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

Chapter 13 | 701

gory of ideal solutions. Another major category of ideal solutions is the dilute
liquid solutions, such as the saline water. It can be shown that the enthalpy of
mixing and the volume change due to mixing are zero for ideal solutions (see
Wark, 1995). That is,

(13–46)

Then it follows thatvi
v–iand h



i
h

• i. That is, the partial molar volume
  and the partial molar enthalpy of a component in a solution equal the specific
  volume and enthalpy of that component when it existed alone as a pure sub-
  stance at the mixture temperature and pressure. Therefore, the specific volume
  and enthalpy of individual components do not change during mixing if they
  form an ideal solution. Then the specific volume and enthalpy of an ideal
  solution can be expressed as (Fig. 13–20)

(13–47)

Note that this is not the case for entropy and the properties that involve
entropy such as the Gibbs function, even for ideal solutions. To obtain a rela-
tion for the entropy of a mixture, we differentiate Eq. 13–45 with respect to
temperature at constant pressure and mole fraction,

(13–48)

We note from Eq. 13–38 that the two partial derivatives above are simply the
negative of the partial molar entropies. Substituting,

(13–49)

Note that ln yiis a negative quantity since yi1, and thus Ruln yiis always
positive. Therefore, the entropy of a component in a mixture is always greater
than the entropy of that component when it exists alone at the mixture temper-
ature and pressure. Then the entropy of mixingof an ideal solution is deter-
mined by substituting Eq. 13–49 into Eq. 13–34 to be

(13–50a)

or, dividing by the total number of moles of the mixture Nm,

(13–50b)

Minimum Work of Separation of Mixtures

The entropy balance for a steady-flow system simplifies to SinSoutSgen

0. Noting that entropy can be transferred by heat and mass only, the

¢smixing,ideal
a
i

yi 1
sisi 2
Ru (^) a
i
yi ln yi¬¬ 1 per unit mole of mixture 2
¢Smixing,ideal
a
i
Ni 1
sisi 2
Ru (^) a
i
Ni ln yi¬¬ 1 ideal solution 2
si,mixture,ideal 1 T, Pi 2
si,pure 1 T, P 2 Ru ln y 1 ¬¬ 1 ideal solution 2
a
0 mi,mixing 1 T, Pi 2
0 T
b
P,y

a
0 mi,pure 1 T, P 2
0 T
b
P,y
Ru ln yi
vmixing,ideal
a
i
yiv
i
a
i
yivi,pure¬and¬hmixture,ideal
a
i
yih
i
a
i
yihi,pure
¢Vmixing,ideal
a
i
Ni 1 v
ivi 2
0 ¬and¬¢Hmixing,ideal
a
i
Ni 1 h
ihi 2
0

V mixing,ideal
0
H (^) mixing,ideal
0
v mixture
yivi,pure
vi,mixture
vi,pure
h
mixture
^ yihi,pure
h
i,mixing
^ hi,pure
i
i
FIGURE 13–20
The specific volume and enthalpy of
individual components do not change
during mixing if they form an ideal
solution (this is not the case for
entropy).