There are a few caveats to Carnot's question. First of all,
the work should not be done
at the expense of the heat engine itself, otherwise the conversion of heat into
work could not continue indefinitely. We can ensure that this is the
case if the heat engine performs some sort of cycle,
by which it periodically returns to the same macrostate, but, in the meantime, has
extracted heat from the reservoir and done an equivalent amount of useful work.
Furthermore,
a cyclic process seems reasonable because we know that both
steam engines and internal
combustion
engines perform continuous cycles. The second caveat is that the work done by
the heat engine should be such as to change a single parameter of
some external device (*e.g.*, by lifting a weight) without doing it at the
expense of affecting the other degrees of freedom, or the entropy, of that device.
For instance, if we are extracting heat from the ocean to
generate electricity, we want to spin the shaft
of the electrical generator without increasing the generator's
entropy; *i.e.*, causing the generator
to heat up or fall to bits.

Let us examine the feasibility of a heat engine using the laws of
thermodynamics. Suppose that a heat engine performs a single cycle. Since
has returned to its initial macrostate, its internal energy is
unchanged, and the first law of thermodynamics tell us that the work done
by the engine must equal the heat extracted from the reservoir , so

(341) |

The improbability of the scenario just outlined is summed up in the second law
of thermodynamics. This says that the total entropy of an isolated system
can never spontaneously decrease, so

(343) |

(344) |

Since we wish the work done by the engine to be positive, the above relation clearly cannot be satisfied, which proves that an engine which converts heat directly into work is thermodynamically impossible.

A perpetual motion device,
which continuously executes a cycle without extracting heat from,
or doing work on, its surroundings,
is just about possible according to Eq. (345). In fact, such a device
corresponds to the equality sign in Eq. (342), which
means that it must be completely
*reversible*. In reality, there is no such thing as a completely reversible engine.
All engines, even the most efficient, have frictional losses which make them,
at least,
slightly irreversible. Thus, the equality sign in Eq. (342) corresponds to
an asymptotic limit which reality can closely approach, but never quite attain.
It follows that a perpetual motion device is thermodynamically impossible.
Nevertheless, the U.S. patent office receives about 100 patent
applications a year regarding perpetual motion devices. The British patent office,
being slightly less open-minded that its American counterpart, refuses to entertain
such applications on the basis that perpetual motion devices are forbidden by
the second law of thermodynamics.

According to Eq. (345), there is no thermodynamic objection to a heat engine which runs backwards, and converts work directly into heat. This is not surprising, since we know that this is essentially what frictional forces do. Clearly, we have, here, another example of a natural process which is fundamentally irreversible according to the second law of thermodynamics. In fact, the statement

is called Kelvin's formulation of the second law.It is impossible to construct a perfect heat engine which converts heat directly into work

We have demonstrated that a *perfect heat engine*, which converts
heat directly into work, is impossible. But, there must be some way of
obtaining useful work from heat energy, otherwise steam engines would not operate.
Well, the reason that our
previous scheme did not work was that it decreased the entropy of a heat reservoir,
at some temperature ,
by extracting an amount of heat per cycle,
without any compensating increase in the entropy of anything else, so the
second law of thermodynamics was violated. How can we remedy this situation?
We still want
the heat engine itself to perform periodic cycles (so,
by definition, its entropy cannot increase over a
cycle), and we also do not
want to increase the entropy of the external device upon which the
work is done. Our only other option is to increase the entropy of some other
body. In Carnot's analysis, this other body is
a second heat reservoir at temperature . We can increase the entropy
of the second reservoir by dumping some of the heat we extracted from the
first reservoir into it. Suppose that the heat per cycle we extract from the first
reservoir
is , and the heat per cycle we reject into the second reservoir
is . Let the
work done on the external device be per cycle. The first law of thermodynamics
tells us that

(346) |

(347) |

(348) |

(349) |

Note that the efficiency is always less than unity. A real engine must always reject some energy into the second heat reservoir in order to satisfy the second law of thermodynamics, so less energy is available to do external work, and the efficiency of the engine is reduced. The equality sign in the above expression corresponds to a completely reversible heat engine (

irrespective of the way in which they operate.

Let us consider how we might construct one of these reversible heat engines.
Suppose that we have some gas in a cylinder equipped with a frictionless piston.
The gas is not necessarily a perfect gas. Suppose that we also have two heat
reservoirs at temperatures and (where ). These
reservoirs might take the form of large water baths.
Let us start off with the
gas in thermal contact with the first reservoir. We now pull the piston out
very slowly so that heat energy flows reversibly into the gas from the
reservoir. Let us now thermally isolate the gas and slowly pull out
the piston some more. During this adiabatic process the temperature of the
gas falls (since there is no longer any heat flowing into it to
compensate for the work it does on the piston). Let us continue this process
until the temperature of the gas falls to . We now place the
gas in thermal contact with the second reservoir and slowly push the piston
in. During this isothermal
process heat flows out of the gas into the reservoir. We next
thermally isolate the gas a second time and slowly compress it some more. In this
process the temperature of the gas increases. We stop the compression when the temperature
reaches . If we carry out each step properly we can return the gas to
its initial state and then repeat the cycle *ad infinitum*.
We now have a set of reversible processes by which a quantity
of heat is extracted from the
first reservoir and a quantity of heat is dumped into the second. We can best
evaluate
the work done by the system during each cycle
by plotting out the locus of the gas in a -
diagram. The locus takes the form of a closed curve--see Fig. 1.
The net work done per cycle is the ``area'' contained inside this curve, since
[if is plotted vertically and horizontally,
then is clearly an element of area under the curve ].
The engine we have
just described is called a *Carnot engine*, and is the simplest conceivable
device capable of converting heat energy into useful work.

For the specific case of an ideal gas, we can actually
calculate the work done per cycle, and, thereby, verify Eq. (351).
Consider the isothermal expansion phase of the gas. For an ideal gas, the internal
energy is a function of the temperature alone. The temperature does not
change during isothermal expansion,
so the internal energy remains constant, and the net heat absorbed by the
gas must equal the work it does on the piston. Thus,

(352) |

(353) |

(354) |

(355) |

(356) |

(357) |

(358) |

(359) |

(360) |

(361) |

The engine described above is very idealized. Of course,
real engines are far
more complicated than this. Nevertheless, the maximum efficiency of an ideal
heat engine places severe constraints on real engines.
Conventional power stations have many different ``front ends''
(*e.g.*, coal fired furnaces, oil fired furnaces, nuclear reactors), but their
``back ends'' are all
very similar, and consist of a steam driven turbine connected to
an electric generator. The ``front end'' heats water extracted
from a local river and turns it into steam, which is then used to
drive the turbine, and, hence, to generate electricity. Finally,
the steam is sent through
a heat exchanger so that it can heat up the incoming river water,
which means that the incoming water does not have to be heated so much by
the ``front end.''
At this stage, some heat is rejected to the environment, usually as clouds
of steam escaping from the top of cooling towers. We can see that a power station
possesses many of the same features as our idealized heat engine. There is a
cycle which operates between two temperatures. The upper temperature is the
temperature to which
the steam is heated by the ``front end,'' and the lower temperature
is the temperature of the environment into which heat is rejected. Suppose
that the steam is only heated to C (or K), and the
temperature of the environment is C (or K). It follows from
Eq. (350) that the *maximum* possible efficiency of the steam cycle is

(362) |

(363) |