermal equilibrium is the same as saying the temperature is equal. But, since entropy might decrease with energy added, temperature can be negative.
But it also means that temperature (T) is mathematically sort of the wrong quantity to use when talking about equilibrium and energy flow. 1/T or 1/(k_B T), usually written \beta, handles this more cleanly.
Let “cold” and “hot” relate to how energy will flow (i.e., from hot to cold). Then T=+0 is the coldest you can get, and then T=\infty is a lot hotter, and then T=-\infty is hotter still, and then T=-0 is the hottest you can get (where suitable limits should obviously be taken in this notation).
So, any negative temperature is hotter than infinitely positive temperature. And, say, -1 Kelvin is way hotter than 1 Kelvin. Which is to say: a system as -1 K will very happily give up heat to a system at 1 K.
If we were to use \beta instead of T, the language gets a lot cleaner. Higher \beta means colder; lower \beta means hotter. Full stop. One is free to talk about -\beta instead to flip the axis over if you want hotter to run in the positive direction, but that’s not the usual convention for \beta, and there’s no particular need to do so.
In summary, \beta is just “change in entropy for a change in energy”, and that can be positive or negative for a given system and that “slope” varies as energy changes, and energy flow ceases when \beta is equal between two systems in thermal contact, and energy always flows from the lower \beta system to the higher \beta system.
When negative \beta's are involved, it can no longer be said that “energy always flows from higher T to lower T”, but that’s mostly because T was a poor choice in the first place.
But, for everyday systems, energy is unbounded and/or there are suitable “continuous” degrees of freedom such that entropy is always strictly increasing with energy, and so we typically experience only the positive temperature region.
Thanks for that write-up. It’s a great explanation of negative temperature.