This principle can be viewed as another statement of the second law of thermodynamics and can be shown to be equivalent to the Kelvin statement and the Clausius statement.
The carnot engine
A Carnot engine has an efficiency of 0.60 and the temperature of its cold reservoir is 300 K. (a) What is the temperature of the hot reservoir? (b) If the engine does 300 J of work per cycle, how much heat is removed from the high-temperature reservoir per cycle? (c) How much heat is exhausted to the low-temperature reservoir per cycle?
Strategy
From the temperature dependence of the thermal efficiency of the Carnot engine, we can find the temperature of the hot reservoir. Then, from the definition of the efficiency, we can find the heat removed when the work done by the engine is given. Finally, energy conservation will lead to how much heat must be dumped to the cold reservoir.
Solution
From
we have
so that the temperature of the hot reservoir is
By definition, the efficiency of the engine is
, so that the heat removed from the high-temperature reservoir per cycle is
From the first law, the heat exhausted to the low-temperature reservoir per cycle by the engine is
Significance
A Carnot engine has the maximum possible efficiency of converting heat into work between two reservoirs, but this does not necessarily mean it is
efficient. As the difference in temperatures of the hot and cold reservoir increases, the efficiency of a Carnot engine increases.
Imagine a Carnot heat pump operates between an outside temperature of
and an inside temperature of
. What is the work needed if the heat delivered to the inside of the house is 30.0 kJ?
Strategy
Because the heat pump is assumed to be a Carnot pump, its performance coefficient is given by
Thus, we can find the work
W from the heat delivered
Solution
The work needed is obtained from
Significance
We note that this work depends not only on the heat delivered to the house but also on the temperatures outside and inside. The dependence on the temperature outside makes them impractical to use in areas where the temperature is much colder outside than room temperature.
In terms of energy costs, the
heat pump is a very economical means for heating buildings (
[link] ). Contrast this method with turning electrical energy directly into heat with resistive heating elements. In this case, one unit of electrical energy furnishes at most only one unit of heat. Unfortunately, heat pumps have problems that do limit their usefulness. They are quite expensive to purchase compared to resistive heating elements, and, as the performance coefficient for a Carnot heat pump shows, they become less effective as the outside temperature decreases. In fact, below about
, the heat they furnish is less than the energy used to operate them.
A photograph of a heat pump (large box) located outside a house. This heat pump is located in a warm climate area, like the southern United States, since it would be far too inefficient located in the northern half of the United States. (credit: modification of work by Peter Stevens)
