The second law
A cup of coffee that spontaneously heats up while the room around it cools would conserve energy exactly. So would a wrecked car reassembling itself. The first law has no objection to either, which is why a second law had to be written down.
Everything up to this point has been accounting. Energy went in, energy went out, and the difference was stored. The accounts have never once told you which way a process would run, and it is worth being clear that this was not an oversight waiting to be corrected by a longer calculation. Nothing in the accounts can tell you. That information has to come from somewhere else.
Direction
Put a cup of coffee at \(70\,^\circ\text{C}\) in a room at \(25\,^\circ\text{C}\). It cools. The heat that leaves the cup is exactly the heat the room receives, so the total energy of cup and room together is unchanged, and the energy balance is satisfied to the last joule.
Now imagine the reverse. The room, which is enormous, gives up a tiny amount of heat — so little that its temperature falls by an immeasurable fraction of a degree — and the coffee takes that heat in and grows hotter. Write the energy balance for that process. It balances. The same \(Q\) appears with the same two signs on the same two systems. There is no arithmetic anywhere in the first law that comes out wrong, and no term you can point to and say that this is the one being violated.
The same is true of every one-way process you can think of. A paddle wheel driven by a falling weight warms the water in a tank; the water has never yet cooled itself and lifted the weight back up, though the energy would balance if it did. A rock dropped on a floor warms the floor slightly; a floor has never yet chilled itself and launched a rock into the air. A car in a wreck converts a great deal of kinetic energy into deformation and heat, and the reverse film — the wreck gathering itself up and reversing off down the road — is absurd without being unbalanced.
What this means is that satisfying the first law is a necessary condition for a process to occur and not a sufficient one. The first law is an equality, and equalities read the same in both directions. Direction is a separate fact about the world, and it needs a separate law to state it. That law cannot be derived from the first; it has to be added, and it is added in the form of a flat assertion about what cannot be built.
Two such assertions are used, both negative, both about machines. Neither has a proof in the sense that a theorem has a proof. They rest on the fact that nobody has contradicted them, that a great many people have tried, and that every consequence drawn from them — including some very unobvious ones, such as an upper limit on the efficiency of a power station that does not depend at all on what the power station burns — has held up under measurement.
Three things come out of the second law, and it is worth knowing what you are getting before the machinery arrives. It tells you which way a process will go. It sets a hard ceiling on how well a device can possibly perform, so that a claimed performance can be checked before any drawing is opened. And it lets you say how much of the difference between the real device and the ceiling is your own fault rather than the universe’s.
A cup of coffee, and a room large enough not to care.
Engines and the Kelvin–Planck statement
The statements are about machines, so the machines need defining first, and the first of them is a body rather than a device. A thermal energy reservoir is anything with a heat capacity large enough that it can absorb or supply finite quantities of heat without its temperature changing. The atmosphere, a lake, the ocean, the ground beneath a building, a furnace held at a set point, a two-phase mixture boiling at constant pressure — all behave as reservoirs on the scale of a laboratory process. The word is a convenience, not a new kind of matter: it means that \(T\) can be treated as a constant on one side of the boundary.
Now the asymmetry the whole subject turns on. Work converts to heat completely and easily: a paddle wheel, a resistor, a brake pad, and every joule of work becomes a joule of internal energy in something. Heat does not convert back to work completely, and no amount of cleverness has ever made it do so. A device that takes heat and delivers work must be built with some care, and it has three fixed features: it receives heat from a high-temperature reservoir, it converts part of that heat to work, and it rejects the remainder to a low-temperature reservoir. It is a heat engine.
The fourth feature is the one that costs you. It operates on a cycle, returning to its initial state, which is what allows it to keep going rather than running once and stopping. Energy is a property, so over a cycle the change in the working fluid’s energy is zero, and the energy balance collapses to a statement about the boundary alone,
where both heats are written as magnitudes with their directions carried by the words rather than by the signs. The fraction that came out as work is the thermal efficiency,
which is worth reading as what you got divided by what you paid for. The numbers are humbling. A petrol engine turns roughly a quarter of the chemical energy released in it into work at the crankshaft; a large steam power plant reaches around forty per cent. Three quarters of the fuel burnt in a car, and more than half of that burnt in a power station, leaves as warm gas and warm water.
The Kelvin–Planck statement is the assertion that this cannot be tidied up: it is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work. Both qualifiers do real work in that sentence. A single isothermal expansion of a gas does turn heat entirely into work, and there is nothing wrong with it; it is simply not a cycle, because the gas ends up somewhere else, and to bring it back you must give some of the work back. And a single reservoir is forbidden, not a single heat transfer, which is why an engine needs a sink as well as a source.
The consequence to take seriously is that \(Q_L\) is not an engineering failure to be designed out by a better arrangement of pipes. It is the condition of the engine running at all. Every heat engine ever built, and every one that ever will be, throws away heat at the bottom of its cycle, and a design that claims otherwise — a perpetual motion machine of the second kind, in the usual phrase — can be dismissed on this paragraph alone, without examining the drawings.
Two reservoirs, and a device that returns to its own starting state.
Refrigerators and the Clausius statement
Heat flows from hot to cold on its own. To move it the other way you need a machine, and the machine is the heat engine run backwards: the same four components in the same loop, with the working fluid — a refrigerant chosen so that its saturation temperatures land where you need them — carried around by a compressor rather than driving a turbine. It absorbs \(Q_L\) from the cold space in an evaporator, receives \(W_{\text{net,in}}\) at the compressor, and rejects \(Q_H\) to the warm surroundings in a condenser. Over a cycle the same argument as before gives \(Q_H = Q_L + W_{\text{net,in}}\), and the hardware is the subject of steady-flow analysis.
Performance is measured the same way as for an engine, as what you want over what you pay, but what you want depends on which side of the wall you are standing. For a refrigerator it is the heat removed from the cold space:
It is called a coefficient of performance rather than an efficiency for a plain reason: it is routinely greater than one, and a quantity called an efficiency that comes out as three would invite the suspicion that something had been created out of nothing. Nothing has. The work is not producing the cooling; it is moving heat that already existed, and there is no rule against moving more energy than you spend doing the moving.
Point the same machine the other way round — leave it outside and let it warm the house instead — and what you want is the heat delivered upstairs, so
for the same device between the same two reservoirs. The relation is arithmetic rather than deep, but it carries a useful floor with it: since \(\mathrm{COP}_R\) cannot be negative, \(\mathrm{COP}_{HP}\) can never fall below one. In the worst imaginable case a heat pump delivers exactly the work you put into it, which is what an electric bar heater does, and any real heat pump does considerably better because it brings in \(Q_L\) from outside for free.
The Clausius statement is the assertion that the work cannot be dispensed with: it is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a lower-temperature body to a higher-temperature body. The phrase no effect other than is the whole content. Heat is moved from cold to hot in every refrigerator in the world; what no refrigerator does is move it while leaving everything else exactly as it was.
Now the fact that makes these two statements one law rather than two. Suppose someone hands you a machine that violates Kelvin–Planck: it takes \(Q_H\) from the hot reservoir and delivers the whole of it as work, with no rejection at the bottom. Connect that work to an ordinary refrigerator running between the same two reservoirs. The refrigerator draws \(Q_L\) from the cold reservoir and dumps \(Q_L + Q_H\) into the hot one. Look at the pair together as a single device. The hot reservoir gives up \(Q_H\) and receives \(Q_L + Q_H\), a net gain of \(Q_L\); the cold reservoir has lost \(Q_L\); no work has crossed the outer boundary at all. That is precisely the device Clausius says cannot exist. The argument runs equally well the other way, so violating either statement lets you build a violation of the other, and the two are the same prohibition seen from two sides.
The same hardware between the same reservoirs, driven the other way.
Reversibility
A reversible process is one that can be reversed without leaving any trace on the system or on the surroundings. Both halves of that matter, and the second half is where the definition bites. Any process can be undone as far as the system is concerned — put the gas back at the pressure and temperature it started at and it is, by the state postulate, in exactly the state it started in. The question is what it cost the surroundings to put it there, and whether they too came out unchanged.
No real process passes that test. The mechanisms that fail it are called irreversibilities, and the same short list turns up everywhere. Friction, wherever two surfaces slide, converting work into heat that will not convert back. Unrestrained expansion, where a gas is let into a vacuum and does no work at all going out, though compressing it back would cost you plenty. Mixing, of two gases or of hot and cold water, which nobody has ever run backwards. Inelastic deformation, current through a resistance, chemical reaction. And heat transfer across a finite temperature difference.
That last one deserves its own paragraph, because it is the one that catches people who accept all the others. Heat crossing a boundary between a body at \(T_1\) and a body at a lower \(T_2\) is irreversible for a reason already established: undoing it would mean carrying that heat back up the temperature difference with no other effect, and that is the device Clausius forbids. Note what follows. Heat transfer is reversible only in the limit of zero temperature difference — and the rate of heat transfer is proportional to that difference, so the reversible limit is also the limit in which nothing happens at all. This is not a quibble. It is the trade every heat exchanger in industry is designed around: a larger area buys a smaller driving difference, which buys less waste, at a price per square metre.
It is useful to split the accounting. A process is internally reversible if no irreversibilities occur within the system boundary, so that the system passes through a series of equilibrium states and its path can be drawn on a property diagram at all. It is externally reversible if none occur outside it, which in practice means no heat crosses the boundary through a finite temperature difference. Totally reversible means both. The division is worth keeping because a great many textbook processes are internally reversible and externally not, and the difference is exactly where the losses will be found.
Internal reversibility is close to the quasi-equilibrium idea but not identical to it, and conflating the two is a common error. A process can be slow enough for the system to be uniform throughout — quasi-equilibrium in good standing — and still be irreversible, because the piston is dragging on the cylinder wall. Slowness removes the pressure gradients. It does nothing about friction.
Which raises the obvious objection: if no process is reversible, why spend time on one? Because it is the yardstick, and there is no other candidate. A reversible process is the theoretical limit of the corresponding real one, so it fixes the best that could conceivably be done, and the comparison is where engineering judgement lives. A turbine that produces eighty-five per cent of the work its reversible counterpart would produce is a good turbine; the same figure quoted against no reference at all means nothing. Idealisations of this kind are also easy to analyse — that is rather the point of them — and the answers they give bracket the real ones from a known side.
Take the weight off a grain at a time and the gas stays in step with it.
The Carnot limit
If reversible processes are the limit, then the best possible engine is made of nothing but reversible processes, and there is one obvious way to build it. Heat may only be received or rejected reversibly at the temperature of a reservoir, so the two heat transfers must be isothermal, and everything in between must be adiabatic. That gives four processes and no room for a fifth. A gas in a cylinder expands isothermally at \(T_H\) while in contact with the hot reservoir, taking in \(Q_H\). Contact is broken and it continues expanding adiabatically, falling to \(T_L\). It is compressed isothermally at \(T_L\) against the cold reservoir, giving up \(Q_L\). Then it is compressed adiabatically back to \(T_H\) and the cycle closes. That is the Carnot cycle, and every step of it is reversible by construction. Run the whole thing backwards and every quantity reverses with it, which gives the Carnot refrigerator.
Two results follow, known as the Carnot principles. The efficiency of an irreversible heat engine is always less than that of a reversible one operating between the same two reservoirs; and all reversible engines operating between the same two reservoirs have the same efficiency. Both are proved by contradiction and by the same trick as before: assume otherwise, couple the better engine to the reverse of the other, and the combination turns out to violate Kelvin–Planck.
The second principle is the surprising one. It says the efficiency cannot depend on the working fluid, on the pressures, on the size of the machine, or on anything else about how it is put together. Two reversible engines between the same reservoirs perform identically whether one uses steam and the other helium. So the ratio \(Q_H/Q_L\) for a reversible engine can depend only on the two reservoir temperatures, and Kelvin took that as the definition of an absolute temperature scale:
which is what the kelvin is, and why it is not merely a shifted version of the Celsius scale but a scale defined by a machine that nobody can build. Substituting it into the efficiency gives the result the whole topic has been heading towards:
with both temperatures absolute, and no exceptions. An engine between reservoirs at \(800\) K and \(300\) K cannot exceed \(1 - 300/800 = 0.625\), whatever it burns. Doing the same sum in degrees Celsius, as \(1 - 27/527 = 0.949\), is the single most common error in this topic and it is off by a factor that would change any conclusion drawn from it. The corresponding limits for the machines of the previous section are \(\mathrm{COP}_{R,\text{rev}} = T_L/(T_H - T_L)\) and \(\mathrm{COP}_{HP,\text{rev}} = T_H/(T_H - T_L)\), which still differ by one; between \(270\) K and \(300\) K they come to \(9\) and \(10\).
Be careful about what has been promised. The result is a ceiling and not a target. No real engine reaches it, because reaching it would require infinitely slow heat transfer through vanishing temperature differences and a machine with no friction anywhere, and the Carnot cycle is in any case impractical as hardware — isothermal heat transfer in a reciprocating cylinder is close to unbuildable at any useful rate. Its value is as a reference: an efficiency above it is impossible and the claim can be rejected, an efficiency equal to it means the device is reversible, and an efficiency below it is everything real.
The formula also says where improvement has to come from, and it is not from the working fluid. Only \(T_H\) and \(T_L\) appear, so raising the temperature at which heat is supplied or lowering the temperature at which it is rejected is the whole of the available gain — which is why turbine inlet temperature is the number gas turbine development has chased for sixty years, and why power stations are built beside rivers and coastlines. It also explains why some tempting sources of heat are not worth much. Ocean thermal power, drawing on surface water at \(293\) K and deep water at \(278\) K, faces a reversible ceiling of \(1 - 278/293 = 0.051\). There is an enormous amount of energy in the ocean and almost no ability to use it, and no improvement in the machinery can change that. The quality of a quantity of heat, not merely its size, is fixed by the temperature at which it is available, and that is the idea the energy accounting had no way of expressing.