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

698 | Thermodynamics

i



 h  Ts  g

 gi,mixture

 hi,mixture Tsi,mixture

Mixture:

Pure substance:

FIGURE 13–18

For a pure substance, the chemical

potential is equivalent to the Gibbs

function.

where yiNi/Nmis the mole fraction of component i(Nmis the total num-

ber of moles of the mixture) and

(13–30)

is the chemical potential,which is the change in the Gibbs function of the

mixture in a specified phase when a unit amount of component i in the same

phase is added as pressure, temperature, and the amounts of all other com-

ponents are held constant. The symbol tilde (as in v

,h

, and s
) is used to

denote the partial molar propertiesof the components. Note that the sum-

mation term in Eq. 13–29 is zero for a single component system and thus the

chemical potential of a pure system in a given phase is equivalent to the

molar Gibbs function (Fig. 13–18) since GNgNm, where

(13–31)

Therefore, the difference between the chemical potential and the Gibbs

function is due to the effect of dissimilar molecules in a mixture on each

other. It is because of this molecular effect that the volume of the mixture of

two miscible liquids may be more or less than the sum of the initial volumes

of the individual liquids. Likewise, the total enthalpy of the mixture of two

components at the same pressure and temperature, in general, is not equal to

the sum of the total enthalpies of the individual components before mixing,

the difference being the enthalpy (or heat) of mixing, which is the heat

released or absorbed as two or more components are mixed isothermally. For

example, the volume of an ethyl alcohol–water mixture is a few percent less

than the sum of the volumes of the individual liquids before mixing. Also,

when water and flour are mixed to make dough, the temperature of the dough

rises noticeably due to the enthalpy of mixing released.

For reasons explained above, the partial molar properties of the compo-

nents (denoted by an tilde) should be used in the evaluation of the extensive

properties of a mixture instead of the specific properties of the pure compo-

nents. For example, the total volume, enthalpy, and entropy of a mixture

should be determined from, respectively,

(13–32)

instead of

(13–33)

Then the changes in these extensive properties during mixing become

(13–34)

where Hmixingis the enthalpy of mixingand Smixingis the entropy of

mixing(Fig. 13–19). The enthalpy of mixing is positive for exothermic mix-

¢Vmixinga

i

Ni 1 vivi 2 , ¢Hmixinga

i

Ni 1 hihi 2 , ¢Smixinga

i

Ni 1 sisi 2

V*a

i

Nivi¬H*a

i

Nihi¬and¬S*a

i

Nisi

Va

i

Nivi¬Ha

i

Ni hi¬and¬Sa

i

Nisi¬¬ 1 mixture 2

ma

0 G

0 N

b

P,T

ghTs¬¬ 1 pure substance 2

mia

0 G

0 Ni

b

P,T,Nj

gihiTsi¬¬ 1 for component i of a mixture 2

Mixing

chamber

A

B

yB

yA A + B

mixture

FIGURE 13–19

The amount of heat released or

absorbed during a mixing process is

called the enthalpy (or heat) of

mixing, which is zero for ideal

solutions.

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