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

finally along the T 2 constant line to P 2 , as we did for the enthalpy. This

approach is not suitable for entropy-change calculations, however, since it

involves the value of entropy at zero pressure, which is infinity. We can

avoid this difficulty by choosing a different (but more complex) path

between the two states, as shown in Fig. 12–17. Then the entropy change

can be expressed as

(12–61)

States 1 and 1* are identical (T 1 T 1 *and P 1 P 1 *) and so are states 2

and 2*. The gas is assumed to behave as an ideal gas at the imaginary states

1* and 2* as well as at the states between the two. Therefore, the entropy

change during process 1*-2* can be determined from the entropy-change

relations for ideal gases. The calculation of entropy change between an

actual state and the corresponding imaginary ideal-gas state is more

involved, however, and requires the use of generalized entropy departure

charts, as explained below.

Consider a gas at a pressure Pand temperature T. To determine how much

different the entropy of this gas would be if it were an ideal gas at the same

temperature and pressure, we consider an isothermal process from the actual

state P,Tto zero (or close to zero) pressure and back to the imaginary ideal-

gas state P*,T* (denoted by superscript *), as shown in Fig. 12–17. The

entropy change during this isothermal process can be expressed as

where vZRT/Pand v* videalRT/P. Performing the differentiations

and rearranging, we obtain

By substituting TTcrTRand P PcrPRand rearranging, the entropy

departure can be expressed in a nondimensionalized form as

(12–62)

The difference (s–* s–)T,Pis called the entropy departureand Zsis called

the entropy departure factor.The integral in the above equation can be

performed by using data from the compressibility charts. The values of Zs

are presented in graphical form as a function of PRand TRin Fig. A–30.

This graph is called the generalized entropy departure chart,and it is

used to determine the deviation of the entropy of a gas at a given Pand T

from the entropy of an ideal gas at the same Pand T. Replacing s* by sideal

for clarity, we can rewrite Eq. 12–61 for the entropy change of a gas during

a process 1-2 as

s 2 s 1  1 s 2 s 12 idealRu 1 Zs 2 Zs 12 (12–63)

Zs

1 ss (^2) T,P
Ru

PR
0
cZ 1 TRa
0 Z
0 TR
b
PR
d d 1 ln PR 2
1 sPsP (^2) T
P
0
c
11 Z 2 R
P

RT
P
a
0 Zr
0 T
b
P
d dP

P
0
a
0 v
0 T
b
P
dP
0
P
a
0 v
0 T
b
P
dP
1 sPsP (^2) T 1 sPs 02 T 1 s 0 sP (^2) T
s 2 s 1  1 s 2 sb 2  1 sbs 2 2  1 s 2 s 1 2  1 s 1 sa 2  1 sas 12
672 | Thermodynamics
s
T
2
b
a
1*
1
2
Alternative
process path
Actual
process path
P 2
P 1
P^0
T 2
T 1
FIGURE 12–17
An alternative process path to evaluate
the entropy changes of real gases
during process 1-2.