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

Chapter 13 | 699

ing processes, negative for endothermic mixing processes, and zero for
isothermal mixing processes during which no heat is absorbed or released.
Note that mixing is an irreversible process, and thus the entropy of mixing
must be a positive quantity during an adiabatic process. The specific volume,
enthalpy, and entropy of a mixture are determined from

(13–35)

where yiis the mole fraction of component iin the mixture.
Reconsider Eq. 13–29 for dG. Recall that properties are point functions, and
they have exact differentials. Therefore, the test of exactness can be applied to
the right-hand side of Eq. 13–29 to obtain some important relations. For the dif-
ferential dz
MdxNdyof a function z(x,y), the test of exactness is
expressed as (M/y)x
(N/x)y. When the amount of component iin a mix-
ture is varied at constant pressure or temperature while other components (indi-
cated by j) are held constant, Eq. 13–29 simplifies to

(13–36)

(13–37)

Applying the test of exactness to both of these relations gives

(13–38)

where the subscript Nindicates that the mole numbers of all components
(and thus the composition of the mixture) is to remain constant. Taking the
chemical potential of a component to be a function of temperature, pressure,
and composition and thus mi
mi(P,T,y 1 ,y 2 ,... ,yj.. .), its total differ-
ential can be expressed as

(13–39)

where the subscript yindicates that the mole fractions of all components
(and thus the composition of the mixture) is to remain constant. Substituting
Eqs. 13–38 into the above relation gives

(13–40)

For a mixture of fixed composition undergoing an isothermal process, it sim-
plifies to

(13–41)

Ideal-Gas Mixtures and Ideal Solutions

When the effect of dissimilar molecules in a mixture on each other is negli-
gible, the mixture is said to be an ideal mixtureor ideal solutionand the
chemical potential of a component in such a mixture equals the Gibbs function

dmi
v
i dP¬¬ 1 T
constant, yi
constant 2

dmi

vi dP
si dTa

i

a

0 mi

0 yi

b

P,T,yj

dyi

dmi
dg
i
a

0 mi

0 P

b

T,y

dPa

0 mi

0 T

b

P,y

dTa

i

a

0 mi

0 yi

b

P,T,yj

dyi

a

0 mi

0 T

b

P,N

a

0 S

0 Ni

b

T,P,Nj


si¬and¬a

0 mi

0 P

b

T,N

a

0 V

0 Ni

b

T,P,Nj

v
i

dG
V dPmi dNi¬¬ 1 for T
constant and Nj
constant 2

dG
S dTmi dNi¬¬ 1 for P
constant and Nj
constant 2

v
a

i

yiv
i h
a

i

yih
i¬and¬s
a

i

yi
si