Physical Chemistry , 1st ed.

Equations 3.7 and 3.9 can be divided to get a new expression for the ratio q 3 /q 1 :



q

q

3

1



T

T

h

lo

ig

w

h



Substituting into equation 3.6, we get an equation for efficiency in terms of the
temperatures:

e 1 

T

T

h

lo

ig

w

h

 (3.10)

Equation 3.10 has some interesting interpretations. First, the efficiency of an
engine is very simply related to the ratioof the low- and high-temperature
reservoirs. The smaller this ratio is, the more efficient an engine is.* Thus, high
efficiencies are favored by high Thighvalues and low Tlowvalues. Second, equa-
tion 3.10 allows us to describe a thermodynamicscale for temperature. It is a
scale for which T0 when the efficiency equals 1 for the Carnot cycle. This
scale is the same one used for ideal gas laws, but it is based on the efficiency of
a Carnot cycle, rather than the behavior of ideal gases.
Finally, unless the temperature of the low-temperature reservoir is absolute
zero, the efficiency of an engine will never be 1; it will always be less than 1.
Since it can be shown that absolute zero is physically unobtainable for a macro-
scopic object, we have the further statement that

No engine can ever be 100% efficient.

When one generalizes by recognizing that every process can be considered an
engine of some sort, the statement becomes

No process can ever be 100% efficient.

It is statements like this that preclude the existence of perpetual motion ma-
chines, devices that purportedly have an efficiency greater than 1 (that is,
100%), producing more work out than the energy coming in. Carnot’s study
of steam engines helped establish such statements, and so much faith is placed
in them that the U.S. Patent Office categorically does not consider any patent
application claiming to be a perpetual motion machine (although some appli-
cations for such machines are considered because they disguise themselves to
cover the fact). Such is the power of the laws of thermodynamics.
The two definitions of efficiency can be combined:

1 + q

q

3

1

 1 

T

T

h

lo

ig

w

h





q

q

3

1



T

T

h

lo

ig

w

h





q

q

3

1

+ 

T

T

h

lo

ig

w

h

 0



T

q

lo

3

w

+ 

T

q

hi

1

gh

 0 (3.11)

Notice that q 3 is the heat that goes to the low-temperature reservoir, whereas
q 1 is the heat that comes from the high-temperature reservoir. Each fraction

nRTlowln 

V

V

A

B



nRThighln 

V

V

A

B

3.3 The Carnot Cycle and Efficiency 71

*In practice, other factors (including mechanical ones) reduce the efficiency of most
engines.