Perpetual Motion
Some people regard heat engines as imperfect because they can never be 100 percent efficient. Even idealized heat engines must always waste some energy. Why? Heat is disorganized energy, and it takes energy to organize things.
Let’s now consider engines with zero efficiency. They can do no useful work. We can make any engine drop to zero efficiency by cutting off its supply of fuel. If there is no fuel the fire goes out and the source of heat cools until its temperature is equal to the temperature of the environment, the temperature at which the engine discharges waste heat. That makes the highest temperature H equal to the lowest temperature L. When two numbers are the same, dividing one by the other yields 1. The ratio 1, subtracted from the 1 that indicates 100 percent efficiency, gives 0. An engine with no fuel has zero efficiency. It cannot do useful work.
What if an engine keeps on moving after its fuel is exhausted? The second law does not forbid perpetual motion. If it did it would be outlawing the stable movement of the electron in the hydrogen atom. The second law puts a limit on the useful work we can extract from heat energy. If there is no difference of temperature between the heat source and the reservoir where the engine discharges used heat energy, then no engine can extract work. That doesn’t mean that the parts of the engine can’t go around and around forever. They can do that if they have no friction. The planets approximate this condition by moving year after year around the Sun in the airless void of space. There is no such thing as friction in engines the size of atoms, either. Friction is a macroscopic phenomenon. Macroscopic means that the parts contain many atoms or molecules. In other words, the fact that objects of ordinary size don’t move forever doesn’t mean that there is no perpetual motion of very large or very small things.
Let’s now consider engines with zero efficiency. They can do no useful work. We can make any engine drop to zero efficiency by cutting off its supply of fuel. If there is no fuel the fire goes out and the source of heat cools until its temperature is equal to the temperature of the environment, the temperature at which the engine discharges waste heat. That makes the highest temperature H equal to the lowest temperature L. When two numbers are the same, dividing one by the other yields 1. The ratio 1, subtracted from the 1 that indicates 100 percent efficiency, gives 0. An engine with no fuel has zero efficiency. It cannot do useful work.
What if an engine keeps on moving after its fuel is exhausted? The second law does not forbid perpetual motion. If it did it would be outlawing the stable movement of the electron in the hydrogen atom. The second law puts a limit on the useful work we can extract from heat energy. If there is no difference of temperature between the heat source and the reservoir where the engine discharges used heat energy, then no engine can extract work. That doesn’t mean that the parts of the engine can’t go around and around forever. They can do that if they have no friction. The planets approximate this condition by moving year after year around the Sun in the airless void of space. There is no such thing as friction in engines the size of atoms, either. Friction is a macroscopic phenomenon. Macroscopic means that the parts contain many atoms or molecules. In other words, the fact that objects of ordinary size don’t move forever doesn’t mean that there is no perpetual motion of very large or very small things.
Information
Boltzmann developed a way of measuring the probability of a state of motion and related it to the entropy. Much later Shannon considered the probabilities of different messages. The probability of a message is related to the information it contains in the way Boltzmann described. Shannon used Boltzmann’s formula, but he prefixed it with a negative sign, and used a different multiplying factor to get a formula for calculating information. We use information to organize and order things. When information is destroyed, there is disorder. The same law, the second law of thermodynamics, applies both to information and to the entropy of heat processes.
Order
Entropy is proportional to negative information. Of course, entropy is itself a negative concept. It refers to disorder. To speak more clearly, we should say that entropy is the negative of information. That way we avoid a double negative.
Shannon had to use a multiplying factor that differs from the one Boltzmann used because we measure entropy and information in different units. The word “bit” we use as a unit of information only coincidentally means “a small fragment.” Originally “bit” was a contraction of “binary digit.” We may measure information in binary digits (bits) or in units mathematicians find “natural,” units called nits. A nit is a little more than 1.442695 bits. To obtain entropy in watt-seconds per kelvin, multiply the number of nits of information by the Boltzmann constant, 0.000 000 000 000 000 000 000 013 806 62 watt-seconds per kelvin.
Shannon had to use a multiplying factor that differs from the one Boltzmann used because we measure entropy and information in different units. The word “bit” we use as a unit of information only coincidentally means “a small fragment.” Originally “bit” was a contraction of “binary digit.” We may measure information in binary digits (bits) or in units mathematicians find “natural,” units called nits. A nit is a little more than 1.442695 bits. To obtain entropy in watt-seconds per kelvin, multiply the number of nits of information by the Boltzmann constant, 0.000 000 000 000 000 000 000 013 806 62 watt-seconds per kelvin.
Entropy or Information and Probability
Information is now a measurable quantity, thanks to the second law of thermodynamics. But what does information or entropy have to do with probability?
Engineers who design heat engines would like the motion of steam molecules to be completely ordered, because then they could design heat engines that are 100 percent efficient. Ideally, all the molecules would move in the same direction, bounce off the piston perpendicularly, give up all their energy, and then fall to the bottom of the cylinder. If the engineers could make the steam molecules do that, the engine would be 100 percent efficient. It would take all the energy out of heat and convert it into useful work.
But engineers know that it is impossible for all the molecules to move in such an orderly fashion. The state of motion of the vapor molecules is likely to be random and have entropy depending on pressure and temperature. This is the relationship between probability and entropy or information.
Engineers who design heat engines would like the motion of steam molecules to be completely ordered, because then they could design heat engines that are 100 percent efficient. Ideally, all the molecules would move in the same direction, bounce off the piston perpendicularly, give up all their energy, and then fall to the bottom of the cylinder. If the engineers could make the steam molecules do that, the engine would be 100 percent efficient. It would take all the energy out of heat and convert it into useful work.
But engineers know that it is impossible for all the molecules to move in such an orderly fashion. The state of motion of the vapor molecules is likely to be random and have entropy depending on pressure and temperature. This is the relationship between probability and entropy or information.