Simple Nature - Light and Matter

Efficiency of the Carnot engine example 21

As an application, we now prove the result claimed earlier for the

efficiency of a Carnot engine. First consider the work done dur-

ing the constant-temperature strokes. Integrating the equation

dW =PdV, we haveW =

∫

PdV. Since the thermal energy of

an ideal gas depends only on its temperature, there is no change

in the thermal energy of the gas during this constant-temperature

process. Conservation of energy therefore tells us that work done

by the gas must be exactly balanced by the amount of heat trans-

ferred in from the reservoir.

Q=W

=

∫

PdV

For our proof of the efficiency of the Carnot engine, we need only

the ratio ofQHtoQL, so we neglect constants of proportionality,

and simply subsitutdeP∝T/V, giving

Q∝

∫

T

V

dV∝Tln

V 2

V 1

∝Tln

P 1

P 2

.

The efficiency of a heat engine is

efficiency = 1−

QL

QH

.

Making use of the result from the previous proof for a Carnot en-

gine with a monoatomic ideal gas as its working gas, we have

efficiency = 1−

TLln(P 4 /P 3 )

THln(P 1 /P 2 )

,

where the subscripts 1, 2, 3, and 4 refer to figures d–g on page

322. We have shown above that the temperature is proportional
     toPbon the insulated strokes 2-3 and 4-1, the pressures must be
     related byP 2 /P 3 =P 1 /P 4 , which can be rearranged asP 4 /P 3 =
     P 1 /P 2 , and we therefore have

efficiency = 1−

TL

TH

.

5.4.5 The arrow of time, or “this way to the Big Bang”

Now that we have a microscopic understanding of entropy, what
does that tell us about the second law of thermodynamics? The
second law defines a forward direction to time, “time’s arrow.” The
microscopic treatment of entropy, however, seems to have mysteri-
ously sidestepped that whole issue. A graph like figure b on page
327, showing a fluctuation away from equilibrium, would look just

Section 5.4 Entropy as a microscopic quantity 337