Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

The entropy change for a process is obtained by integrating this relation
between the end states:

(7–31)

A second relation for the entropy change of an ideal gas is obtained in a
similar manner by substituting dh
cpdTand v
RT/Pinto Eq. 7–26 and
integrating. The result is

(7–32)

The specific heats of ideal gases, with the exception of monatomic gases,
depend on temperature, and the integrals in Eqs. 7–31 and 7–32 cannot be
performed unless the dependence of cvand cpon temperature is known.
Even when the cv(T) and cp(T) functions are available, performing long
integrations every time entropy change is calculated is not practical. Then
two reasonable choices are left: either perform these integrations by simply
assuming constant specific heats or evaluate those integrals once and tabu-
late the results. Both approaches are presented next.

Constant Specific Heats (Approximate Analysis)

Assuming constant specific heats for ideal gases is a common approxima-
tion, and we used this assumption before on several occasions. It usually
simplifies the analysis greatly, and the price we pay for this convenience is
some loss in accuracy. The magnitude of the error introduced by this
assumption depends on the situation at hand. For example, for monatomic
ideal gases such as helium, the specific heats are independent of tempera-
ture, and therefore the constant-specific-heat assumption introduces no
error. For ideal gases whose specific heats vary almost linearly in the tem-
perature range of interest, the possible error is minimized by using specific
heat values evaluated at the average temperature (Fig. 7–32). The results
obtained in this way usually are sufficiently accurate if the temperature
range is not greater than a few hundred degrees.
The entropy-change relations for ideal gases under the constant-specific-
heat assumption are easily obtained by replacing cv(T) and cp(T) in Eqs.
7–31 and 7–32 by cv,avgand cp,avg, respectively, and performing the integra-
tions. We obtain

(7–33)

and

(7–34)

Entropy changes can also be expressed on a unit-mole basis by multiplying
these relations by molar mass:

s 2 s 1 
cv,avg ln¬ (7–35)

T 2

T 1

Ru ln¬

v 2

v 1

¬¬ 1 kJ>kmol#K 2

s 2 s 1 
cp,avg ln¬

T 2

T 1

R ln¬

P 2

P 1

¬¬ 1 kJ>kg#K 2

s 2 s 1 
cv,avg ln¬

T 2

T 1

R ln¬

v 2

v 1

¬¬ 1 kJ>kg#K 2

s 2 s 1

2

1

cp 1 T 2 ¬

dT

T

R ln¬

P 2

P 1

s 2 s 1

2

1

¬cv 1 T 2

dT

T

R ln¬

v 2

v 1

Chapter 7 | 355

Pv= RT

du = Cv dT

dh = Cp dT

FIGURE 7–31

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T 1 T 2 T

cp,avg

Tavg

Average cp

cp

Actual cp

FIGURE 7–32

Under the constant-specific-heat

assumption, the specific heat is

assumed to be constant at some

average value.