AN ATTEMPT TO MAKE AN ARBITRARY QUANTITY
OF ARTEFACTS FROM A FINITE QUANTITY OF WORK AND WHY IT FAILS
Once upon a time a town's sole source of work energy was an enchanted mountain. Out of the side of the mountain came a rotating shaft. This shaft provided exactly W work energy per day, no more. Every energy requiring process in the town used up some of this work.
The Limits to Growth
Eventually there came a time when the town reached a size and activity where all of the work was being used up and further economic growth was impossible.
A Clever Plan
The townsfolk hit upon a plan to multiply up the work available to them, by any chosen amount, enabling the town to continue to grown indefinitely. The plan did not violate the laws of thermodynamics.
The proposal was to cover the town with a dome. The dome to be made of an insulating material.
The various activities inside the town dissipate the work from the mountain, turning it into heat. Inside the new dome the heat accumulates raising the temperature, Thot, of the town. Clearly the temperature can be raised to an arbitrarily high level. (The citizenry are unaffected by temperature, no matter how high.)
Cake and the Ha'penny
A hole is made in the dome and a heat engine installed therein. The heat engine takes the heat from the town and, using the temperature difference between the inside of the dome and the surroundings (at temperature Tcold), it converts a proportion of this heat into work and sends the remainder to the surroundings.
The townspeople have devised a way of turning their waste heat back into work again so that it can be re-used. And when they have re-used it and it has turned again to heat, they can turn this heat back into work again. And so on.
Mathematically: In the steady state, when the temperature is constant, the amount of heat leaving to the surroundings through the cold end of the heat engine equals the energy input to the town, W. By the Carnot efficiency 1 - Qout/Qin = 1 - Tcold/Thot (where Qin is the total heat into the heat engine and Qout is the heat it ejects to the surroundings, Qout = W). Hence Qin = Qout(Thot/Tcold) and the work output of the engine is: W(Thot/Tcold - 1). Adding the work from the mountain, W, to this gives the total work available to the town: W.Thot/Tcold.
The townspeople may have as much work as they wish simply by raising the temperature to the appropriate level! This merely involves waiting the appropriate time before switching on the heat engine.
Infinite Riches
The town has an inexhaustible supply of raw materials so it would seem the people can use their unlimited work supply to fabricate an infinite quantity of goods. This is certainly breaching the spirit of, if not the letter of, the second law.
Spoke In the Wheel
In fact, Sporus would argue, the townspeople may only make things, say cars, at the same rate as they did pre-dome.
Creating a car from raw materials represents a reduction in the entropy of the materials. It is the production of "negentropy", or negative entropy - a la Erwin Schrödinger, in his hugely seminal book, "What is Life." Since entropy cannot be destroyed the production of negentropy must involve the production of an equal quantity of positive entropy. This positive entropy leaves the dome through the cold end of the heat engine.
The amount of entropy the dome emits to its surroundings is W/Tcold. This quantity is independent of the temperature inside the dome.
Thus the quantity of manufactures that may be made per day is measured by W/Tcold and is not increased by raising the temperature of the town.
Sod It
The higher the temperature the more violently the atoms inside the town move. When work is used to try to push atoms into position to make an artifact, the pushing is resisted by the pattering of the other atoms against the one being pushed. The higher the temperature, therefore, the harder the push needs to be and the more work is used.
In other words: the higher the temperature the more work is needed to make things. This effect exactly cancels out the townspeoples' plan.