How did Carnot know?

During his short life, Sadi Carnot made important contributions to our understanding of heat engines. His contributions are particularly impressive considering the primitive understanding of heat that people had in his day. The life and times of Carnot help us understand his thinking, thinking that has left it’s mark on our present understanding of this sophisticated and abstract concept.

Carnot was born 1796 to one of the important and influential families of Paris. When he was 3 years old, for example, his father was appointed Napoleon's minister of war. The family's fortunes varied, however, with that of Napoleon. It is probably fortunate for us that the father of this brilliant boy resigned from public service to devote himself to the education of Sadi and his brother. At the age of 16, Sadi entered the Ecole Polytechnique and studied under some best scientists of the day. At the age of 36, he fell ill and then, unfortunately weakened by his illness, died in the cholera epidemic of 1832. 

You might wonder how this young man could figure out the theoretical efficiency of a perfect engine without a correct understanding of the theory of heat. We know from his notes that Carnot was only starting to understand heat as a form of energy. We know that, when he died, he was planning to do certain experiments that would be done twenty years latter by Joule to show that energy is a conserved quantity so long frictional losses to heat are accounted for. Instead, Carnot based his advances on a concept of heat as a fluid that flows from a hot place to a cold place.

Entropy, the ghost of the caloric fluid theory

Carnot did understand how gasses expand and contract, and he was very familiar with water wheels that provide motive power by taking water from a high place and allowing it to flow to a low place.

In a water wheel, water coming from a high place has great potential energy. When it flows to a low place, where it has less potential energy, it extracts the energy difference  as work.

We can look at the potential energy of a fluid as proportional to both the pressure and the volume of the fluid.

With the same volume of fluid delivered to the low place as comes from the high place, the difference in energies is in the relative pressures.

Using this reasoning, it is easy to see that ideal efficiency of a water wheel, or any other water engine for that matter, depends on the pressures available. Efficiency, after all, is just the ratio of the work you get out of the engine to the energy you put in.

Carnot, at that point in his career, thought of a heat engine as working in much the same way. Heat from a hot place passes through the engine to a low place and, in passing does a certain amount of work. He thought of temperature as a sort of pressure that causes the heat to flow. His idea was that energy of the caloric fluid was proportional to both the temperature and the amount of this fluid.

Now, we could just replace the energy, pressure and volume in
the above discussion, starting with

with heat energy, temperature and
entropy, and we would come out
with the correct result for the
efficiency of a heat engine.

It is more common in modern day
thinking, however, to start with
entropy as being, at best, equal
at the end, after producing work,
as it is in the beginning.

Thinking as entropy as the amount
 of caloric fluid, or "stuff,"
delivered to the cold place, we can
reason that it would be,
under ideal conditions,
the same as that which came
from the hot place.

Maybe some of the "stuff" would leak around whatever mechanism in the engine that produces work, so that you would have more stuff at the end than you started with, but under ideal conditions we could say

If we express this caloric fluid in
terms of the ratio of heat to temperature

w

We can use this ratio to find the theoretical limit on the possible efficiency of a heat engine in terms of the temperatures of hot and cold places. Efficiency, after all, is just the ratio of the work we get out to the energy we put in.

Now the first law of thermodynamics says that the work you get out of a heat engine is the difference between the heat energy you take in from the hot place and the heat energy you dump into the cold place.

So the efficiency depends on the ratio of the heats, and therefore the temperatures.

Thus, the temperature of the cold place as well as the temperature of the hot place are both important in that they limit the ideal efficiency of any energy cycle.

The grand importance of this conclusion lies in the fact that it applies to all energy cycles. Perhaps the most important energy cycle for us humans is the Earth itself. Our Earth takes energy form the sun and dumps it into the cold night sky. Both the energy source and the energy sink are essential to the operation of this cycle.