Lecture 9 - McMaster Physics and Astronomy

Lecture 9 (Schroeder, chapter 4.1-4.2).
Reversible and Irreversible Processes.
Carnot Cycle.
Carnot Cycle with Ideal Gas.
First Carnot Theorem.
Second Carnot Theorem.
We defined a quasistatic process as a process that is a succession of
equilibrium states. It could stop at any stage and the system would stay where it is. Let
me consider an even narrower class of processes. A reversible process is a process
under which the system can be returned to its original state without any change in
the surroundings.
Reversible processes are always quasistatic but the converse is not always true.
For example, an infinitesimal compression of a gas in a cylinder where there exists
friction between the piston and the cylinder is a quasistatic, but not reversible process.
Although the system has been driven through the succession of equilibrium states, heat
has been irreversibly lost due to friction, and cannot be recovered by simply moving the
piston infinitesimally in the opposite direction.
Finally, we can say that a reversible process is a quasistatic process where no
dissipative forces are present.
The Carnot cycle is a cyclic process in which a working substance takes heat
reversibly from a reservoir at a constant temperature T h , expands adiabatically and
reversibly to a lower temperature T c , gives up heat reversibly to a reservoir at this
temperature, and finally compressed reversibly and adiabatically to its original state. The
working substance may be a solid, liquid or gas (ideal or not). It may even change from
one phase to another during the cycle.

Let me consider the Carnot engine using an ideal gas as the working substance
(see figure below), and determine its efficiency, e (problem 4.5 in the textbook).
⎛V
⎞
W ' =
⎜
b
ab nRTh
ln
⎟ , W ' b c = CV
( Th
−Tc
),
⎝Va
⎠
⎛V
⎞
⎜
b
Qab
= nRTh
ln ⎟ , Q bc = 0 ,
⎝Va
⎠
⎛V
⎞
⎜
d
W ' cd = nRTc
ln
⎟ ,
⎝ Vc
⎠
W ' da = CV
( Tc
−Th
);
⎛V
⎞
⎜
d
Qcd
= nRTc
ln ⎟ ,
⎝ Vc
⎠
Q da = 0 .
b
d
Th
ln Tc
W ab W bc W cd W
⎜
da
V
⎟ + ln
⎜
a V
⎟
' net ' + ' + ' + '
c Th
−Tc
Tc
=
=
⎝ ⎠ ⎝ ⎠
= = − .
h
Qab
⎛Vb
⎞ Th
Th
Th
ln
⎜
V
⎟
a
W
e = 1
Q
We used that points b and c lie on the same adiabat, so do points d and a, i.e.
T
T
( γ −1) 1/
V T
( γ 1
=
)
V ,
1/
−
h b c
( γ −1) 1/
V T
( γ 1
=
)
V
1/
−
h a c
and, as a result, V V = V V .
b
a
c
d
c
d
.
⎛V
If the cycle is operated in the reverse direction, so that we have a Carnot
Qc
Tc
refrigerator, the coefficient of performance is COP = = .
W T − T
⎞
⎝
net
⎠
⎛V
h
⎞
c

First Carnot Theorem:
all Carnot engines operating between two reservoirs at constant temperatures have the
same efficiency.
Proof: Let us suppose that there are two Carnot engines having efficiencies e and e ' ,
both of which operate between reservoirs at temperatures T h and T c . Furthermore let us
suppose that
e > e'
. Carnot cycle is reversible. Therefore, the second engine e ' can be
operated as a refrigerator. It can also be arranged that the work output of the heat engine
provides the work input to the refrigerator.
Since
e > e'
by assumption, we have
Qh
− Q
Q
h
c
Q' h −Q'
>
Q'
In writing e ' in this way we are using the fact that
h
c
.
Q' h and
Q' c in the refrigerator mode
are numerically equal to the heats absorbed and expelled at the heat reservoirs when the
device is operated in the heat engine mode.

Qh
− Q
Then:
Q
Since
Q
Q' h −Q'
>
Q'
W
Q
W
> →
'
1 1 > → Q h Q'
Q h Q'
h
h
c
c
→
h
h h Q h
h − Q c = Q' h −Q'
c , we have Q'
h − Qh
= Q'
с −Qс
< → Q ' Q > 0 .
h − h
. Hence our assumption that
e > e' leads to the conclusion that both Q' h −Qh
and Q'
с −Qс
are positive quantities. This
implies that the engine-refrigerator combination, regarded as a single device, has the net
effect of transferring an amount of heat
Q'
с
−Q
с
per cycle from the cold reservoir to the
hot reservoir, thereby violating the Clausius statement of the second law. Therefore, the
original assumption
assumption
possibility is that
e > e'
must be false. The same line of reasoning starting from the
e ' > e also leads to a contradiction of the second law. Therefore, the only
e = e'
.
In particular, the first Carnot theorem states that the efficiency of a Carnot engine
doesn’t depend on a working substance.
Second Carnot Theorem:
no heat engine operating in cycles between two reservoirs at constant
temperatures can have a greater efficiency than a Carnot engine operating
between the same two reservoirs: ( Qh
Qc
) Qh
≤ ( Th
−Tc
) Th
− .
Proof: Suppose that there existed an irreversible engine of efficiency e irr such that
e irr > e , where e is the efficiency of any reversible engine operating between the same
temperatures T h and T c . If the preceding proof, summarized in Figure 2, is modified so
that the reversible engine operating in the heat-engine mode is replaced by the
irreversible engine, then the same line of argument starting from
e irr > e leads to a
contradiction of the Clausius statement of the second law. Hence, we must conclude that
e irr ≤ e .
Notice that although a Carnot cycle is very efficient, it is also horribly impractical.
The heat flows so slowly during the isothermal steps that it takes forever to get a
significant amount of work out of the engine. So don’t bother installing a Carnot engine
in your car; while it would increase your gas mileage, you’d be passed by pedestrians.