The Thermodynamic Origins of Entropy: Carnot and The Heat Engine

Carnot, Caloric Theory and The Heat Engine

In 1823 when Sadi Carnot (1796–1832) began this task, less than thirty years had passed since Rumford’s cannon-boring experiments led him to declare “heat is motion”. And although this should have been the end of caloric theory, it and its principle of heat (caloric) conservation were mostly undaunted. Further, a more complete understanding of energy would have to wait for some thirty years for the first law to be established.

Thus it’s not surprising that Carnot adopted (the incorrect) caloric theory and that his description of the work done by a heat engine insisted on heat conservation. In addition, he subscribed to another axiom: the impossibility of perpetual motion, which had been around a long time in application to mechanical systems. Carnot extended this notion to also include heat engines.

Carnot realized that heat flows from hot to cold, and that a heat engine allows one to obtain work from this flow. He imagined this temperature difference as similar to the height difference needed to run a water engine. An example of a water engine would be a water wheel positioned at the bottom of a waterfall. The water falling from the top to the bottom turns the water wheel when it hits it on its descent, and this motion can be harnessed to do work. A water wheel is the most efficient when every single drop of water falling from the top hits the water wheel and drives it on its way to the bottom; water falling past the water wheel does not contribute to this motion, and therefore lowers the efficiency.

Likewise, Carnot envisioned something similar occurring in heat engines. Moreover, he assumed that it’s impossible to extract work from heat without having a temperature difference; there must be a hot reservoir (source) and a cold reservoir (sink) for heat to flow and drive a heat engine, just as there must be a height difference (from high to low) for water to flow to drive a water engine.

He was also convinced that heat must necessarily be discarded in this effort. As in a water engine, where the water falls from the top to join the rest of the water below, Carnot concluded so must heat in a heat engine “fall”, from high temperature to low temperature, finally being tossed into the cold reservoir.

In short, Carnot’s Theorem says: the work output (and thus efficiency) of any reversible (ideal) heat engine depends only on the temperatures of the hot reservoir and the cold reservoir. Although an idealization, Carnot’s ingenious model provides much to engineers in the way of real heat engine design.

Now, moving a system reversibly is the best-case scenario, and the resulting work output and efficiency of a reversible heat engine provide an upper bound that no real heat engine can surpass. Nonetheless, according Carnot’s Theorem we still can’t get one hundred percent efficiency, even for the best-case scenario of the reversible heat engine. This is because a temperature difference is still needed to get the required heat flow. To be sure, one must discard a portion of heat to the cold reservoir, and therefore not all of the heat withdrawn from the hot reservoir can be used for work. In other words, if you want to use a heat engine to do work for you, nature requires its compensation.

Carnot’s Final Days

In 1824, a year after his father’s death, Carnot wrote Reflections on the Motive Power of Fire, where he emphasized, in a similar fashion to his father’s work, his general theory applicable to all types of heat engines regardless of the details of their design. The memoir was published by a leading scientific publisher, received only one, yet enthusiastic review, and a decade later was cited in an important journal – and then spent the next twenty years in obscurity.

In June 1832, Carnot contracted scarlet fever. Feeling momentarily better he writes to a friend:

“I have been sick for a long time, and in a very wearisome way. I have had an inflammation of the lungs, followed by scarlet-fever. (Perhaps you know what this horrible disease is.) I had to remain twelve days in bed, without sleep or food, without any occupation, …”

The scarlet fever eventually spread to his brain, and then in August he contracted cholera and died within hours. He was only thirty-six years old. As was standard with cholera victims, his clothes, his personal effects, and almost all his papers were burned. Among the surviving papers we see Carnot beginning to abandon caloric theory as he realizes its inherent challenges in the face of Rumford’s prior work.

Carnot attended the École Polytechnique were he was literally surrounded by renowned physicist, chemists and mathematicians, with several providing his formal scientific training. Nonetheless, other than during his academic studies, Carnot was never a member of this distinguished group, doing his important work as an outsider. Regardless, Carnot’s contributions to the foundations of thermodynamics are undeniable, and firmly establish him as both a pioneer and as one of the greats in this field. Moreover, his theoretical model for the heat engine provided early insight into the thermodynamic origins of entropy, thus lighting the way for others such as Clausius.



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