Conceptual Physics

Step-by-step derivation

We start with the entropy inequality, and use that to write an inequality with the ratio of reservoir temperatures on one side.

The inequality in step 4 is a concise statement of the entropy changes during an engine cycle: The increase in entropy of the cold reservoir

outweighs the decrease in entropy of the hot reservoir.

Now we use the equation for engine efficiency to prove the inequality that describes the maximum engine efficiency.

The inequality derived in step 8 shows the maximum possible efficiency for an engine with given hot and cold reservoir temperatures. The

efficiency limit 1 íTc/Th is maximized when the cold reservoir is as cold as possible, and the hot reservoir is as hot as possible. For instance,

an engine that has a cold reservoir of 300 K and a hot reservoir of 1000 K has an efficiency limit of 1 í 0.3, or 70%. One appeal of diesel

engines is that they run “hotter” than gasoline engines, which is a reason why diesel engines are more fuel-efficient.

There are practical limits to engine efficiency. Reservoirs at 10 K and 10,000 K could make for an extremely efficient engine in principle, but it

is not possible to build a cost-efficient engine that would maintain reservoirs at these temperatures. Factors other than efficiency, such as cost

and the ability to supply power, also affect the design of engines.

Step Reason

1. entropy never decreases

2. definition of entropy, cold reservoir

3. definition of entropy, hot reservoir

4. substitute equations 2 and 3 into inequality 1

5. multiply by Tc/Qh

Step Reason

6. equation stated above

7. add inequality 5 to equation 6

8. Simplify

21.8 - Carnot cycle and efficiency

Sadi Carnot devised a theoretical engine cycle that is often discussed in

thermodynamics. The details of its working are less important than the conclusions that

he drew from it.

Carnot proved that any fully reversible engine, like his, was the most efficient possible.

In his argument, this French scientist showed that a more efficient engine cycle would

violate the second law of thermodynamics, which meant it was not possible to construct

such an engine. In an irreversible process, energy is lost from the system to its

environment in unrecoverable ways, such as through friction or sound energy. All real

engines operate irreversibly and are less efficient than a Carnot engine.

Earlier, we showed that the maximum efficiency of a heat engine was less than or

equal to one minus the ratio of the temperature of the reservoirs. The closer an engine

cycle comes to having no increase in entropy, the closer its efficiency will be to this limit.

The Carnot cycle attains the theoretical maximum efficiency for any engine functioning

with reservoirs at two particular temperatures. It can be shown that the efficiency of this

cycleequals one minus the ratio of the temperature of the cold reservoir to the hot

reservoir. This important conclusion is shown in Equation 1.

Even the ideal Carnot engine is not 100% efficient. To make a Carnot engine operate at

Carnot engine efficiency

(^390) Copyright 2007 Kinetic Books Co. Chapter 21