Physical Chemistry , 1st ed.

for 1 g of benzene. Since this represents the entropy change for 1 g of ben-

zene, we can also write this Sas 1.12 J/gK. The entropy of the system—

the benzene—is increasing in this example.

Other cyclic processes having different steps or conditions can be defined.
However, it has been found that no known process is more efficient than a
Carnot cycle, which is defined in terms ofreversiblesteps. This means that any
irreversible change is a less efficient conversion of heat to work than a re-
versible change, since a Carnot cycle is defined in terms of reversible processes.
So, for any arbitrary process:

earb eCarnot

where earbis the efficiency for that arbitrary cycle and eCarnotis the efficiency
of a Carnot cycle. If the arbitrary process is a Carnot-type cycle, then the
“equals” part of the sign applies. If the cycle is an irreversible cycle, the “less
than” part of the sign applies. Substituting for efficiency:

1 + 

q

q

o

i

u

n

t

,a

,a

r

r

b

b 1 + q

q

3

1

,

,

C

C

a

a

r

r

n

n

o

o

t

t





q

q

o

i

u

n

t

,a

,a

r

r

b

b q

q

3

1

,

,

C

C

a

a

r

r

n

n

o

o

t

t



where the 1s have canceled. The fraction on the right is equal to Tlow/Thigh,
as demonstrated earlier. Substituting:



q

q

o

i

u

n

t

,a

,a

r

r

b

b



T

T

h

lo

ig

w

h



and rearranging:



q

q

o

i

u

n

t

,a

,a

r

r

b

b+ 

T

T

h

lo

ig

w

h

 

0

This equation can be rearranged to get the heat and temperature variables that
are associated with the two reservoirs into the same fractions (that is,qinwith
Thighand qoutwith Tlow). It is also convenient to relabel the temperatures
and/or the heats to emphasize which steps of the Carnot cycle are involved.
Finally, we will drop the “arb” designation. (Can you reproduce these steps?)
The above expression thus simplifies to



T

q 3

3

+ 

T

q 1

1



0

For the complete cycle of many steps, we can write this as a summation:



0

all steps



T

qs

s

t

t

e

e

p

p



0

As each step gets smaller and smaller, the summation sign can be replaced by
an integral sign, and the above expression becomes

d

T

q 

0 (3.15)

for any complete cycle. Equation 3.15 is one way of stating what is called
Clausius’s theorem, after Rudolf Julius Emmanuel Clausius, a Pomeranian
(now part of Poland) and German physicist who first demonstrated this rela-
tionship in 1865.

3.4 Entropy and the Second Law of Thermodynamics 73