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

700 | Thermodynamics

of the pure component. Many liquid solutions encountered in practice, espe-

cially dilute ones, satisfy this condition very closely and can be considered to

be ideal solutions with negligible error. As expected, the ideal solution

approximation greatly simplifies the thermodynamic analysis of mixtures. In

an ideal solution, a molecule treats the molecules of all components in the

mixture the same way—no extra attraction or repulsion for the molecules of

other components. This is usually the case for mixtures of similar substances

such as those of petroleum products. Very dissimilar substances such as

water and oil won’t even mix at all to form a solution.

For an ideal-gas mixture at temperature Tand total pressure P, the partial

molar volume of a component iis vi
vi
RuT/P. Substituting this relation

into Eq. 13–41 gives

(13–42)

since, from Dalton’s law of additive pressures,Pi
yiPfor an ideal gas

mixture and

(13–43)

for constant yi. Integrating Eq. 13–42 at constant temperature from the total

mixture pressure Pto the component pressure Piof component igives

(13–44)

For yi
1 (i.e., a pure substance of component ialone), the last term in the

above equation drops out and we end up with mi(T,Pi) 
mi(T,P), which is

the value for the pure substance i. Therefore, the term mi(T,P) is simply the

chemical potential of the pure substance iwhen it exists alone at total mix-

ture pressure and temperature, which is equivalent to the Gibbs function

since the chemical potential and the Gibbs function are identical for pure

substances. The term mi(T,P) is independent of mixture composition and

mole fractions, and its value can be determined from the property tables of

pure substances. Then Eq. 13–44 can be rewritten more explicitly as

(13–45)

Note that the chemical potential of a component of an ideal gas mixture

depends on the mole fraction of the components as well as the mixture tem-

perature and pressure, and is independent of the identity of the other con-

stituent gases. This is not surprising since the molecules of an ideal gas

behave like they exist alone and are not influenced by the presence of other

molecules.

Eq. 13–45 is developed for an ideal-gas mixture, but it is also applicable to

mixtures or solutions that behave the same way—that is, mixtures or solutions

in which the effects of molecules of different components on each other are

negligible. The class of such mixtures is called ideal solutions(or ideal mix-

tures), as discussed before. The ideal-gas mixture described is just one cate-

mi,mixture,ideal 1 T, Pi 2 
mi,pure 1 T, P 2 RuT ln yi

mi 1 T, Pi 2 
mi 1 T, P 2 RuT ln

Pi

P

mi 1 T, P 2 RuT ln yi¬¬ 1 ideal gas 2

d ln Pi
d ln 1 yiP 2
d 1 ln yiln P 2
d ln P¬¬ 1 yi
constant 2

dmi

RuT

P

dP
RuTd ln P
RuTd ln Pi 1 T
constant, yi
constant, ideal gas 2