A glass of hot chai left on the table cools until it is as warm as the room. Nobody has ever seen a glass of chai at room temperature grow hot by itself while the room around it cools a little. Energy would be conserved in both cases. Yet only one of them ever happens.
What the first law does not forbid
The first law says that energy is conserved. It does not say which way a process runs. A block sliding on a rough floor stops, and its kinetic energy becomes heat in the block and the floor. The reverse would also conserve energy: a warm block and floor could cool and push the block into motion. The first law allows both films.
In the same way, the first law would allow an engine with \(Q_C = 0\). It would take heat from the sea and turn every joule into work. Cooling one cubic kilometre of sea water by \(1\,\text{K}\) releases \(4.2 \times 10^{15}\,\text{J}\), with \(c_\text{water} = 4200\,\text{J/(kg K)}\). That would run a \(1000\,\text{MW}\) power station for about \(49\) days, with no fuel. No such engine has ever worked.
The failure of every such attempt is itself a law of nature. It is the second law of thermodynamics, and it has two classic forms. The first form, due to Kelvin and Planck, is about engines.
No engine working in a cycle can take heat from a single reservoir and turn all of it into work. Some heat must always be given to a colder reservoir.
The words "in a cycle" matter. A gas expanding at constant temperature does turn all the heat it takes into work, as the chapter on the First Law showed. But it ends with a larger volume and a lower pressure, so it cannot repeat the stroke without being compressed. Bringing it back costs work and rejects heat. The statement forbids only the perfect engine that runs for ever.
A machine that would break this law is a perpetual motion machine of the second kind. It breaks no energy law, since it only turns heat into an equal amount of work. It fails because it has no sink. So the Kelvin-Planck statement is often put in one line: the efficiency of a heat engine is always less than \(1\).
An inventor proposes a ship that takes in warm sea water and extracts heat from it to drive the propeller. The cooled water, now ice, is dropped overboard. No fuel is burnt. What is wrong with it?
Heat flows downhill
The second form of the law, due to Clausius, is about the direction of heat flow. Heat flows by itself from a hot body to a cold one, as in the cooling chai. The reverse never happens on its own.
No device working in a cycle can move heat from a colder body to a hotter body with no other effect. Heat does not flow from cold to hot by itself.
A refrigerator does move heat from its cold inside to the warmer kitchen. It does not break the law, because it has another effect. Its motor takes electrical work from the mains, and the kitchen receives that work as extra heat. The law forbids only a transfer from cold to hot that happens with nothing else changing.
A refrigerator takes \(300\,\text{J}\) of heat from its inside and gives \(400\,\text{J}\) to the kitchen in the same time. Why does it not break the Clausius statement?
Two statements, one law
The two statements look different: one is about engines, the other about heat flow. In fact each one implies the other. If either could be broken, so could the other. Fig 2.1 shows one half of the proof with numbers.
Suppose a perfect refrigerator existed, one that breaks the Clausius statement. Join it to an ordinary engine as in Fig 2.1. The cold sink receives \(600\,\text{J}\) from the engine and loses \(600\,\text{J}\) to the refrigerator, so nothing changes there. The hot source gives \(1000\,\text{J}\) and gets \(600\,\text{J}\) back, a net \(400\,\text{J}\). The combined machine turns that \(400\,\text{J}\) wholly into work.
That combined machine is a perfect engine, forbidden by the Kelvin-Planck statement. So a perfect refrigerator cannot exist either. The other half runs the same way. A perfect engine could take \(300\,\text{J}\) from the source and turn it into \(300\,\text{J}\) of work. That work could drive an ordinary refrigerator that moves \(900\,\text{J}\) from the cold body and gives \(1200\,\text{J}\) to the hot one.
Count the heat of the hot body in that second machine. It gives \(300\,\text{J}\) to the perfect engine and receives \(1200\,\text{J}\) from the refrigerator. Net, \(900\,\text{J}\) has moved from the cold body to the hot body, and no work came from outside. That breaks the Clausius statement. So each statement guards the other.
Which statements about the two forms of the second law are true?
Both films in Fig 2.2 keep the total energy at \(8\,\text{J}\). But only the forward film is ever seen. The backward film needs the random jiggling of the molecules in a warm block to line up and push the whole block the same way. That never happens. The sliding stop is an irreversible process.
A reversible process is one that can be run backwards so that the system and everything around it return exactly to their first states. A process is reversible only if it is quasi-static, so that each stage is an equilibrium state. It must also have no friction, and heat may flow only across a vanishingly small temperature difference.
Three things make a process irreversible. Friction turns ordered motion into heat. A free expansion lets a gas rush into a vacuum, and the gas never gathers back into its corner by itself. Heat flowing across a finite difference of temperature, as from hot chai to the room, never flows back. Every real process has at least one of these.
A reversible process is like an ideal slope with no friction. It is never quite reached, yet it is useful, because real processes can come as close to it as we please. Squeeze a gas very slowly with a well-oiled piston, touching a reservoir only a hair warmer or colder than the gas. The process is then almost reversible.
Which of these processes comes closest to being reversible?
Energy is conserved both when a warm block and floor cool while the block starts to slide, and when a sliding block stops and warms. Explain why only the second is ever seen.
When the block stops, the ordered motion of the whole block is shared out as random jiggling among an enormous number of molecules. To reverse it, all those random motions would have to line up and push the block the same way at the same moment. The first law allows this, but the chance is so small that it never happens. The second law of thermodynamics states this fact as a law: the process is irreversible.
Now return to the glass of chai. The first law cannot decide between a cooling glass and a warming one. The second law does. Heat flows by itself only from hot to cold, so the chai cools and never warms again. The next lesson finds how close a real engine can come to the ideal, reversible limit.
In 1850 the German physicist Rudolf Clausius showed that Carnot's ideas survive if heat never passes by itself from cold to hot. A year later in Glasgow, William Thomson, later Lord Kelvin, wrote a second form. No engine can draw work from a body by cooling it below its coldest surroundings. Both men were trying to reconcile Carnot's engine with the new idea that heat is energy. Together their papers founded the second law.
A plant for ocean thermal energy conversion uses the warm surface of a tropical sea as its source. Its sink is cold water pumped up from hundreds of metres below. Such a plant obeys the second law because it has two reservoirs. With only about \(20\,\text{K}\) between them, its efficiency is only a few percent. In Lakshadweep, plants of the National Institute of Ocean Technology use the same warm and cold water to make fresh water from the sea.
- Thinking that a refrigerator breaks the second law.The Clausius statement forbids moving heat from cold to hot with no other effect. A refrigerator uses work.
- Believing that an isothermal expansion breaks the Kelvin-Planck statement.The statement is about a cycle. A single expansion does not repeat; bringing the gas back rejects heat.
- Calling any slow process reversible.It must also be free of friction, and heat may cross only a tiny temperature difference.