Prelecture 31: Slide 4
On the last slide, we defined the efficiency of a heat engine to be the ratio of the work done by the engine to the heat that was extracted from the hot reservoir.
We can rewrite this expression in terms of the heat transfers with the two reservoirs by applying the first law of thermodynamics to the engine. Consequently, we see that a 100% efficient heat engine would convert all of the heat extracted from the hot reservoir into work. That is, no heat would flow into the cold reservoir. We shall soon see that this possibility cannot happen since it would violate the second law of thermodynamics.
To see how this works, let's look at the entropy change of the engine plus the reservoirs during one cycle. Since the working substance of the engine is returned to its initial state at the completion of one cycle, its entropy does not change! Now, the entropies of each reservoir, however, do change. We can determine these entropy changes from our definition of temperature.
Since the reservoirs do no work, the first law of thermodynamics tells us that the change in the internal energy of each reservoir is just equal to the heat that flows into the reservoir.
Consequently, the change in the entropy of each reservoir is equal to the ratio of the heat flowing into the reservoir to the temperature of that reservoir.
Now, the second law of thermodynamics says this total entropy change must always be ≥ 0! Consequently, we see that the ratio of the heat transfers from the reservoirs must always be ≥ the ratio of the temperatures of the reservoirs.
Going back to our expression for the efficiency of a heat engine, we see that this efficiency can never be greater than 1 minus the ratio of the temperatures of the reservoirs. This maximum efficiency is usually referred to as the Carnot efficiency, after Sadi Carnot, a French physicist and engineer of the 19th century.