I thought we'd talk about the laws of thermodynamics, and maybe the second law in particular. So there are four laws: there are laws one to three, and then there's law zero, which I suspect was slotted in after the first three laws, and is actually, in some sense, also the most trivial of them. Which is why it's probably called the zeroth law. So the zeroth law just basically says: If you've got three bodies -- call them A, B and C -- if A and B are in thermal equilibrium with each other, which is just a fancy way of saying they're the same temperature, which means if you put them together no heat flows from one to the other. So if A and B are in thermal equilibrium, and if A and C are in thermal equilibrium, then B and C will be in thermal equilibrium. It does seem fairly obvious. It's a kind-of a formal way of saying really there is only one definition of temperature. And that there is no way you can come up with a system where temperature is defined in a funny way such that one of these bodies has some peculiar property, such that its temperature as far as body A is concerned is one thing, but its temperature as far as body C is concerned is another thing. And I think because it is stating the obvious I think that's why it is the zeroth law. So the first law is also fairly obvious, in that its basically a sort-of restatement of the conservation of energy. In thermodynamics there's kind-of two sorts of energy: There's useful energy, which is work. So that's when if you push something, or whatever, that's sort-of applying energy to it, and that's useful energy. And then there's useless energy, which is heat. And what the first law says, is that if you've got a body, and you apply a certain amount of work to it, so that's one sort of energy, and then you add a certain amount of heat to it, the total energy of that body will be whatever it was to start with, plus the energy you added from the work, plus the energy you added from the heat. So it's really just the statement of the conservation of energy really. So they are kind-of obvious laws, but you have to start with the basic building blocks in order to build up the entire subject of thermodynamics, and these are, really, the most fundamental building blocks. The third is difficult to talk about until we introduce the concept of a thing called entropy. Which is to do with the disorder of the system. And the third law is basically saying that, as you cool a system down towards absolute zero, the degree of disorder, so this quantity entropy, drops towards zero or at least drops to a very low value. So classically it will drop to zero... there are some sorts of systems it'll just drop to a low value... but it will decrease until you get to absolute zero and then it will stop. So the second law has various formulations... The first formulations were put together really before this idea of entropy and disorder was invented. And they were very kind-of operational definitions of it. So there are two classical statements of the second law of thermodynamics: The first is one due to a guy called Clausius, who said that heat wont travel from a cold body to a hot body. And that's sort-of trivially known if you've just got a hot body and a cold body, and you put them in contact with each other, the heat will flow from the hot thing to the cold thing, so they'll end up at the same temperature, they'll meet somewhere in the middle. And so it's kind-of obvious if you're just putting things in contact with each other but his statement is stronger than
that, which is there is nothing you can do to make heat flow from a cold body to
a hot body, without putting some work in. So if you're prepared to inject some
energy you can make hot things colder and cold things hotter. And so a fridge is an example where you pump heat out of something which is already cold and add it to the air which is warmer. So fridges, in some sense, they violate the
purist view of this law that you can't get heat to travel from
something cold something hot, but the get-out is "but you're putting work in, so you're allowed to do it". So if you've just got an isolated
system you can't do it. It's a one-way street, and there's no way around it is the important point, that no matter how clever a machine you come up with, you can never make it so it will extract the heat from the colder thing and add it to the hotter thing without having to put some work in along the way So that was Clausius' statement and then there's the other statement and that's, in some sense, that's a sort of physicists statement because that's about, you-know, heat flow and observations of what happens when you put bodies in contact with each other. The second version of this is due to
Lord Kelvin, and Kelvin's statement is more of an engineer's statement. This comes back to this fact that there are these two sorts of energy: this kind-of useful energy, that you can do work with, and this useless energy, heat, that you can't really do work with directly. And Kelvin's statement basically says that you can't turn heat directly into work. You can't just take heat and turn it into into useful energy. Now of course again, you do this... a steam engine, for example, takes heat and turns it into work... But in order to do that, you have to have waste heat left over. So his statement is that you can't, again, have a kind-of sealed box, where you just put heat in at the top and the only thing that comes out is work. You're always going to have to have: heat going in, work coming out, but then some waste heat coming out as well. There is a fundamental limit as to how little the amount of waste heat there is that you can get out of it... And you can figure out what that limit is... But it's always a finite amount. You can never reduce that to some, you-know, completely infinitesimal zero level. Those two things sound like completely different statements, like they're talking about completely different things, but it turns out they're exactly equivalent to one another. And I can show you, if you like. So the way we're going to do this is by saying if one of these versions of the second law isn't true then the other one isn't true either. So, let's assume... lets violate Clausius's
statement of the second law which basically says "you can't get heat to
flow from the cold body to the hot body". So let's assume we have a magic fridge,
which violates that law by actually allowing us to suck energy out of the cold body, put it into the hot body without actually having to plug the fridge into the mains. Okay? And then over here I'm going to just have an engine, like a steam engine or something, which doesn't violate any of the laws at
all, and I'm going to take an amount of energy out of this hot body. And because this isn't breaking any laws there has to be some waste heat from it. So we'll set it up so that amount of weight heat is actually equal to Q1. [Brady] Keeping Lord Kelvin Happy. [Professor Merrifield] So, yes exactly, that's not violating any of the laws and we're going to get some work out of this, which is just the difference between those two. This is the first law again that says
that you have to conserve energy overall, so if we've got Q2 flowing in and then the amount of work we get out of there is the difference between the two. Now, the neat part here is that what we've managed effectively to do... If I just draw a box around this little lot here... And so I'm going to stick all that in a box and I'm not going to care what's inside that box. And what that box is actually doing is it's taking an amount of energy... well how much energy is coming in? it's taking heat out of this body... and it's taking
an amount... well theres Q2 flowing out and Q1 flowing back in, so the amount of
energy that's coming out of this hot body is Q2 minus Q1... And this box then, the only thing that's coming out of it is work. Q2 minus Q1. So this now violates Kelvin's version of the second law, because actually all we've done in this process is we've taken an amount of heat out of the hot body and turned it directly into work, and Kelvin's statement of the law says you can't do that. [Brady] Professor, that shouldn't be a surprise because we built the box by breaking the rules... [Brady] we just broke Clausius's wording of the rules. [Professor Merrifield] Exactly! But what we're trying to show is that those two statements of the second law are the same as each other, they're really stating the same law. And so, indeed, if they are the same law, you'd expect if you broke one version of the law you'd break the other version of the law as well. But if they're actually different laws, or stating different things, then that wouldn't necessarily be the case. But that's kind of halfway there... that shows that if we violate Clausius's version then we're violating Kelvin's version, but to really show they're equivalent we've also got the show that if we violate Kelvin's version we're also violating Clausius's version. Okay, so Kelvin's version of the law says that you can't just turn energy into work. So you can't just take heat and turn it directly into work: there has to be a by-product. So let's violate that by coming up with
a magical engine which directly turns energy, turns heat, into work. An engine which which does nothing but take this useless energy, heat, and turn it directly is useful energy, work, An infinitely efficient steam engine... In a
real steam engine there'd be waste heat coming out of the bottom. But we're violating Kelvin's law here so we don't have to do that. Okay, and then we're going to take that work, and we're going to use it to power a fridge. And that fridge will... because this is now a proper fridge, it's not violating Clausius's law or anything... ...we can get it to work. So we've plugged it into the mains and we can actually use it to suck energy, Q2, out. So we can figure out how much heat's going to come out the other end. It's just going to be the two lots of
energy going in. That's Q1 and Q2. So now, if we just put a box around all this
lot, and say this is this is now my magic machine... And I don't really want to
look inside the box I just want to know what it's doing... What's happening? It's
sucking an amount of energy, Q2, out of the cold, and then, on the hot side, you've
got Q1 coming out and Q1 plus Q2 going in, so the net amount of energy flowing into
the hot is Q2. So effectively we've created a machine here where... with nothing, you-know, no external wires or anything... within that box, it takes in an amount of heat, Q2, from a cold body and adds it to a hot body. And that violates Clausius's version of the second law. So what we've shown is that if Kelvin's version of the second law isn't true, then Clausius's version isn't true, and if Clausius's version of the second law isn't true then Kelvin's version of the second law isn't true. And so, just to kind-of finish it off... So we've got these two laws: Kelvin's version and Clausius's version. And there are sort-of four possibilities: They might both not be true, one might be true and the other one might be false, or the other way around, or they both might be true, And what we've ruled out by showing this is we've shown if Kelvin's law is false then Clausius's has to be false too. If Clausius's version of the law is false, then Kelvin's can't be true. It has to be false as well. So we ruled out those two in the middle. So the only two possibilities is they're either both wrong or they're both right. And that's just another way of saying that they're saying exactly the same thing. We haven't been able to prove that they're true, but we've shown that if one of them is true the other's true, and it's one of them is false, the other one's false. In other words, they're kind-of equivalent to each other.
Is there any reason the zeroth law of thermodynamics has to be stated as a law? Is there any reason to not assume such transitivity in the real world?